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\title
{Brauer groups and Galois cohomology of function fields of varieties}

\author[Starr]{Jason Michael Starr}
\address{Department of Mathematics \\
  Stony Brook University \\ Stony Brook, NY 11794}
\email{jstarr@math.sunysb.edu} 

\date{\today}

\begin{document}

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%%
%% Abstract
%%
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%% \begin{abstract}
%% These notes accompany lectures presented at the 
%% XX \'{E}scola de Algebra held in IMPA, Rio de Janeiro, August 2008. 
%% These lectures introduce the subject of Brauer groups and Galois
%% cohomology, and then describe some recent results about Brauer groups
%% and Galois cohomology of function fields of varieties over
%% algebraically closed fields which are proved using \emph{rational
%%   connectedness} and \emph{rational simple connectedness}.
%% \end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Body
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\maketitle

\tableofcontents

\ 

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Acknowledgments
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Acknowledgments} \label{sec-ack}
\marpar{sec-ack}

\mni
Chapters \ref{pt-TL} and \ref{pt-RC} notes are largely adapted 
from notes for a similar lecture
series presented at the Clay Mathematics Institute Summer School in
G\"ottingen, Germany in Summer 2006.  
I am grateful to the Clay Mathematics
Institute and the Universit\"at G\"ottingen
for sponsoring that school.  

\mni
Chapters \ref{pt-PI} and \ref{pt-S2} are largely adapted from notes
for a lecture series presented at the Etats de la Recherche 2008 de la
Soci\'et\'e Math\'ematique de France conference 
\emph{Vari\'et\'es rationnellement connexes :
aspects g\'eom\'etriques et arithm\'etiques} presented at the
Universit\'e Louis Pasteur Strasbourg in May 2008.  I am grateful to
the Soci\'et\'e Math\'ematique de France and the CNRS for sponsoring
that school.  

\mni
I am
grateful to Brendan Hassett, Yuri Tschinkel and A. J. de Jong for
useful conversations on the content and exposition of these notes.
These notes report on joint work with Joe Harris and Tom Graber, as
well as joint work with A. J. de Jong and Xuhua He.  Moreover, the
results in Chapter \ref{pt-S2} are deeply motivated by a strategy of
Philippe Gille for proving the split case of Serre's ``Conjecture II''
which he explained to us some years ago.  I express my deepest
gratitude to Gille for showing us that rational simple connectedness
is related to Serre's conjecture.

\mni  
I am partially supported by grants from the National Science
Foundation as well as an Alfred P. Sloan research fellowship.
Finally, I
very gratefully acknowledge the support of 
INSTITUTO NACIONAL DE MATEM\'{A}TICA PURA E APLICADA and the Brazilian
government for making the XX Escola de Algebra possible.

\newpage

\

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Introduction
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction} \label{sec-intro}
\marpar{sec-intro}

\mni
These are the lecture notes for a minicourse presented in the XX
Escola de Algebra at the 
Instituto Nacional de Matem\'{a}tica Pura e Aplicada in Rio de Janeiro,
  Brazil, in August of 2008.  
The minicourse begins with an introduction to Brauer groups and
Galois cohomology followed by the proofs of the Chevalley-Warning and
Tsen-Lang theorems and their applications to Brauer groups and Galois
cohomology.  The minicourse concludes with some more recent
theorems about Brauer groups and Galois cohomology of function fields
of varieties over an algebraically closed field: de Jong's
Period-Index theorem and the split case of Serre's "Conjecture II"
over function fields of surfaces due to de Jong, He and myself and
following a strategy proposed by Philippe Gille.  The split case
completes the proof of the general case of Serre's "Conjecture II"
over function fields of surfaces
following the work of Merkurjev, Suslin, Bayer-Fluckiger, Parimala,
Colliot-Th\'el\`ene, and Gille.

\mni 
In addition to introducing students and post-docs to the classical
subjects of Brauer groups and Galois cohomology, a second goal of the
minicourse is to
show by example how results from geometry -- specifically the geometry
of rationally connected varieties and the geometry of spaces of
curves -- can be used to prove results in algebra -- namely
the Period-Index
theorem and the split case of Serre's "Conjecture II".

\mni
Here is a short description of the contents of the minicourse.

\mni
\textbf{Chapter 1: Brauer groups and Galois cohomology.}
Definition of division algebras and central simple algebras.
Definition of the Brauer group.  Short sketch of Galois cohomology and
its relation to the Brauer group.

\mni
\textbf{Chapter 2: The Chevalley-Warning and Tsen-Lang theorems.}
The statement and proofs of Chevalley's theorem and Tsen's theorem
which prove that the Brauer group of a finite field, resp. the
function field of a curve over an algebraically closed field, is
trivial.  Statement of the generalized results of Chevalley-Warning
and Tsen-Lang with applications to Brauer groups and Galois
cohomology.


\mni
\textbf{Chapter 3: Rationally connected fibrations.}
A sketch of rationally connected varieties.  The statement of the
conjecture of Koll\'ar-Miyaoka-Mori that every rationally connected
fibration over a curve has a section.  A sketch of the proof of this
conjecture by Graber, Harris and myself.

\mni
\textbf{Chapter 4: The Period-Index theorem of de Jong}
Explanation of de Jong's Period-Index theorem: for every division
algebra over the function field of a surface, the period equals the
index.  Sketch of a second proof, by de Jong and myself, using
rational simple connectedness.  Comparison to a third proof by
Lieblich, 
which also proves generalizations of de Jong's theorem.

\mni
\textbf{Chapter 5: Rational simple connectedness and Serre's
  "Conjecture II"} 
Description of known results about Serre's ``Conjecture II'':
theorems of Merkurjev and Suslin, Bayer-Fluckiger
and Parimala, Colliot-Th\'el\`ene, Gille and Parimala, and Gille.
Sketch of the proof of the split case of Serre's "Conjecture II" over
function fields of surfaces using rational simple connectedness.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Part: Brauer groups and Galois cohomology
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Brauer groups and Galois cohomology} \label{pt-Br}
\marpar{pt-Br}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Abelian Galois cohomology
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Abelian Galois cohomology} \label{sec-Ab}
\marpar{sec-Ab}

\mni
The first lecture gives a quick survey of some results in Galois
cohomology and Brauer groups.  Most results are stated without proof
or with only an outline of a proof.  The main references for this
lecture are \cite{GalCoh} and \cite{LocalFields}.  For results about
homological algebra, the reference is \cite{HomAlg}.

\mni
Let $L/K$ be a Galois extension of fields, e.g., a separable closure
$K^s$ of $K$.  
The Galois group 
$\G(L/K)$ is a profinite group, namely the inverse limit of the Galois
groups $\G(L_i/K)$ of all \emph{finite} Galois subextensions $L_i$, i.e.,
$K\subset L_i \subset L$.  

\begin{defn} \label{defn-mod}
\marpar{defn-mod}
An \emph{Abelian $\G(L/K)$-module} is an Abelian group $A$ together
with a left action of $\G(L/K)$ by group homomorphisms.  The module is
\emph{discrete} if every element of $A$ is stabilized by $\G(L_i/K)$
for some finite Galois subextension $L_i/K$, or equivalently, if
the canonical map of Abelian $\G(L/K)$-modules
$$
A \rightarrow \lim_{K\subset L_i \subset L} A_i, \ \ A_i := A^{\G(L/L_i)},
$$ 
is a bijection.

\mni
The Abelian category of all Abelian $\G(L/K)$-modules is denoted
$\G(L/K)-\lmod$.  And the full Abelian subcategory of discrete Abelian
$\G(L/K)$-modules is denoted $\G(L/K)-\lmod_{\text{discrete}}$.
\end{defn}

\mni
Let $L_i/K$ be a finite Galois subextension of $L/K$.
There is a left-exact additive functor
$$
H^0(\G(L_i/K),-): \G(L_i/K)-\lmod \rightarrow \ZZ-\lmod, 
$$
$$
B \mapsto
B^{\G(L_i/K)},
$$
the functor of $\G(L_i/K)$-invariants.  

\begin{defn} \label{defn-GC}
\marpar{defn-GC}
The sequence of right derived
functors of $H^0(\G(L_i/K),-)$ considered as a cohomological
$\delta$-functor is the \emph{Galois cohomology} or
\emph{$\G(L_i/K)$-cohomology}, 
$$
H^p(\G(L_i/K),-): \G(L_i/K)-\lmod \rightarrow \ZZ-\lmod, 
$$
$$
H^p(\G(L_i/K),B) := R^p H^0(\G(L_i/K),B).
$$
\end{defn}

\mni
For a discrete Abelian $\G(L/K)$-module $A$, for every integer $p\geq
0$ the sequence of group cohomologies $(H^p(\G(L_i/K),A_i))_{L_i}$
form a directed system of Abelian groups.  
Similarly, for every exact sequence of discrete Abelian
$\G(L/K)$-modules,
$$
\begin{CD}
0 @>>> A' @>>> A @>>> A'' @>>> 0,
\end{CD}
$$
the collection of connecting maps
$$
\delta^p_{L_i}:H^p(\G(L_i/K),A''_i) \rightarrow
H^{p+1}(\G(L_i/K),A'_i)
$$
is a morphism of compatible systems. 

\begin{defn} \label{defn-GC2}
\marpar{defn-GC2}
The \emph{Galois
  cohomology} or 
\emph{$\G(L/K)$-cohomology} is the cohomological $\delta$-functor from
$\G(L/K)-\lmod_{\text{discrete}}$ to $\ZZ-\lmod$ defined by the colimits
$$
H^p(\G(L/K),-): \G(L/K)-\lmod_{\text{discrete}} \rightarrow \ZZ-\lmod,
$$
$$
H^p(\G(L/K),A) := \lim_{L_i} H^p(\G(L_i/K),A_i),
$$
and
$$
\delta^p:H^p(\G(L/K),A'') \rightarrow H^{p+1}(\G(L/K),A'),
\ \ \delta_p = \lim_{L_i} \delta^p_{L_i}.
$$
\end{defn}

\mni
This is a
universal $\delta$-functor.  In fact the category
$\G(L/K)-\lmod_{\text{discrete}}$ has enough injective objects,
and one can prove that the usual
sequence of right derived functors of $A^{\G(L/K)}$ is canonically
isomorphic as a $\delta$-functor to the one from Definition
\ref{defn-GC2}.  
Definition \ref{defn-GC2} reflects one technique of studying
$\G(L/K)$-cohomology: reduction to the case of finite
Galois extensions $L_i/K$.  

\mni
For the reader
who knows of such things, $\G(L/K)$-cohomology is nothing other than
continuous group cohomology for the group $\G(L/K)$ with its profinite
topology.  This raises the question, how is the study of Galois
cohomology different from the study of profinite group cohomology?  
The answer
has to do mainly with the particular Galois modules of interest.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Low degree Galois cohomology 
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Low degree Galois cohomology} \label{ssec-low}
\marpar{ssec-low}

The low degree Galois cohomology groups have interpretations of
special interest.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: The zeroth group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The zeroth group} \label{sssec-zero}
\marpar{sssec-zero}

\mni
Let $L/K$ be a Galois extension and let $A$ be a discrete
$\G(L/K)$-module.  

\begin{prop} \label{prop-H0}
\marpar{prop-H0}
There is a canonical bijection between $H^0(\G(L/K),A)$ and the normal
subgroup $A^{\G(L/K)}$ of elements of $A$ left invariant under
$\G(L/K)$.
\end{prop}

\begin{proof}
By definition, for every finite, Galois subextension 
$K\subset L_i \subset L$
$$
H^0(\G(L_i/K),A_i) = A_i^{\G(L_i/K)}.
$$
Since $A_i$ is defined to be $A^{\G(L/L_i)}$, this gives
$$
H^0(\G(L_i/K),A_i) = A^{\G(L/K)}.
$$
Thus also
$$
H^0(\G(L/K),A) = A^{\G(L/K)},
$$
i.e., $H^0(\G(L/K),A)$ equals
the subgroup of elements of $A$ left invariant under $\G(L/K)$.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: The first group, Interpretation I.
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The first group, Interpretation I.} 
\label{sssec-firstI}
\marpar{sssec-firstI}

\mni
Let $L/K$ be a Galois extension and let $A$ be a discrete, Abelian
$\G(L/K)$-module.  
A \emph{crossed homomorphism} from $\G(L/K)$ into $A$ is a set map
$$
a:\G(L/K) \rightarrow A
$$
such that for every $g,h \in \G(L/K)$, 
$$
a_{g\cdot h} =  a_g\cdot \lsp{g}{a_h}.
$$
The crossed homomorphism is \emph{continuous} if it factors through
$\G(L_i/K)$ for some finite subextension $K\subset L_i \subset L$.  

\mni
There is a distinguished continuous crossed homomorphism $e$, namely
$$
e_g = e,
$$
the identity element of $A$ for every $g$ in $\G(L_i/K)$.  
The set of continous 
crossed homomorphisms from $\G(L/K)$ to $A$ is denoted
$Z^1_{\text{cts.}}(\G(L/K),A)$.  
There is a left action of $A$ on $Z^1_{\text{cts.}}(\G(L/K),A)$ 
associating to  
a crossed homomorphism $a$ and an element $b$ of $A$,
the new crossed homomorphism $a^b$,
$$
a^b_g := b^{-1}\cdot a_g \cdot \lsp{g}{b}.
$$

\begin{prop} \label{prop-H1a}
\marpar{prop-H1a}
The standard description of group cohomology establishes a
bijection between the set $Z^1_{\text{cts.}}(\G(L/K),A)/A$ of
$A$-orbits and the set $H^1(\G(L/K),A)$.  This bijection identifies
the crossed homomorphism $e$ with the additive identity of
$H^1(\G(L/K),A)$.  
\end{prop}

\begin{proof}
It suffices to consider the case when $L=L_i$ is a finite Galois
extension of $K$. 
The ``standard description'' of group cohomology arises 
from the standard resolution (or \emph{unnormalized bar resolution}
of the trivial (left)
$\ZZ[\G(L_i/K)]$-module
by free $\ZZ[\G(L_i/K)]$-modules,
$$
\hdots \rightarrow \ZZ[\G(L_i/K)](\G(L_i/K)^{p}) \xrightarrow{d_p} \dots
\rightarrow \ZZ[\G(L_i/K)](\G(L_i/K)\times \G(L_i/K)) 
$$
$$
\xrightarrow{d_2}
\ZZ[\G(L_i/K)](\G(L_i/K)) \xrightarrow{d_1} \ZZ[\G(L_i/K)] \rightarrow \ZZ
\rightarrow 0,
$$
$$
d_p(\mathbf{e}_{(g_1,\dots,g_p)}) = g_1\mathbf{e}_{g_2,\dots,g_p} +
\sum_{i=1}^{p-1}(-1)^p\mathbf{e}_{g_1,\dots,g_{i-1},g_i\cdot
  g_{i+1},g_{i+2},\dots,g_p} + (-1)^p\mathbf{e}_{g_1,\dots,g_{p-1}}.
$$
Applying $\text{Hom}_{\ZZ[\G(L_i/K)]-\lmod}(-,B)$ to this chain complex
gives a cochain complex computing
$$
H^p(\G(L_i/K),A) = \text{Ext}^p_{\ZZ[\G(L_i/K)]-\lmod}(\ZZ,A).
$$
Writing out the codifferential dual to the differential above gives
the ``standard description'' of group
cohomology in terms of cocycles and coboundaries.  The group of
$1$-cocycles is precisely the group
$Z^1(\G(L_i/K),A)$ of crossed homomorphisms with $e$ being the
additive identity.  And two crossed homomorphisms are cohomologous if
and only if they are equivalent.  
\end{proof} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: The first group, Interpretation II.
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The first group, Interpretation II.} 
\label{sssec-firstII}
\marpar{sssec-firstII}

\mni
There is a second interpretation of $H^1(\G(L/K),A)$
in terms of discrete $\G(L/K)$-equivariant
$A$-torsors.  

\begin{defn} \label{defn-torsor}
\marpar{defn-torsor}
A right \emph{$A$-torsor} is a nonempty set $E$ 
together with a right action of $A$ which is free and transitive.
A \emph{$\G(L/K)$-equivariant} $A$-torsor is a right
$A$-torsor together with a compatible left action of $\G(L/K)$ on $E$,
i.e.,  
for every $g$ in $\G(L/K)$, for every $a$ in $A$ and for every
$m$ in $E$,
$$
\lsp{g}{(m\cdot a)} = (\lsp{g}{m})\cdot(\lsp{g}{a}).
$$
The $\G(L/K)$-equivariant $A$-torsor is \emph{discrete} if every
element of $E$ is stabilized by $\G(L/L_i)$ for some finite Galois
subextension $L_i/K$.  
An \emph{isomorphism} of $\G(L_i/K)$-equivariant $A$-torsors is a 
bijection which is simultaneously left $\G(L_i/K)$-equivariant and right
$A$-equivariant.  
\end{defn}

\mni
Let $E$ be a $\G(L_i/K)$-equivariant $A$-torsor.  For every element
$m$ of $E$, there is a bijection of right $A$-torsors
$$
R_m:A\rightarrow E, \ \ R_m(a) = m\cdot a.
$$
This is left $\G(L_i/K)$-equivariant if and only if $m$ is
$\G(L_i/K)$-invariant.  The failure of $m$ to be $\G(L_i/K)$-invariant
is measured by the map
$$
a:\G(L_i/K) \rightarrow A, \ \ \forall g, \lsp{g}{m} = m\cdot a_g.
$$
It is straightforward to verify $a$ is a crossed homomorphism which is
continuous if $E$ is discrete.  
Moreover, replacing $m$ by $m\cdot b^{-1}$ replaces the crossed
homomorphism $a$ by $a^b$.  Thus the $B$-orbit of $a$ is independent
of the choice of $m$.  

\begin{prop} \label{prop-H1b}
\marpar{prop-H1b}
Let $L/K$ be a Galois extension and let $A$ be a discrete
$\G(L/K)$-module.  The rule associating to every discrete
$\G(L/K)$-equivariant $A$-torsor $E$ the $A$-orbit of $a$ in
$Z^1_{\text{cts.}}(\G(L/K),A)/A$ determines a bijection between the
set of isomorphism classes of discrete $\G(L/K)$-equivariant
$A$-torsors and $Z^1_{\text{cts.}}(\G(L/K),A)/A$.  
\end{prop}

\begin{proof}
For every pair 
$(E_1,m_1)$ and $(E_2,m_2)$, there is a unique isomorphism of
$A$-torsors 
$$
u:E_1 \rightarrow E_2
$$
such that $u(m_1)$ equals $u(m_2)$.  This isomorphism is
$\G(L_i/K)$-equivariant if and only if the crossed homomorphisms
associated to $(E_1,m_1)$ and $(E_2,m_2)$ are equal.

\mni
Moreover, for every
crossed homomorphism $a$, there is a pair $(E,m)$ giving rise to $a$
defined as follows.
Define $(E,m)$ to equal $(A,e)$ as a right $A$-torsor.  
But endow this with a new
$\G(L/K)$-action 
$$
g\ast_a b := a_g \cdot \lsp{g}{b}.
$$
If $A$ is discrete and $a$ is continuous, the new action is also
discrete.  And the crossed homomorphism associated to $(E,m)$ is
precisely $a$.

\mni
This shows the rule associating to $(E,m)$ the
continuous crossed homomorphism $a$ gives a bijection between the set
of isomorphism classes of pairs $(E,m)$ and the set
$Z^1_{\text{cts.}}(\G(L/K),A)$.  
Since changing $m$ to $m\cdot b^{-1}$ changes $a$ to $a^b$
this rule also gives a canonical bijection
between the set of isomorphism classes of discrete
$\G(L/K)$-equivariant $A$-torsors and the set
$Z^1_{\text{cts.}}(\G(L/K),A)/A$. 
\end{proof} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: The second group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The second group} 
\label{sssec-second}
\marpar{sssec-second}

\mni
Let $L_i/K$ be a finite
Galois extension and let $A$ be an Abelian
$\G(L_i/K)$-module.  An \emph{extension} of $\G(L_i/K)$ by $A$ is an 
exact sequence of groups
$$
\begin{CD}
0 @>>> A @>>> E @>>> \G(L_i/K) @>>> 1,
\end{CD}
$$
i.e., a group $E$ together with an isomorphism of $A$ with a normal
subgroup of $E$ and an isomorphism of $\G(L_i/K)$ with the quotient
group $E/A$ such that the induced conjugation action of $E$ on $A$
factors through the given action of $\G(L_i/K)$ on $A$.  An
isomorphism of extensions is a commutative diagram of short exact
sequences which is the identity on $A$ and $\G(L_i/K)$.  

\begin{prop} \label{prop-H2}
\marpar{prop-H2}
There is a canonical bijection between the set of equivalence classes
of extensions of $\G(L_i/K)$ by $A$ and $H^2(\G(L_i/K),A)$.  The
bijection associates the additive identity in $H^2(\G(L_i/K),A)$ to
the equivalence class of the semidirect product $A \rtimes
\G(L_i/K)$.  
\end{prop}

\mni
This is best checked by using the standard resolution to describe
$H^2(\G(L_i/K),A)$ in terms of cocycles, cf. \cite[Theorem 6.6.3,
  p. 183]{HomAlg}.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Cohomological dimension
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Cohomological dimension} 
\label{sssec-CD}
\marpar{sssec-CD}

\mni
Let $L/K$ be a Galois extension.  Let $p$ be a prime number and let
$n$ be a nonnegative integer.  

\begin{defn} \label{defn-cd}
\marpar{defn-cd}
The \emph{$p$-cohomological dimension of $\G(L/K)$},
$\text{cd}_p(L/K)$, is the smallest nonnegative integer $n$ such that
for every torsion, discrete, Abelian $\G(L/K)$-module $A$,
the $p$-primary component of $H^q(\G(L/K),A)$ is $\{0\}$ for all
$q>n$.  If there is no such integer $n$, then $\text{cd}_p(L/K)$ is
defined to be $\infty$.  The \emph{cohomological dimension of
  $\G(L/K)$}, $\text{cd}(L/K)$, is defined to be $\sup
\{\text{cd}_p(L/K)| p \text{ prime }\}$.  When $L = K^s$, a separable
closure of $K$, these are denoted simply by $\text{cd}_p(K)$ and
$\text{cd}(K)$.  
\end{defn}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Non-Abelian Galois cohomology and the long exact sequence
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Non-Abelian Galois cohomology and the long exact sequence} 
\label{sec-non}
\marpar{sec-non}

\mni
Let $L/K$ be a Galois extension.  

\begin{defn} \label{defn-mod2}
\marpar{defn-mod2}
A \emph{$\G(L/K)$-module} is a (not
necessarily Abelian) group $A$ together with a left action of
$\G(L/K)$ acting by group homomorphisms.  The module is \emph{discrete}
if every element of $A$
is stabilized by $\G(L/L_i)$ for some finite Galois subextension
subextension $L_i/K$.  
\end{defn}

\mni
The category of $\G(L/K)$-modules is not Abelian, thus one cannot
define Galois cohomology in the same way as above.
There is a theory
of
non-Abelian cohomology,
developed in 
\cite{Giraud} for instance.  However, for $H^0$ and $H^1$ it is
simpler to define
$H^0(\G(L/K),A)$ and $H^1(\G(L/K),A)$ so that Propositions
\ref{prop-H0} and \ref{prop-H1a} continue to hold.  

\begin{defn} \label{defn-adhoc}
\marpar{defn-adhoc}
For every discrete $\G(L/K)$-module $A$, the \emph{zeroth Galois
  cohomology} is the subgroup 
$$
H^0(\G(L/K),A) := A^{\G(L/K)},
$$
of elements of $A$ left invariant under $\G(L/K)$.  And the
\emph{first Galois cohomology} is the set of right orbits,
$$
H^1(\G(L/K),A) := Z^1(\G(L/K),A)/A.
$$
\end{defn}

\mni
Let $u:A\rightarrow B$ be a homomorphism of discrete
$\G(L/K)$-modules.  Since $u$ is $\G(L/K)$-equivariant, $u$ maps
$A^{\G(L/K)}$ to $B^{\G(L/K)}$.  And for every crossed homomorphism
$$
a:\G(L/K) \rightarrow A,
$$
the composition
$$
u\circ a:\G(L/K) \rightarrow B
$$
is a crossed homomorphism.  Moreover, $u\circ(a^b)$ equals $(u\circ
a)^{u(b)}$.  Thus $u\circ a$ maps $A$-orbits into $B$-orbits.  And
$u\circ e_A$ equals $e_B$, so $u$ preserves the distinguished points.   

\begin{defn} \label{defn-adhoc2}
\marpar{defn-adhoc2}
For every homomorphism $u:A\rightarrow B$ of discrete
$\G(L/K)$-modules, the group homomorphism
$$
H^0(\G(L/K),u): H^0(\G(L/K),A) \rightarrow H^0(\G(L/K), B)
$$
and the map of pointed sets
$$
H^1(\G(L/K),u): H^1(\G(L/K),A) \rightarrow H^1(\G(L/K), B)
$$
are as defined above. 
\end{defn}

\mni
These maps complete the definition of the functors,
$$
H^0(\G(L/K),-):\text{non-Abelian} \G(L/K)-\lmod_{\text{discrete}}
\rightarrow \text{Groups},
$$
$$
H^1(\G(L/K),-):\text{non-Abelian} \G(L/K)-\lmod_{\text{discrete}}
\rightarrow \text{Pointed Sets}.
$$

\begin{rmk} \label{rmk-adhoc}
\marpar{rmk-adhoc}
\textbf{(i)}
Note that the proof of Proposition ~\ref{prop-H1b} is still valid.
Thus there is a canonical bijection between $H^1(\G(L/K),A)$ and the
pointed set of equivalence classes of 
discrete $\G(L/K)$-equivariant $A$-torsors.  
\linebreak
\noindent
\textbf{(ii)}
Note also that for a product $\G(L/K)$-module, $B=A\times C$, 
the functorialities of $H^0$ and $H^1$ establish and isomorphism
$$
H^0(\G(L/K),A\times C) \cong H^0(\G(L/K),A)\times H^0(\G(L/K),C)
$$
and a bijection of pointed sets,
$$
H^0(\G(L/K),A\times C) \cong H^0(\G(L/K),A)\times H^0(\G(L/K),C).
$$
\end{rmk}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: The long exact sequence  
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The long exact sequence} \label{ssec-long}
\marpar{ssec-long}

\mni
Notice that $H^1(\G(L/K),A)$ typically has no natural structure of
group, although it does have the distinguished element $[e]$.  Thus it
may seem meaningless to ask about a long exact sequence of non-Abelian
Galois cohomology associated to a short exact sequence of groups.
However the situation is not as bad as it might at first seem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step I. Twisting
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step I. Twisting} 
\label{sssec-StepI}
\marpar{sssec-StepI}

\mni
The first ingredient in describing the ``long exact sequence'' is the
notion of twisting.  Let $E$ be a $\G(L/K)$-equivariant $A$-torsor and
let $F$ be a set with a left $\G(L/K)$-action and a
$\G(L/K)$-equivariant right action of $A$, i.e.,
$$
\lsp{g}(f\cdot a) = \lsp{g}{f} \cdot \lsp{g}{a}
$$
for every $g$ in $\G(L/K)$, for every $a$ in $A$ and for every $f$ in
$F$.  
There is a left $\G(L/K)$-action on $E\times F$ by $\lsp{g}{(e,f)} :=
(\lsp{g}{e}, \lsp{g}{f})$.  
And there is a $\G(L/K)$-equivariant 
right action of $A$ on $E\times F$ by $(e,f)\cdot a := (e\cdot a,
f\cdot a)$.  
The set of right $A$-orbits
$$
\lsb{E}{F} := (E\times F)/A
$$
has a well-defined left action of $\G(L/K)$ such that
$[\lsp{g}{(e,f)}]$ equals $\lsp{g}{[(e,f)]}$.  

\begin{defn} \label{defn-twistE}
\marpar{defn-twistE}  
The $\G(L/K)$-set $\lsb{E}{F}$ is 
the \emph{twist} of $F$ by $E$.  
If $E$ and $F$ are both discrete,
$\lsb{E}{F}$ is also discrete.
\end{defn}

\mni
Twisting is functorial in both $E$ and $F$.  Since pairs $(E,m)$ are
unique up to unique isomorphism, for every element $a$ in
$Z^1_{\text{cts.}}(\G(L/F),A)$, the twist $\lsb{a}{F}$ is also
well-defined up to unique isomorphism.  But since two representatives
$a$ and $a'$ of the same $A$-orbit in $Z^1_{\text{cts.}}(\G(L,F),A)$
may be conjugate under many elements of $A$, there is no
canonical isomorphism of $\lsb{a}{F}$ and $\lsb{a'}{F}$.  

\mni
If $A$ is non-Abelian then
typically $\lsb{E}{F}$ has no natural structure of $A$-set.  However,
if the action of $A$ on $F$ preserves some ``structure'', then the
twist $\lsb{E}{F}$ typically also has this ``structure''.  
Here are two examples of this.

\begin{ex}[Translation by $F$] \label{ex-twist1}
\marpar{ex-twist1}
Let $A$ and $B$ be $\G(L/K)$-modules and let $F$ be a $\G(L/K)$-set
that has commuting $\G(L/K)$-equivariant right actions of both $A$ and
$B$.  Then for every $\G(L/K)$-equivariant $A$-torsor $E$,
$\lsb{E}{F}$ also has a $\G(L/K)$-equivariant right $B$-action.  In
particular, if $F$ is a $B$-torsor, then also $\lsb{E}{F}$ is a
$B$-torsor.  Thus, associated to every $\G(L/K)$-equivariant
$B$-torsor $F$ with a commuting $\G(L/K)$-equivariant right $A$-action,
there is an induced set map
$$
\ast F: H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),B), \ \ [E]\mapsto
[\lsb{E}{F}].
$$
This maps sends the trivial $A$-torsor to $F$, and thus can be
considered a ``translation by $F$''.  
\end{ex}

\mni
The canonical example of this comes from a homomorphism
$u:A\rightarrow B$ of $\G(L/K)$-modules.  There is a right $A$-action
on $B$ via $b\cdot a := bu(a)$.  And there is a commuting ``right''
$B$-action on $B$ via $b\cdot \beta := \beta^{-1}b$.  Thus, as above,
there is a set map
$$
u_*:H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),B), \ \ [E] \mapsto
[\lsb{E}{B}]. 
$$
Via
Proposition ~\ref{prop-H1b}, the map $u_*$ is precisely the same as
the map $H^1(\G(L/K),u)$.  

\mni
A second example comes from the case that
$A$ is Abelian.  Let $B$ be $A$.  For every $A$-torsor $F$, let the
$B$-torsor structure be precisely the same as the given $A$-torsor
structure.  The ``two'' actions commute because multiplication in $A$
is commutative.  And the induced map $\ast F$ literally is translation
by $[F]$ in the Abelian group $H^1(\G(L/K),A)$.  

\mni
If $F$ is itself a $\G(L/K)$-module and if $A$ acts by group
homomorphisms,  
then $\lsb{E}{F}$ is also a group,
i.e., a $\G(L/K)$-module.

\begin{ex}[Inner twists] \label{ex-twist2}
For instance, $A$ acts on itself by inner
automorphisms.  Thus associated to every discrete $\G(L/K)$-equivariant
$A$-torsor $E$, there is an associated $\G(L/K)$-module $\lsb{E}{A}$,
$$
\lsb{E}{A} := (E\times A)/\sim, \ \ (m,a) \sim (m\gamma,
\gamma^{-1}a\gamma).
$$
To make this more explicit, observe there is a well-defined
composition,
$$
\ast:(E\times A) \times (E\times A) \rightarrow E\times A, 
$$
$$
(m_1,a_1)\ast (m_2,a_2) = (m_1\cdot (a_1 b), b^{-1}a_1b_1 a_2),
\ \ m_2 = m_1 
\cdot b.
$$
Observe that 
$$
(m_1\cdot \alpha, \alpha^{-1}a_1\alpha) \ast (m_2\cdot \beta,
\beta^{-1}a_2\beta) = ((m_1 \cdot (a_1b))\cdot
\beta,\beta^{-1}(b^{-1}a_1ba_2)\beta).
$$
Thus the composition gives a well-defined composition on
$\lsb{E}{A}$.  The identity element is the class of $E\times\{e_A\}$.
And the inverse of the class $(m,a)$ is $(m\cdot a^{-1},a)$.  It is
important to note that the composition above gives a well-defined
\emph{left} action of $\lsb{E}{A}$ on $E = E\times\{e_A\}$ by
$$
[(m_1,a_1)] \ast m_2 := m_1a_1b, \ \ m_2 = m_1b.
$$
Since $\lsb{E}{A}$ is a group, we can change this into a ``right''
action by
$$
m_2 \ast [(m_1,a_1)] := [(m_1a_1^{-1},a_1^{-1})]\ast m_2 = m_1a^{-1}b.
$$
With respect to this action, $E$ is both a $\G(L/K)$-equivariant
$A$-torsor and a $\G(L/K)$-equivariant 
$\lsb{E}{A}$-torsor.  Thus, by the mechanism in Example
~\ref{ex-twist1}, there are well-defined set maps
$$
\ast E: H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),\lsb{E}{A}),
$$
$$
\ast E: H^1(\G(L/K),\lsb{E}{A}) \rightarrow H^1(\G(L/K),A).
$$
Each of these maps sends the trivial torsor to $E$.  Observe that if
$A$ is Abelian, then $\lsb{E}{A}$ equals $A$ and the map $\ast E$
agrees with the map from Example ~\ref{ex-twist1}, i.e., it is simply
translation by $[E]$ in the Abelian group $H^1(\G(L/K),A)$.  
\end{ex}

\begin{prop} \label{prop-astE}
\marpar{prop-astE}
The maps
$$
\ast E: H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),\lsb{E}{A}),
$$
$$
\ast E: H^1(\G(L/K),\lsb{E}{A}) \rightarrow H^1(\G(L/K),A).
$$
are both bijections.
\end{prop}

\begin{proof}
Let $F$ be a $\G(L/K)$-equivariant $A$-torsor.  Denote by $I$ the set
$$
I = \text{Isom}_A(E,F) = \{\phi:E\rightarrow F | \forall e\in E,
\forall a\in A, \phi(e\cdot a) = \phi(e)\cdot a\}.
$$
Make this into a $\G(L/K)$-set by defining,
$$
(\lsp{g}{\phi})(e) := \lsp{g}{(\phi(\lsp{g^{-1}}{e}))}, \text{ i.e., } 
(\lsp{g}{\phi})(\lsp{g}{e}) = \lsp{g}{(\phi(e))}.
$$
Make this into a right $\lsb{E}{A}$-set by defining
$$
(\phi\cdot \alpha)(e) := \phi(e\cdot \alpha^{-1}), \text{ i.e., }
(\phi\cdot \alpha)(e\cdot \alpha) = \phi(e).
$$
Putting these together,
$$
(\lsp{g}{(\phi\cdot\alpha)})(\lsp{g}{(e\cdot \alpha)}) = \phi(e)
=(\lsp{g}{\phi}\cdot \lsp{g}{\alpha})(\lsp{g}{e}\cdot
\lsp{g}{\alpha}).
$$
Since $\lsp{g}{(e\cdot\alpha)}$ equals $\lsp{g}{e}\cdot
\lsp{g}{\alpha}$, it follows that $\lsp{g}{(\phi\cdot\alpha)}$ equals
$\lsp{g}{\phi}\cdot \lsp{g}{\alpha}$, i.e., the $\lsb{E}{A}$-action is
$\G(L/K)$-equivariant.  

\mni
On the underlying level of groups and group-sets ignoring the
$\G(L/K)$-actions, $E$ equals $A$ as $A$-sets, $\lsb{E}{A}$
equals $A$ as groups, and $F$ equals $A$ as $A$-sets.  By considering
this case, it is clear that the right $\lsb{E}{A}$-action on $I$ makes
$I$ into an $\lsb{E}{A}$-torsor.  Therefore $I$ is a $\G(L/K)$-equivariant
$\lsb{E}{A}$-torsor. 

\mni
There
is a canonical map
$$
c:I\times E \rightarrow F, \ \ (\phi,e)\mapsto \phi(e).
$$
This map is $\lsb{E}{A}$-invariant for the diagonal right action of
$\lsb{E}{A}$ on $I\times E$.  Thus it factors through a map
$$
c':\lsb{I}{E} \rightarrow F.
$$
Since $c$ is $\G(L/K)$-equivariant and $A$-equivariant for the right
$A$-action on $E$, $c'$ is a morphism of $\G(L/K)$-equivariant
$A$-torsors.  Thus it is an isomorphism.  So $\ast E$ sends $I$ to
$F$.

\mni
Conversely, for every $\G(L/K)$-equivariant $\lsb{E}{A}$-torsor $J$ and
for every morphism
$$
d':\lsb{J}{E} \rightarrow F
$$
of $\G(L/K)$-equivariant $A$-torsors, the composition
$$
d:J\times E \twoheadrightarrow \lsb{J}{E} \xrightarrow{d'} F
$$
is $A$-equivariant for the right $A$-action on $E$, and thus induces a
set map
$$
d'':J\rightarrow I.
$$
Since $d$ is $\G(L/K)$-equivariant and $\lsb{E}{A}$-equivariant, so is
$d''$.  Thus $d''$ is a map of $\G(L/K)$-equivariant
$\lsb{E}{A}$-torsors, which is automatically an isomorphism.
Therefore, up to isomorphism, $I$ is the unique $\G(L/K)$-equivariant
$\lsb{E}{A}$-torsor such that $\lsb{I}{E}$ is isomorphic to $F$, i.e.,
the map
$$
\ast E:H^1(\G(L/K),\lsb{E}{A}) \rightarrow H^1(\G(L/K),A)
$$
is bijective.  The argument that the other map $\ast E$ 
is bijective is similar.
\end{proof}

\mni
Let $u:A\rightarrow B$ be a homomorphism of $\G(L/K)$-modules and let
$E$ be a discrete $\G(L/K)$-equivariant $A$-torsor.  There is a
well-defined homomorphism of $\G(L/K)$-modules,
$$
\lsb{E}{u}:\lsb{E}{A} \rightarrow \lsb{u_*E}{B}
$$
associated to the map
$$
(E\times A) \rightarrow (E\times B), \ \ (m,a) \mapsto (m,b).
$$
The proof of the following is a straightforward exercise for the reader.

\begin{lem} \label{lem-astE}
\marpar{lem-astE}
With respect to this map, both the diagram
\marpar{eqn-1}
\begin{equation} \label{eqn-1}
\begin{CD}
H^1(\G(L/K),A) @> u_* >> H^1(\G(L/K),B) \\
@V \ast E VV @VV \ast u_* E V \\
H^1(\G(L/K),\lsb{E}{A}) @> \lsb{E}{u}_* >> H^1(\G(L/K),\lsb{u_*E}{B})
\end{CD}
\end{equation}
and the diagram
\marpar{eqn-2}
\begin{equation} \label{eqn-2}
\begin{CD}
H^1(\G(L/K),\lsb{E}{A}) @> \lsb{E}{u}_* >> H^1(\G(L/K),\lsb{u_*E}{B}) \\
@V \ast E VV @VV \ast u_* E V \\
H^1(\G(L/K),A) @> u_* >> H^1(\G(L/K),B) 
\end{CD}
\end{equation}
commute.
\end{lem}

\begin{ex} \label{ex-twist3}
\marpar{ex-twist3}
Next, consider an \emph{Abelian extension} of discrete
$\G(L/K)$-modules,
$$
\begin{CD}
0 @>>> A' @>>> A @>>> A'' @>>> 1,
\end{CD}
$$
i.e., $A'$ is Abelian (although neither $A$ nor $A''$ need be Abelian). 
The conjugation action of $A$ on the normal subgroup $A'$ determines a
well-defined action of $A''$ on $A'$.  Thus, associated to every
$\G(L/K)$-equivariant $A''$-torsor $E''$ 
there is a discrete Abelian
$\G(L/K)$-module $\lsb{E''}{A'}$.  
\end{ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step II. The first terms
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step II. The first terms} 
\label{sssec-StepII}
\marpar{sssec-StepII}

\mni
This is enough to begin describing the long exact sequence.  Let
$u:A\rightarrow B$ be an injective homomorphism of discrete
$\G(L/K)$-modules. Denote by $C=B/A$ the set of right $A$-cosets of
$B$ with its natural discrete structure of $\G(L/K)$-set.  Denote by
$v:B\rightarrow C$ the natural map, which is $\G(L/K)$-equivariant.
The
homomorphism $u$ determines an injective homomorphism
$$
H^0(\G(L/K),u):H^0(\G(L/K),A) \rightarrow H^0(\G(L/K),B),
$$
and in particular an action of $H^0(\G(L/K),A)$ on $H^0(\G(L/K),B)$.
Because $v$ is $\G(L/K)$-equivariant, it also determines a map
$$
v_*:H^0(\G(L/K),B) \rightarrow C^{\G(L/K)}.
$$
Clearly the fibers of $v_*$ are precisely the
$H^0(\G(L/K),A)$-orbits.  
In this sense, the sequence of sets
$$
\begin{CD}
1 @>>> H^0(\G(L/K),A) @>>> H^0(\G(L/K),B) @>>> C^{\G(L/K)}
\end{CD}
$$
is an exact sequence of sets.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step III. The connecting map
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step III. The connecting map} 
\label{sssec-StepIII}
\marpar{sssec-StepIII}

\mni
The $\G(L/K)$-invariant elements of $C$ are the same as the
$A$-orbits $E$ of $B$ which are fixed as a set (but not necessarily
pointwise) by the action of $\G(L/K)$.  
Such a $\G(L/K)$-invariant $A$-orbit $E$ need
not contain a $\G(L/K)$-invariant element, i.e., $v_*$ need not be
surjective.  For every $\G(L/K)$-invariant $A$-orbit $E$, the
restrictions to $E$ of the $\G(L/K)$-action and the $A$-action make
$E$ into a $\G(L/K)$-invariant $A$-torsor.  Forming the equivalence
class of this torsor gives
a well-defined set map
$$
\delta:C^{\G(L/K)} \rightarrow H^1(\G(L/K),A).
$$
Observe that for every $\G(L/K)$-invariant $A$-orbit $E$ and every
$\G(L/K)$-invariant element $b$ of $B$, left multiplication by $b$
defines an isomorphism of the $\G(L/K)$-equivariant $A$-torsor $E$ and
the $\G(L/K)$-equivariant $A$-torsor $b\cdot E$.  Thus the map
$\delta$ is constan on each $H^0(\G(L/K),B)$-orbits in
$C^{\G(L/K)}$.  Conversely, if $E_1$ and $E_2$ are
$\G(L/K)$-invariant $A$-orbits in $B$ and if $v:E_1\rightarrow E_2$ is
an isomorphism of $\G(L/K)$-invariant $A$-torsors, then there exists a
unique element $b$ of $B$ such that $v(m) = b\cdot m$ for every $m$ in
$E_1$.  Since $v$ is $\G(L/K)$-equivariant, $b$ is
$\G(L/K)$-invariant.  Therefore $\delta(E_1)$ equals $\delta(E_2)$ if
and only if $E_1$ and $E_2$ differ by left multiplication by an
element of $H^0(\G(L/K),B)$.  In particular, the fiber of $\delta$
over $[A]$ is precisely the image of $H^0(\G(L/K),B)$.  

\mni
In all of these senses, the set map
$$
1 \rightarrow H^0(\G(L/K),A) \rightarrow H^0(\G(L/K),B) \rightarrow
C^{\G(L/K)} \xrightarrow{\delta} H^1(\G(L/K),A)
$$
is ``exact''.  The first map is injective.  The fibers of the
second map are precisely the orbits for the action of $H^0(\G(L/K),A)$
on $H^0(\G(L/K),B)$.  And the fibers of the third map are precisely
the orbits for the action of $H^0(\G(L/K),B)$ on $(B/A)^{\G(L/K)}$.   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step IV. The map of torsors
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step IV. The map of torsors} 
\label{sssec-StepIV}
\marpar{sssec-StepIV}

\mni
Let $E$ be a $\G(L/K)$-equivariant $A$-torsor and let $u_*E$ be the
associated $\G(L/K)$-equivariant $B$-torsor as constructed in Example
~\ref{ex-twist1}, $u_*E = E\times B/A$.  Taking inverse images,   
a $\G(L/K)$-invariant element of $u_*E$ is the same as a
$\G(L/K)$-invariant $A$-orbit $F$ of $E\times B$.  The projection
$\text{pr}_E:F\rightarrow E$ is an isomorphism of $\G(L/K)$-invariant
$A$-torsors.  Inverting this, $F$ is the graph $F=\Gamma_v$ of a
unique map $v:E\rightarrow B$ which is simultaneously
$\G(L/K)$-equivariant and $A$-equivariant.  But then $E$ is isomorphic
to $v(E)$, which is a $\G(L/K)$-equivariant left $A$-orbit in $B$.
Thus $u_*E$ is isomorphic to the distinguished element of
$H^1(\G(L/K),B)$ if and only if $E$ is isomorphic to an $A$-torsor in
the image of $\delta$.  In summary, this gives the following.

\begin{prop} \label{prop-les1}
\marpar{prop-les1}
Associated to an injective homomorphism of $\G(L/K)$-modules,
$u:A\rightarrow B$, and denoting by $v:B\rightarrow C$ the quotient
map of $\G(L/K)$-sets, $B\rightarrow B/A$, the sequence of set maps
$$
1 \rightarrow H^0(\G(L/K),A) \xrightarrow{u_*} H^0(\G(L/K),B) \xrightarrow{v_*}
C^{\G(L/K)} 
$$
$$
\xrightarrow{\delta} H^1(\G(L/K),A) \xrightarrow{u_*}
H^1(\G(L/K),B) 
$$
is ``exact'' in the following sense:
the first map $u_*$ is injective, the fibers of $v_*$ equal
the orbits of $\text{Image}(u_*)$, the fibers of $\delta$ equal the
orbits of $H^0(\G(L/K),B)$, and the fiber of $u_*$ over the
distinguished element equals the image of $\delta$.
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step V. Fibers of $u_*$
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step V. Fibers of $u_*$} 
\label{sssec-StepV}
\marpar{sssec-StepV}

\mni
Using the twists from Example ~\ref{ex-twist2}, we can push this a bit
further.
Let $E$ be a $\G(L/K)$-equivariant $A$-torsor and let 
$$
\lsb{E}{u}:\lsb{E}{A} \rightarrow \lsb{u_*E}{B}
$$
be the associated morphism of inner twist $\G(L/K)$-modules as in
Example ~\ref{ex-twist2}.
This gives rise to a
sequence of sets,
$$
1 \rightarrow H^0(\G(L/K),\lsb{E}{A}) \xrightarrow{\lsb{E}{u}_*}
H^0(\G(L/K),\lsb{u_*E}{B}) \xrightarrow{\lsb{E}{v}_*}
\lsb{E}{C}^{\G(L/K)} 
$$
$$
\xrightarrow{\lsb{E}{\delta}} H^1(\G(L/K),\lsb{E}{A})
\xrightarrow{\lsb{E}{u}_*} H^1(\G(L/K),\lsb{u_*E}{B}) 
$$
By the diagram in
Equation ~\ref{eqn-2}, the map $\ast E$ defines a map from the fiber
of $\lsb{E}{u}_*$ containing the trivial $\lsb{E}{A}$-torsor to the
fiber of $u_*:H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),B)$ containing
$[E]$.  

\begin{prop} \label{prop-les2}
\marpar{prop-les2}
The fiber of
$$
u_*:H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),B)
$$
containing $[E]$ equals the image of the map
$$
\ast E \circ \lsb{E}{\delta}:(\lsb{E}{C})^{\G(L/K)} \rightarrow
H^1(\G(L/K),A).
$$
\end{prop}

\begin{proof}
This follows from Lemma ~\ref{lem-astE}, Proposition ~\ref{prop-astE}
and Proposition ~\ref{prop-les1} applied to $\lsb{E}{u}$ as discussed
above.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step VI. The image of $u_*$
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step VI. The image of $u_*$} 
\label{sssec-StepVI}
\marpar{sssec-StepVI}

\mni
Let $F$ be a $\G(L/K)$-equivariant $B$-torsor.

\begin{prop} \label{prop-les3}
\marpar{prop-les3}
There is $\G(L/K)$-equivariant $A$-torsor $E$ such that $[F]$ equals $u_*[E]$
if and only if $F/u(A)$ has a $\G(L/K)$-invariant element.
\end{prop}

\begin{proof}
This is straightforward.  Certainly $\lsb{E}{B}=(E\times B)/\sim$ has
a $\G(L/K)$-invariant $u(A)$-orbit, namely the image of
$E\times\{e_B\}$.  So if $[F]$ equals $u_*[E]$, then $F/u(A)$ has
$\G(L/K)$-invariant element.  

\mni
Conversely, if $F/u(A)$ has a
$\G(L/K)$-invariant element, the inverse image in $F$ is a
$\G(L/K)$-invariant $u(A)$-orbit $E$.  The restrictions of the
$\G(L/K)$-structure and the $A$-structure on $F$ to $E$ make $E$ into
a $\G(L/K)$-equivariant $A$-torsor.  And the inclusion
$E\hookrightarrow F$ induces a morphism $\lsb{E}{B} \rightarrow F$ of
$\G(L/K)$-equivariant $B$-torsors.  Thus $[F]$ equals $u_*[E]$.  
\end{proof}

\mni
Observe
that $G/u(A)$ is the same as the twist $\lsb{G}{C}$.  In the sense
that the twist $\lsb{G}{C}$ should be considered ``trivial'' if it
has a $\G(L/K)$-invariant element, this extends the long exact
sequence of sets one more place to the right. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step VII. Quotient by a normal subgroup
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step VII. Quotient by a normal subgroup} 
\label{sssec-StepVII}
\marpar{sssec-StepVII}

\mni
If $A$ is a normal subgroup, then $C$ is naturally a discrete
$\G(L/K)$-module.  
In this case, for every $\G(L/K)$-equivariant $A$-torsor $E$, the
quotient $\lsb{E}{C} = \lsb{u_*E}{B}/\lsb{E}{A}$ 
is canonically isomorphic as a
$\G(L/K)$-set to $C$.  In particular, the rule associating to every
element $E$ the set map
$$
\ast E\circ \lsb{E}{\delta}:H^0(\G(L/K),C) \rightarrow H^1(\G(L/K),A)
$$
is an action of $H^0(\G(L/K),C)$ on $H^1(\G(L/K),A)$.  The previous
step shows that the fibers of
$$
u_*:H^1(\G(L/K),A) \rightarrow H^1(\G(L/K),B)
$$
are precisely the orbits of $H^0(\G(L/K),C)$.  Finally, for every
$B$-torsor $G$, the twist $\lsb{G}{C}$ from the previous paragraph
still has a left $C$-action, and thus is a discrete
$\G(L/K)$-equivariant $C$-torsor.  As above, it is a trivial
$C$-torsor if and only if it has a $\G(L/K)$-invariant element if and
only if $G$ is in the image of $u_*$.  
In summary, this gives the following.

\begin{prop} \label{prop-les4}
\marpar{prop-les1}
Associated to an injective homomorphism of $\G(L/K)$-modules,
$u:A\rightarrow B$ whose image is normal, and denoting by
$v:B\rightarrow C$ the quotient map of $\G(L/K)$-modules,
$B\rightarrow B/A$, 
the sequence of set maps
$$
1 \rightarrow H^0(\G(L/K),A) \xrightarrow{u_*} H^0(\G(L/K),B) \xrightarrow{v_*}
H^0(\G(L/K),C) 
$$
$$
\xrightarrow{\delta} H^1(\G(L/K),A) \xrightarrow{u_*}
H^1(\G(L/K),B) \xrightarrow{v_*} H^1(\G(L/K),C)
$$
is ``exact'' in the following sense:
the first map $u_*$ is injective with normal image, the fibers of $v_*$ equal
the orbits of $\text{Image}(u_*)$, the fibers of $\delta$ equal the
orbits of $H^0(\G(L/K),B)$, the fibers of the second map $u_*$ equal
the orbits of $H^0(\G(L/K),C)$, and the image of the second map $u_*$
equals the fiber of the second map $v_*$ containing the neutral element.
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step VIII. Quotient by an Abelian normal subgroup
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step VIII. Quotient by an Abelian normal subgroup} 
\label{sssec-StepVIII}
\marpar{sssec-StepVIII}

\mni
The next step is when $A$ is both normal and Abelian.  Then the
conjugation action of $B$ on $A$ descends to a well-defined
action of
$C$ on $A$.  
Given $c$ in $C$ and $a$ in $A$, we will denote this
action by $cac^{-1}$ as a reminder that the action is $bab^{-1}$ for
any representative $b$ of the $A$-coset $c$ in $B$.

\mni
Let 
$$
c:\G(L/K) \rightarrow C
$$
be a continuous crossed homomorphism.
This in turn determines a new discrete action of
$G$ on $A$, namely
$$
g\ast_c a := c_g\cdot \lsp{g}{a} \cdot c_g^{-1}.
$$
The action of $\G(L/K)$ on $B$ defines a semidirect product $B\rtimes
\G(L/K)$.  And $c$ defines an action of $B\rtimes \G(L/K)$ on $C$ by
$$
(b,g)\ast_c d := c_g\cdot \lsp{g}{d} \cdot v(b)^{-1}.
$$
The stabilizer of the identity $e_C$ is a subgroup denoted 
$\lsb{c}{H}$.  
Of course $\lsb{c}{H}$
contains $A\times\{1\}$ as a subgroup.  The induced quotient 
$$
H\subset B\rtimes \G(L/K) \xrightarrow{\pi_2} \G(L/K)
$$
is surjective and has kernel $A$.  The conjugation action of $H$ on
$A$ factors through an action of $\G(L/K)$ on $A$.  And of course this
is the new action defined above.  Thus, altogether $\lsb{c}{H}$ is an
extension 
$$
\begin{CD}
0 @>>> \lsb{c}{A} @>>> \lsb{c}{H} @>>> \G(L/K) @>>> 0.
\end{CD}
$$
This gives a well-defined element
$$
\Delta(c) \in H^2(\G(L/K),\lsb{c}{A}).
$$

\mni
For an element $b$ of $B$, conjugation by $b$ defines an automorphism
of $A$ intertwining the actions $\ast_c$ and $\ast_{c^{v(b)}}$.  
Similarly conjugation by $(b,1) \in B\rtimes \G(L/K)$ gives an
isomorphism of $\lsb{c}{H}$ and $\lsb{c^{v(b)}}{H}$ which gives a
morphism of extensions
$$
\begin{CD}
0 @>>> \lsb{c}{A} @>>> \lsb{c}{H} @>>> \G(L/K) @>>> 0 \\
& &  @V \cong VV @VV \cong V @VV = V \\
0 @>>> \lsb{c^{v(b)}}{A} @>>> \lsb{c^{v(b)}}{H} @>>> \G(L/K) @>>> 0.
\end{CD}
$$
In this sense, $\Delta(c)$ is independent of the choice of $c$.

\mni
Finally, every splitting of the extension comes from a lift of $c$ to
a crossed homomorphism $\G(L/K)\rightarrow B$.  This proves the
following.

\begin{prop} \label{prop-les5}
\marpar{prop-les5}
For every element $c$ in $Z^1_\text{cts.}(\G(L/K),C)$, $\Delta(c)$
equals $0$ in $H^2(\G(L/K),A)$ if and only if $[c]$ is in the image of
$v_*$.
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Step IX. Central extensions
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Step IX. Central extensions} 
\label{sssec-StepIX}
\marpar{sssec-StepIX}

\mni
The final step is when $A$ is a central subgroup of $B$.  Then
$\ast_c$ is the usual action of $\G(L/K)$ on $A$.  Therefore
$\Delta(c)$ is an element of $H^2(\G(L/K),A)$.  Similarly, conjugation
by $b$ on $A$ is trivial, and thus conjugation by $(b,1)$ defines an
isomorphism of extensions of $\lsb{c}{H}$ and $\lsb{c^{v(b)}}{H}$.
Therefore $\Delta(c)$ is independent of the representative of $[c]$,
i.e., there is a well-defined set map
$$
\Delta:H^1(\G(L/K),C) \rightarrow H^2(\G(L/K),A).
$$
Finally, twisting $A$, $B$ and $C$ by an element $c$ again gives a
well-defined map
$$
\lsb{c}{\Delta}:H^1(\G(L/K),\lsb{c}{C}) \rightarrow H^1(\G(L/K),A).
$$
For the same reason as in Proposition ~\ref{prop-les2}, 
$$
\ast c: H^1(\G(L/K),\lsb{c}{C}) \rightarrow H^1(\G(L/K),C)
$$
defines a bijection between the fiber of $\Delta$ containing $[c]$ and
the fiber of $\lsb{c}{\Delta}$ over $0$.  Therefore
$$
\ast c \circ \lsb{c}{v}_*:H^1(\G(L/K),\lsb{c}{B}) \rightarrow
H^1(\G(L/K),C)
$$
maps surjectively onto the fiber of $\Delta$ containing $[c]$.  
In summary, this gives the following.

\begin{prop} \label{prop-les6}
\marpar{prop-les6}
Associated to a central extension of $\G(L/K)$-modules,
$$
\begin{CD}
0 @>>> A @> u >> B @> v >> C @>>> 1
\end{CD}
$$
the sequence of set maps
$$
0 \rightarrow H^0(\G(L/K),A) \rightarrow H^0(\G(L/K),B) \rightarrow
H^0(\G(L/K),C) \xrightarrow{\delta} 
$$
$$
H^1(\G(L/K),A) \xrightarrow{u_*}
H^1(\G(L/K),B) \xrightarrow{v_*} H^1(\G(L/K),C) \xrightarrow{\Delta}
H^2(\G(L/K),A)
$$
is exact in the following sense:
the first map $u_*$ is injective with central image, the fibers of $v_*$ equal
the orbits of $\text{Image}(u_*)$, the fibers of $\delta$ equal the
orbits of $H^0(\G(L/K),B)$, the fibers of the second map $u_*$ equal
the orbits of $H^0(\G(L/K),C)$, the fibers of the second map $v_*$
equal the orbits of the Abelian group $H^1(\G(L/K),A)$, and the fiber
of $\Delta$ containing the element $[c]$ equals the image of the map
$$
\ast c \circ \lsb{c}{v}_*:H^1(\G(L/K),\lsb{c}{B}) \rightarrow
H^1(\G(L/K),C).
$$
\end{prop} 

\begin{rmk} \label{rmk-comm}
\marpar{rmk-comm}
The long exact sequence is functorial in the short exact sequence.  To
be precise, given a commutative diagram of homomorphisms of
$\G(L/K)$-modules 
$$
\begin{CD}
A @> u >> B \\
@V f VV @VV g V \\
A' @> u' >> B'
\end{CD}
$$
with $u$ and $u'$ injective, the diagram
$$
\begin{CD}
C^{\G(L/K)} @> \delta_u >> H^1(\G(L/K),A) \\
@V g_* VV @VV f_* V \\
(C')^{\G(L/K)} @> \delta_v >> H^1(\G(L/K),A')
\end{CD}
$$
is commutative, where $C=B/A$ and $C'=B'/A'$.  Moreover, if $A$,
resp. $A'$, is a central subgroup of $B$, resp. of $B'$, then also the
diagram 
$$
\begin{CD}
H^1(\G(L/K),C) @> \Delta_u >> H^2(\G(L/K),A) \\
@V g_* VV @VV f_* V \\
H^1(\G(L/K),C') @> \Delta_v >> H^2(\G(L/K),A')
\end{CD}
$$
is commutative.  The proof of each of these assertions is a good
exercise for the reader.
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Galois cohomology of smooth group schemes  
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Galois cohomology of smooth group schemes} 
\label{sec-smooth}
\marpar{sec-smooth}

\mni
Most of the flavor of Galois cohomology comes from the specific Galois
modules one studies.  For instance, one could consider 
the action of $\G(K^s/K)$ on the \'etale cohomology groups of
an algebraic variety defined over $K$.  Or, when $K$ is a number
field, 
one could consider the
action of $\G(L/K)$ on the ideal class group of the ring of integers
in $L$.  There is another source of Galois modules which are somewhat
simpler than the previous 2 examples, but which still play an
important role.

\mni
Let $K$ be a field, let $G$ be a quasi-compact,
smooth group scheme over $K$, and let
$L/K$ be a Galois extension.  Then the set $G(L)$ of $L$-points of $G$
is a group with a natural action of $\G(L/K)$ by group homomorphisms.  
Since $G$ is locally
finitely presented over $K$, $G(L)$ is a discrete $\G(L/K)$-module.
In fact, this is even better than a usual Galois module.  For every
extension field $K'/K$ and for every Galois extension $L'/K'$, the set
of $L'$-points $G(L')$ is a discrete $\G(L'/K')$-module.  The Galois
cohomologies of these modules satisfy all the functorialities one
would expect. It is a good exercise to formulate and prove these for
oneself.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Galois descent of torsors  
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Galois descent of torsors} 
\label{ssec-descent}
\marpar{ssec-descent}

\mni
It is straightforward that $H^0(\G(L/K),G(L))$ equals $G(K)$.  There
is also an alternative description of $H^1(\G(L/K),G(L))$ whose proof
uses \emph{descent}.

\begin{defn} \label{defn-descent}
\marpar{defn-descent}
A right 
\emph{$G$-torsor} over $K$ is a nonempty, locally finitely presented
$K$-scheme $T$ together with a right action of $G$ on $T$,
$$
\mu:T\times_K G \rightarrow T
$$
in the category of $K$-schemes such that the induced morphism
$$
(\text{pr}_T,\mu):T\times_K G \rightarrow T\times_K T
$$
is an isomorphism.  A \emph{morphism of $G$-torsors} is a morphism of
$K$-schemes which is equivariant for the action of $G$ (such a
morphism is automatically an isomorphism).  
\end{defn}

\mni
Let $T$ be a $G$-torsor over $K$.  There is a bijective correspondence
between $T(K)$ and the set of morphisms of $G$-torsors from the
trivial $G$-torsor $G$ to $T$ via the rule associating to $t$ in
$T(K)$ the morphism
$$
\mu_t:G \rightarrow T, \ \ g\mapsto \mu(t,g).
$$
In particular, $T$ is trivial if and only if $T(K)$ is nonempty.  By
the same token, for every field extension $L/K$, the base change
$T\otimes_K L$ is trivial as a $G\otimes_K L$-torsor if and only if
$T(L)$ is nonempty.

\mni
Let $L/K$ be a Galois extension and let $T$ be a $G$-torsor such that
$T(L)$ is nonempty.  Then the set $T(L)$ is
naturally a $\G(L/K)$-equivariant $G(L)$-torsor.  Moreover, 
to every morphism 
$$
\phi: T \rightarrow T'
$$
of $G$-torsors over $K$, the induced set map 
$$
\phi(L): T(L) \rightarrow T'(L)
$$
is a morphism of $\G(L/K)$-equivariant $G(L)$-torsors.  This defines a
set map
$$
p^*:\text{Hom}_{G-\text{equiv.}}(T,T') \rightarrow
\text{Hom}_{\G(L/K)-G(L)-\text{equiv.}}(T(L),T'(L)), \ \ \phi \mapsto
\phi(L) 
$$
from the set of morphisms of $G$-torsors to the set of morphisms of
$\G(L/K)$-equivariant $G(L)$-torsors.

\begin{prop}[Galois descent] \label{prop-descent}
\marpar{prop-descent}
Let $L/K$ be a finite
Galois extension and let $G$ be a quasi-compact, smooth
group scheme over $K$.  
\begin{enumerate}
\item[(0)]
Every $G$-torsor $T$ over $K$ is a quasi-projective, smooth 
$K$-scheme.  
\item[(i)]
For every pair $T,T'$ of $G$-torsors over $K$ such that $T(L)$ and
$T'(L)$ are nonempty, the set map $p^*$ defined above is a bijection.
\item[(ii)]
Every $\G(L/K)$-equivariant $G(L)$-torsor is isomorphic to one
associated to a $G$-torsor over
$K$.  
\end{enumerate}
\end{prop}

\begin{proof}
This follows by the technique of \emph{descent}, cf. \cite[Section
  I]{WeilDescent}, \cite[no. 190]{FGA}, \cite[\S 6.5]{Ner}.  
\end{proof}

\mni
When $L/K$ is a Galois extension which is not necessarily finite,
Proposition ~\ref{prop-descent} still holds so long as we restrict
attention to discrete torsors.  In particular, the theorem establishes
a bijection between $H^1(\G(K^s/K),G(L))$ and the set of isomorphism
classes of $G$-torsors.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Weil restriction of scalars
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Weil restriction of scalars} 
\label{sssec-Weil}
\marpar{sssec-Weil}

\mni
Let $L/K$ be a finite extension of fields (not necessarily
separable).  Let $X$ be an $L$-scheme.  

\begin{defn} \label{defn-Weil}
\marpar{defn-Weil}
A \emph{Weil restriction} of $X$ with respect to $L/K$ is a pair
$(\Wr{L/K}(X),f)$ of a
$K$-scheme
$\Wr{L/K}(X)$ together with a morphism of $L$-schemes
$$
f:\Wr{L/K}(X)\otimes_K L \rightarrow X
$$
which is universal among all such pairs.
\end{defn}

\begin{prop}[Weil Restriction] \label{prop-Weil}
\marpar{prop-Weil}
For every quasi-projective $L$-scheme $X$ there exists a Weil
restriction of $X$ with respect to $L/K$.  Moreover, $\Wr{L/K}(X)$ is
a quasi-projective $K$-scheme.  And if $X$ is $L$-smooth, then
$\Wr{L/K}(X)$ is $K$-smooth.
\end{prop}

\begin{proof}
This follows, for instance, from \cite[Part IV.4.c, p. 221-19]{FGA}.
See also \cite[Theorem 4, p. 194]{Ner}.  The smoothness assertion is
an easy application of the infinitesimal lifting criterion for
smoothness, cf. \cite[Th\'eor\`eme 17.5.1]{EGA4} or \cite[Exercise
  II.8.6]{H}.
\end{proof}

\mni
For every quasi-compact, smooth group scheme $G_L$ over $L$, 
the Weil restriction
$\Wr{L/K}(G_L)$ has a natural structure of a group scheme over
$K$.  For every $G_L$-torsor $T_L$ over $L$, the Weil restriction
$\Wr{L/K}(T_L)$ has a natural structure of a
$\Wr{L/K}(G_L)$-torsor.  

\mni
Also, for every $K$-scheme $Y$, denoting by
$Y_L$ the base change $Y\otimes_K L$, the identity morphism
$$
Y\otimes_K L \rightarrow Y_L
$$
determines a $K$-morphism
$$
i_Y:Y \rightarrow \Wr{L/K}(Y_L).
$$
For every open affine $U = \SP R$ of $Y$, for every $K$-algebra $A$
and for every pair of morphisms of $K$-algebras, 
$$
\phi,\phi':R\rightarrow A,
$$
if the associated $A\otimes_K L$-points,
$$
\phi\otimes \text{Id}_L, \phi'\otimes \text{Id}_L: R\rightarrow
A\otimes_K L
$$
are equal, then $\phi$ and $\phi'$ are equal; indeed, the map of
$K$-vector spaces
$$
\text{Hom}_{K-\text{v. space}}(R,A)\rightarrow
\text{Hom}_{K-\text{v. space}}(R,A\otimes_K L)
$$
is injective since $A\rightarrow A\otimes_K L$ is injective.  Thus,
for a quasi-projective $K$-scheme $Y$, it follows that $i_Y$ is an
unramified monomorphism.  

\begin{prop}\label{prop-bound}
\marpar{prop-bound}
Let $L/K$ be a finite Galois extension.  
\begin{enumerate}
\item[(i)]
For every quasi-compact, smooth group scheme $G$ over $K$, 
$$
i_G:G\rightarrow \Wr{L/K}(G_L)
$$
is both a closed immersion and a
homomorphism of quasi-compact, smooth group schemes
over $K$.
\item[(ii)]
The connecting map in Galois cohomology,
$$
\delta:(\Wr{L/K}(G_L)/G)(K) \rightarrow H^1(\G(L/K),G(L))
$$
is surjective, i.e., every $G$-torsor over $K$ having an $L$-point is
isomorphic to a $G$-orbit in $\Wr{L/K}(G_L)$.
\end{enumerate}
\end{prop}

\begin{proof}
\textbf{(i).}  First of all, since $G$ is a quasi-compact, smooth
group scheme over $K$, it is quasi-projective, cf. \cite[Theorem 1,
  p. 153]{Ner} for instance.  Thus $G_L$ is also a quasi-projective,
smooth group scheme over $L$.  Thus, by Proposition \ref{prop-Weil},
$\Wr{L/K}(G_L)$ is a quasi-projective, smooth group scheme over $K$.  
For every $K$-scheme $T$, the pullback map $G(T)\rightarrow
G(T\otimes_K L)$ is a group homomorphism.  Thus also $i_G(T)$ is a
group homomorphism, i.e., $i_G$ is a homomorphism of group schemes.
And by the usual argument, the image of $i_G$ is a closed subgroup scheme
of $\Wr{L/K}(G_L)$: the closure of $i_G(G)$ is a group and $i_G(G)$ is
a dense, equivariant open in this group, and therefore the entire
group.  
A homomorphism of group schemes which is an unramified
monomorphism is automatically a closed immersion; the closure of the
image is a group scheme which contains $i_G(G)$ is an equivariant,
dense open, hence as all of the closure.


\mni
\textbf{(ii).}
Let $T$ be a
$G$-torsor over $K$.  If $T(L)$ is nonempty, then there is an
isomorphism of $G_L$ and $T_L$ as $G_L$-torsors over $L$.  Thus there
is an isomorphism 
$$
u:\Wr{L/K}(T_L) \xrightarrow{\cong} \Wr{L/K}(G_L)
$$
as
$\Wr{L/K}(G_L)$-torsors over $K$.  The morphism
$$
i_G:G\rightarrow \Wr{L/K}(G_L)
$$
is a homomorphism of group schemes over $K$.  And the composition
$$
u\circ i_T: T \rightarrow \Wr{L/K}(G_L)
$$
embeds $T$ as an $i_G(G)$-orbit in $\Wr{L/K}(G_L)$.  
\end{proof}

\mni
Part (ii) can also be proved by writing out explicitly the map
$$
H^1(\G(L/K),G(L)) \xrightarrow{i_{G,*}} H^1(\G(L/K),
\Wr{L/K}(G_L)(L)) 
$$
and verifying it is constant.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Examples  
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Examples} 
\label{ssec-examples}
\marpar{ssec-examples}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: The multiplicative group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{The multiplicative group} 
\label{sssec-Gm}
\marpar{sssec-Gm}

\mni
One particularly important case is when $G = \Gm{m,K}$, the
multiplicative group scheme $\SP K[t,t^{-1}]$.  

\begin{thm}[Hilbert's Theorem 90] \label{thm-HT90}
\marpar{thm-HT90}
For the multiplicative group, $H^0(\G(K^s/K),\Gm{m,K}(K^s))$
equals $K^*$ and $H^1(\G(K^s/K),\Gm{m,K}(K^s))$ is
$\{[\Gm{m,K}(L)]\}$.  
\end{thm}

\begin{proof}
The first assertion is obvious.  For the second assertion, it suffices
to verify $H^1(\G(L/K),\Gm{m,K}(L))$ is trivial for every
finite Galois extension $L/K$, i.e., every
$\Gm{m,K}$-torsor having an $L$-point also has a $K$-point.  
By Proposition ~\ref{prop-bound}, every $\Gm{m,K}$-torsor
having an $L$-point is a $\Gm{m,K}$-orbit in
$\Wr{L/K}(\Gm{m,L})$.      
Of course $\Gm{m,K}$ is an
open subscheme of the affine line $\AAA_K^1$, and the immersion
is equivariant.  Thus $\Wr{L/K}(\Gm{m,L})$ is an
equivariant open subscheme of $\Wr{L/K}(\AAA_L^1)$.  Up to
choosing a basis $\{b_1,\dots,b_n\}$ for $L$ as a $K$-vector space,
$\Wr{L/K}(\AAA_L^1)$ is isomorphic to $\AAA_K^n$ as a
$K$-scheme.  Moreover, the action of $\Gm{m,K}$ on
$\Wr{L/K}(\AAA_L^1)$ induced by the homomorphism
$$
i_{\Gm{m,K}}:\Gm{m,K} \rightarrow
\Wr{L/K}(\Gm{m,L})
$$
equals the standard action of $\Gm{m,K}$ on $\AAA_K^n$
via this isomorphism.  So the $\Gm{m,K}$-orbits in
$\Wr{L/K}(\AAA^1_K)$ are the same as the
$\Gm{m,K}$-orbits in $\AAA_K^n$, i.e., the set of
nonzero elements in $1$-dimensional $K$-linear subspaces of
$\AAA_K^n$.  Therefore Hilbert's Theorem 90 reduces to the
statement that every $1$-dimensional $K$-linear subspace of
$\AAA_K^n$ contains a nonzero vector, which is obvious.
\end{proof}

\mni
Of course there is another proof which forms a Poincar\'e series out
of a $1$-cocycle and then uses linear independence of automorphisms to
conclude this Poincar\'e series is nonzero.  In fact the proof above
uses the full machinery of descent, parts of which are proved by using
arguments very similar to the Poincar\'e series argument.  

\mni
Because of Hilbert's Theorem 90, the first nontrivial cohomology group
is $H^2(\G(K^s/K),\Gm{m,K}(K^s))$.

\begin{defn} \label{defn-Brprime}
\marpar{defn-Brprime}
For every Galois extension $L/K$, the
\emph{cohomological
  Brauer group of $L/K$}, denoted $\text{Br}'(L/K)$, is
$H^2(\G(L/K),\Gm{m,K}(L))$.    
When $L$ is a separable closure $K^s$ of $K$,
this is called the \emph{cohomological Brauer group of $K$},
$\text{Br}'(K)$.
\end{defn}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Groups of roots of unity
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Groups of roots of unity} 
\label{sssec-mun}
\marpar{sssec-mun}

\mni
Let $n$ be a positive integer not divisible by $\text{char}(K)$.  Then
the group scheme $\mun{n,K} := \SP K[t,t^{-1}]/\langle t^n -1
\rangle$ is a smooth group scheme over $K$.  There is a short exact
sequence of smooth, commutative group schemes over $K$, the
\emph{Kummer sequence},
$$
\begin{CD}
1 @>>> \mun{n,K} @>>> \Gm{m,K} @> (-)^n >>
\Gm{m,K} @>>> 1.
\end{CD}
$$

\begin{prop} \label{prop-mu}
\marpar{prop-mu}
The Kummer sequence induces isomorphisms of Abelian groups,
$$
H^1(\G(K^s/K),\mun{n,K}(K^s)) =
K^*/(K^*)^n
$$
and
$$
H^2(\G(K^2/K),\mun{n,K}(K^s)) =
\text{Br}'(K)[n]. 
$$
\end{prop}

\begin{proof}
This follows immediately from the long exact sequence of (Abelian)
Galois cohomology
and Theorem ~\ref{thm-HT90}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Additive and unipotent groups
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Additive and unipotent groups} 
\label{sssec-Ga}
\marpar{sssec-Ga}

\mni
Another important case is when $G = \Gm{a,K} = \SP K[t]$, the
additive group.  

\begin{prop} \label{prop-Ga}
\marpar{prop-Ga}
For the additive group $\Gm{a,K}$,
$H^0(\G(K^s/K),\Gm{a,K}(K^s))$ equals $K$ and for every $q>0$,
$H^q(\G(K^s/K),\Gm{a,K}(K^s))$ equals $\{0\}$.
\end{prop}

\begin{proof}
For every finite Galois extension $L/K$ with Galois group $\Gamma$, 
the normal basis theorem
gives an isomorphism between $L$ and $K[\Gamma]$ as
$K[\Gamma]$-modules.  Since $K[\Gamma]$ is a free module over itself,
it has no higher Galois cohomology.
\end{proof}

\mni
If
$\text{char}(K)= p$, then there is a short exact sequence of smooth,
commutative group schemes, the \emph{Artin-Schreier sequence},
$$
\begin{CD}
0 @>>> \ZZ/p\ZZ @>>> \Gm{a,K} @> F >> \Gm{a,K} @>>> 0
\end{CD}
$$
where $F(a) = a^p - a$.  

\begin{cor} \label{cor-Zp}
\marpar{cor-Zp}
If $\text{char}(K)=p$, then $H^0(\G(K^s/K),\ZZ/p\ZZ)$ equals
$\ZZ/p\ZZ$, $H^1(\G(K^s/K),\ZZ/p\ZZ)$ equals $K/F(K)$, and for every
$q>1$, $H^q(\G(K^s/K),\ZZ/p\ZZ)$ equals $\{0\}$.
\end{cor}

\begin{cor} \label{cor-unipot}
\marpar{cor-unipot}
If a quasi-compact, smooth group scheme $U$ over $K$ has a composition
series by normal, closed subgroup schemes whose subquotients are each
isomorphic to $\Gm{a,K}$, then $H^1(\G(K^s/K),U(K^s))$ equals
$\{*\}$.  
\end{cor}

\mni
The first corollary follows from Proposition ~\ref{prop-Ga} and the
long exact sequence of Abelian Galois cohomology.  The second
corollary follows by inductively applying Proposition ~\ref{prop-les4}
to the terms in the composition series, using Proposition
~\ref{prop-Ga} for the base case of the induction.

\begin{defn} \label{defn-unipot}
\marpar{defn-unipot}
A connected, quasi-compact, 
smooth group scheme $U$ over $K$
is \emph{unipotent} if for some integer $N$,
$U$ is isomorphic to a
closed subgroup scheme of $\GL{N,K}$ whose $K^s$-points are
all unipotent (i.e., $1$ is the only eigenvalue).  
\end{defn}

\begin{rmk} \label{rmk-unipot}
\marpar{rmk-unipot}
If $K$ is a
\emph{perfect} field, then every connected, unipotent group over $K$
has a composition series whose subquotients are each isomorphic to
$\Gm{a,K}$.  Thus $H^1(\G(K^s/K),U(K^s))$ equals $\{*\}$ 
for every
connected, unipotent group $U$ over a perfect field $K$.  
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsubection: Matrix groups
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Matrix groups} 
\label{sssec-GLn}
\marpar{sssec-GLn}

\mni
For every integer $n$, the affine group scheme 
$\GL{n,K}$ over $K$, resp. $\SL{n,K}$, $\PGL{n,K}$ over $K$, 
represents the
functor
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\text{Isom}_{A-\lmod}(A^{\oplus n}, A^{\oplus n}),
$$
resp.
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\{\phi \in \text{Isom}_{A-\lmod}(A^{\oplus n}, A^{\oplus n}) |
\text{det}(\phi) = 1\},
$$
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\text{Isom}_{A-\lmod}(A^{\oplus n}, A^{\oplus n})/\sim, \ \ \phi\sim
\lambda\cdot \phi, \forall \lambda \in A^*.
$$
Basically the same, for every finite dimensional $K$-vector space $V$,
the group schemes $\GL{V,K}$, resp. $\SL{V,K}$, $\PGL{V,K}$,
represents the functor
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\text{Isom}_{A-\lmod}(A\otimes_K V, A\otimes_K V),
$$
resp.
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\{\phi \in \text{Isom}_{A-\lmod}(A\otimes_K V, A\otimes_K V) |
\text{det}(\phi) = 1\},
$$
$$
K-\text{algebras} \rightarrow \text{Groups}, \ \ A \mapsto
\text{Isom}_{A-\lmod}(A\otimes_K V, A\otimes_K V)/\sim, \ \ \phi\sim
\lambda\cdot \phi, \forall \lambda \in A^*.
$$
Of course up to choosing a basis for $V$, these are the same groups.

\begin{thm}[Speiser's Theorem] \label{thm-Speiser}
\marpar{thm-Speiser}
For every integer $n\geq 1$, every $\GL{n,K}$-torsor over
$K$, 
resp. every $\SL{n,K}$-torsor over $K$, is trivial.
\end{thm}

\begin{proof}
This is almost the same as the proof of Theorem ~\ref{thm-HT90}.  By
the same argument as there, for every finite Galois extension $L/K$,
every $\GL{n,K}$-torsor over $K$ with an $L$-point is
isomorphic as a $\GL{n,K}$-torsor to a free (right)
$\GL{n,K}$-orbit in $\text{Hom}_{K-\text{v. space}}(K^{\oplus
  n}, L^{\oplus n})$.  Every free orbit is of the form
$\text{Isom}_{K-\text{v. space}}(K^{\oplus n},V)$ where $V$ is an
$n$-dimensional $K$-linear subspace of $L^{\oplus n}$.  Choosing a
basis for $V$, such an orbit has a $K$-point.

\mni
Consider the short exact sequence of group schemes
$$
\begin{CD}
1 @>>> \SL{n,K} @>>> \GL{n,K} @> \text{det} >>
\Gm{m,K} @>>> 0.
\end{CD}
$$
This is ``split'' by the homomorphism
$$
s:\Gm{m,K} \rightarrow \GL{n,K},
$$
$$
\lambda \mapsto \lt[
\begin{array}{rrrr}
\lambda & 0 & \hdots & 0 \\
0       & 1 & \hdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \hdots & 1
\end{array} \rt].
$$
Thus the connecting map in the associated long exact sequence of
Galois cohomology is constant. Therefore
$H^1(\G(K^s/K),\SL{n,K}(K^s))$ is a subset of
$H^1(\G(K^s/K),\GL{n,K}(K^s))$, which is a singleton set by
the previous paragraph. 
\end{proof}

%% \mni
%% For every integer $n\geq 1$ there is a
%% commutative diagram of flat, linear group schemes over $K$
%% with exact rows and
%% columns,
%% $$
%% \begin{CD}
%% & & 1 & & 1 & & 1 \\
%% & & @VVV  @VVV @VVV \\
%% 1 @>>> \mun{n,K} @>>> \Gm{m,K} @> (-)^n >>
%% \Gm{m,K} @>>> 1 \\
%% & & @VVV  @VVV @VV = V \\
%% 1 @>>> \SL{n,K} @>>> \GL{n,K} @> \text{det} >>
%% \Gm{m,K} @>>> 1 \\
%% & & @VVV @VVV @VVV \\
%% 1 @>>> \PGL{n,K} @> = >> \PGL{n,K} @>>> 1 \\
%% & & @VVV @VVV \\
%% & & 1 & & 1
%% \end{CD}
%% $$
%% The columns are central extensions.  In particular, the middle column
%% is a central extension of smooth, linear group schemes over $K$.  This
%% gives a central extension of $\G(L/K)$-modules.  

\mni
The long exact sequence of Galois cohomology associated to the central
extension 
$$
\begin{CD}
1 @>>> \Gm{m,K} @> (-)\cdot \text{Id}_{n\times n} >>
\GL{n,K} @> q >> \PGL{n,K} @>>> 1
\end{CD}
$$
gives a connecting map
$$
\Delta_n:H^1(\G(L/K),\PGL{n,K}(L)) \rightarrow
H^2(\G(L/K),\Gm{m,K}(L)) = \text{Br}'(L/K).
$$

\begin{thm} \label{thm-BrBrprime}
\marpar{thm-BrBrprime}
\textbf{(i)} The fiber of $\Delta_n$ over the neutral element is
$\{*\}$.
\linebreak
\noindent\textbf{(ii)}
The image of $\Delta_n$ is contained in the $n$-torsion subgroup,
$\text{Br}'(L/K)[n]$.  
\end{thm}

\begin{proof}
\textbf{(i).}  This follows from Proposition ~\ref{prop-les6} and Theorem
~\ref{thm-Speiser}. 

\mni
\textbf{(ii).}
By Remark ~\ref{rmk-comm},
associated to the following commutative diagram of central extensions
of smooth, linear group schemes,
$$
\begin{CD}
1 @>>> \Gm{m,K} @> (-)\cdot\text{Id}_{n\times n} >> 
\GL{n,K} @> q >> \PGL{n,K} @>>> 1 \\
& & @V (-)^n VV @VV (\text{det},q) V @VV = V \\
1 @>>> \Gm{m,K} @> e_1 >> \Gm{m,K} \times
\PGL{n,K} 
@> \pi_2 >> \PGL{n,K} @>>> 1 
\end{CD}
$$
there is a commutative diagram
$$
\begin{CD}
H^1(\G(L/K),\PGL{n,K}(L)) @> \Delta_n >> 
H^2(\G(L/K),\Gm{m,K}(L)) \\
@V = VV @VV (-)^n V \\
H^1(\G(L/K),\PGL{n,K}(L)) @> \Delta_{e_1} >> 
H^2(\G(L/K),\Gm{m,K}(L))
\end{CD}.
$$
Since $e_1$ is split, $\Delta_{e_1}$ is the constant
map.  Thus the image of $\Delta_n$ is in the kernel of the raising to
the $n^\text{th}$ power map, i.e., it is in $\text{Br}'(L/K)[n]$.  
\end{proof}

\mni
For two finite dimensional $K$-vector spaces $V$ and $W$, there is a
morphism of smooth group schemes,
$$
\otimes:\GL{V,K}\times \GL{W,K} \rightarrow \GL{V\otimes_K W,K}, \ \
(\phi,\psi) \mapsto \phi\otimes \psi.
$$
This induces morphisms of smooth group schemes,
$$
\otimes: \SL{V,K}\times \SL{W,K} \rightarrow \SL{V\otimes_K W,K},
$$
and
$$
\otimes: \PGL{V,K}\times \PGL{W,K} \rightarrow \PGL{V\otimes_K W,K}.
$$
In particular, by the functoriality of Galois cohomology and by Remark
~\ref{rmk-adhoc}, this induces a map of pointed sets,
$$
\otimes: H^1(\G(L/K),\PGL{V,K}(L))\times H^1(\G(L/K),\PGL{W,K}(L))
\rightarrow 
H^1(\G(L/K),\PGL{V\otimes_K W,K}(L)).
$$
Up to choosing bases, this is the same as a map
$$
\otimes: H^1(\G(L/K),\PGL{l,K}(L))\times H^1(\G(L/K),\PGL{n,K}(L))
\rightarrow 
H^1(\G(L/K),\PGL{ln,K}(L)).
$$

\begin{lem} \label{lem-BrBrprime2}
\marpar{lem-BrBrprime2}
For every $\PGL{l,K}$-torsor $E$, resp. $\PGL{n,K}$-torsor $F$, with
an $L$-point, $\Delta_{ln}(E\otimes F)$ equals $\Delta_l(E)+\Delta_n(F)$.
\end{lem}

\begin{proof}
By Remark ~\ref{rmk-comm}, 
associated to the following commutative diagram of central extensions
of smooth, linear group schemes,
$$
\begin{CD}
1 @>>> \Gm{m,K}\times \Gm{m,K} @>>> \GL{l,K}\times \GL{n,K} @>>>
\PGL{l,K} \times \PGL{n,K} @>>> 1 \\
& & @V \times VV @VV \otimes V @VV \otimes V \\
1 @>>> \Gm{m,K} @>>> \GL{lm,K} @>>> \PGL{lm,K} @>>> 1
\end{CD},
$$
there is a commutative diagram
$$
\begin{CD}
H^1(\G(L/K),\PGL{l,K}(L))\times H^1(\G(L/K),\PGL{n,K}(L)) @>
\Delta_l\times \Delta_n >> \text{Br}'(L/K) \times \text{Br}'(L/K) \\
@V \otimes VV @VV \times V \\
H^1(\G(L/K),\PGL{ln,K}(L)) @> \Delta_{ln} >> \text{Br}'(L/K)
\end{CD}.
$$
The lemma is equivalent to the commutativity of this diagram.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: The Brauer group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Brauer group} 
\label{sec-Brauer}
\marpar{sec-Brauer}


\begin{defn} \label{defn-CSA}
\marpar{defn-CSA}
Let $L/K$ be a Galois extension. 
A \emph{central simple algebra} over $L/K$ is a finite dimensional
$K$-algebra $A$ (so $K$ is contained in the center of $A$) such that
$A\otimes_K L$ is isomorphic as an $L$-algebra
to the matrix algebra $\Mat{n\times n,L}$ for some integer $n$.
When $L$ is a separable closure $K^s$, this is simply called a central
simple algebra over $K$.
\end{defn}

\mni  
In particular,
matrix algebras $\Mat{n\times n,K}$ are central simple
algebras over $K$.  If $A$ is a central simple algebra over $K$, then
so is the \emph{opposite algebra} $A^{\text{opp}}$ defined to be $A$ but
with opposite multiplication $a\ast b := ba$.  And if $A$ and $A'$ are
central simple algebras over $K$, so is $A\otimes_K A'$.  Thus the set
of isomorphism classes of central simple algebras is an associative,
commutative semigroup with identity element $[K]$.  But it is better
than this.  

\begin{defn} \label{defn-Morita}
\marpar{defn-Morita}
Two central simple algebras $A$ and $A'$ are 
\emph{Morita equivalent} if there exist positive integers $n$ and $n'$
such that $\Mat{n\times n,K}\otimes_K A$ is isomorphic to
$\Mat{n'\times n',K}\otimes_K A'$.  
\end{defn}

\begin{prop} \label{prop-Morita}
\marpar{prop-Morita}
Let $A$, $A'$ and $B$ be central simple algebras over $L/K$.  If $A$
and $A'$ are Morita equivalent, so are $A^{\text{opp}}$ and
$(A')^{\text{opp}}$, and so are $A\otimes_K B$ and $A'\otimes_K B$.
Also, the natural map
$$
A\otimes_K A^{\text{opp}} \rightarrow
\text{Hom}_{K-\text{v. space}}(A,A),
$$
$$
(a,b) \mapsto (c\mapsto acb)
$$
is an isomorphism of central simple algebras.  In fact the composition
with the trace,
$$
A\otimes_K A^{\text{opp}} \rightarrow
\text{Hom}_{K-\text{v. space}}(A,A) \xrightarrow{\text{Trace}} K
$$
is a perfect pairing of $K$-vector spaces.
\end{prop}

\begin{proof}
The assertions about Morita equivalence are straightforward.  The last
two statements say that a certain maps of $K$-vector spaces are
isomorphisms.  These may be checked after base change from $K$ to $K^s$,
and then $A$ is isomorpic to a matrix algebra.  The assertions are easy
to verify for matrix algebras (using the bases of elementary matrices,
for instance).
\end{proof}



\mni
Because of the proposition, the set of Morita equivalence classes is an
Abelian group under $[A] + [B] = [A\otimes_K B]$ and $-[A] =
[A^{\text{opp}}]$.  The identity element is $0 = [\Mat{n,K}]$ for any
integer $n\geq 1$.

\begin{defn} \label{defn-Br}
\marpar{defn-Br}
Let $L/K$ be a Galois extension.  The group of Morita equivalence
classes of central simple algebras over $L/K$ is the \emph{Brauer group
  of $L/K$}, denote $\text{Br}(L/K)$.  When $L$ is a separable closure
$K^s$ of $K$, this is referred to as the \emph{Brauer group of $K$},
$\text{Br}(K)$.  
\end{defn}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Relation to $\PGL{}$-torsors
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relation to $\PGL{}$-torsors} 
\label{ssec-PGL}
\marpar{ssec-PGL}

\mni
Let $n$ be a positive integer.
The group $\PGL{n,K}$ acts by algebra automorphisms
on $\Mat{n\times n,K}$ via conjugation.  Thus, for every
$\PGL{n,K}$-torsor $E$ over $K$, the twist
$$
A_E := \lsb{E}{\Mat{n\times n,K}} = (E\times \Mat{n\times
  n,K})/\PGL{n,K}
$$
is a central simple algebra of dimension $n^2$.  For a field extension
$L/K$, 
if $E$ has an
$L$-point, then $A_E\otimes_K L$ is isomorphic to $\Mat{n\times
  n,L}$.  This defines a set map,
$$
A_{-}:H^1(\G(L/K),\PGL{n,K}(L)) \rightarrow \mc{A}_n(L/K)
$$
where $\mc{A}_n(L/K)$ is the set of isomorphism classes of central
simple algebras $A$ over $K$ such that $A\otimes_K L \cong
\Mat{n\times n,L}$.

\begin{prop}[Skolem-Noether] \label{prop-SN}
\marpar{prop-SN}
The set map above is a bijection.  Moreover, for $\PGL{n,K}$-torsors
$E$ and $F$, the induced map from the set of morphisms in the category
of torsors to the set of morphisms in the category of central simple
algebras, 
$$
A_{-}:\text{Hom}_{\PGL{n,K}}(E,F) \rightarrow
\text{Hom}_{K-\text{CSA}}(A_E,A_F), 
$$
is a bijection.
\end{prop}

\begin{proof}
Let $A$ be a central simple algebra in $\mc{A}_n(L/K)$.  Define $E$ to
be the affine $K$-scheme representing the functor
$$
K-\text{algebras} \rightarrow \text{Sets}, \ \ R \mapsto
\text{Isom}_{R-\text{algebra}}(\Mat{n\times n,R},A\otimes_K R).
$$ 
There is a right action of $\PGL{n,K}$ on $E$ by precomposing
isomorphisms with automorphisms of $\Mat{n\times n,K}$.  The
Skolem-Noether theorem says that this action makes $I$ a
$\PGL{n,K}$-torsor.  In fact this can be checked after base change
from $K$ to $K^s$, so it reduces to the classical formulation
of the Skolem-Noether theorem: every $K$-algebra endomorphism 
of $\Mat{n\times n,K^s}$ comes from $\PGL{n,K}(K^s)$.  The natural map
$$
E \times \Mat{n\times n,K} \rightarrow A
$$
is invariant for the natural $\PGL{n,K}$-action and thus factors
through a $K$-algebra homomorphism
$$
\phi:A_E \rightarrow A.
$$
Since this is a $K$-algebra homomorphism of central simple algebras,
it is automatically an isomorphism (this can be checked after base
change to $K^s$ where it follows from Skolem-Noether).

\mni
For every $\PGL{n,K}$-torsor $F$ and every $K$-algebra homomorphism
$$
\psi:A_F \rightarrow A,
$$
the composition
$$
F\times \Mat{n\times n,K} \rightarrow A
$$
induces a morphism of schemes from $\tau:F\rightarrow E$.  
It is straightforward
to verify this is an isomorphism of $\PGL{n,K}$-torsors, and is the
unique isomorphism such that $\psi = \phi\circ A_\tau$.  The second
part of the proposition follows.    
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: The cohomological Brauer group equals the Brauer group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The cohomological Brauer group equals the Brauer group} 
\label{ssec-Br=Br'}
\marpar{ssec-Br=Br'}

\mni
Because of Proposition ~\ref{prop-SN}, the map from Theorem
~\ref{thm-BrBrprime} factors through a well-defined set map,
$$
\Delta_n:\mc{A}_n(L/K) \rightarrow \text{Br}'(L/K), \ \ \Delta_n(A_E)
:= \Delta_n(E).
$$

\begin{lem} \label{lem-Delta}
\marpar{lem-Delta}
For every $[A]$ in $\mc{A}_{l}(L/K)$ and for every $[B]$ in
$\mc{A}_{n}(L/K)$, $\Delta_{ln}(A\otimes_K B)$ equals
$\Delta_{l}(A)+\Delta_n(B)$.  In particular, if $A$ and $B$ are Morita
equivalent, then $\Delta_l(A)$ equals $\Delta_n(B)$.  Thus there is a
well-defined group homomorphism,
$$
\Delta:\text{Br}(L/K) \rightarrow \text{Br}'(L/K), \ \ \Delta([A]) =
\Delta_l(A).
$$
This group homomorphism is injective.
\end{lem}

\begin{proof}
For a $\PGL{l,K}$-torsor $E$ and a $\PGL{n,K}$-torsor $F$,
$A_{E\otimes F}$ equals $A_E\otimes_K A_F$ as central simple algebras
(this is easiest to verify using cocycles).  Thus Lemma
~\ref{lem-BrBrprime2} implies that $\Delta_{ln}(A_E\otimes_K A_F)$
equals $\Delta_l(A_E) + \Delta_n(A_F)$.  Together with Proposition
~\ref{prop-SN}, this implies the first assertion.

\mni
Since $\Delta_n(E)$ equals $0$ for the trivial $\PGL{n,K}$-torsor, also
$\Delta_n(\Mat{n\times n,K})$ equals $0$.  Thus, by the previous
paragraph, $\Delta_{ln}(A\otimes_K \Mat{n\times n,K})$ equals
$\Delta_l(A)$.  Therefore $\Delta_l(A)$ equals $\Delta_n(B)$ if $A$
and $B$ are Morita equivalent.  Thus there is a well-defined map
$$
\Delta:\text{Br}(L/K) \rightarrow \text{Br}'(L/K)
$$
such that for every integer $l\geq 1$ and every $A$ in
$\mc{A}_{l}(L/K)$, $\Delta([A])$ equals $\Delta_l(A)$.  

\mni
Finally, assume $\Delta([A])$ equals $\Delta([B])$.  Then
$A^{\text{opp}}\otimes_K B$ and $\Mat{ln\times ln,K}$ are elements of
$\mc{A}_{ln}(L/K)$ with the same image under $\Delta_{ln}$.  By
Theorem ~\ref{thm-BrBrprime}(i) and Proposition ~\ref{prop-SN}, this
implies that $A^{\text{opp}}\otimes_K B$ is isomorphic to $\Mat{ln
  \times \ln,K}$ as central simple algebras, i.e., $[B]-[A]$ equals
$0$ in $\text{Br}(L/K)$.  Therefore $[A]$ equals $[B]$ in
$\text{Br}(L/K)$.  So $\Delta$ is injective. 
\end{proof}

\begin{thm} \label{thm-Br=Br'}
\marpar{thm-Br=Br'}
Let $L/K$ be a finite Galois extension of degree $n$.  Then 
$$
\Delta_n:\mc{A}_n(L/K) \rightarrow \text{Br}'(L/K)
$$
is surjective.  Thus for every Galois extension $L/K$ (not necessarily
finite), the homomorphism
$$
\Delta:\text{Br}(L/K) \rightarrow \text{Br}'(L/K)
$$
is surjective, and hence bijective.  In other words, the Brauer group
and the cohomological Brauer group are canonically isomorphic.  
\end{thm}

\begin{proof}
Using Proposition ~\ref{prop-H2}, every element in $\text{Br}'(L/K)$
comes from an extension
$$
\begin{CD}
1 @>>> L^* @> u >> M @> v >> \G(L/K) @>>> 1.
\end{CD}
$$
The group algebra $K[M]$ is an associative $K$-algebra which is a free
$K$-vector space with basis $\{\mathbf{e}_{m}|m\in M\}$.  
There is an action of $M$ on $L$ by
$K$-algebra homomorphisms through the quotient $v$.  This defines a
multiplication on the tensor product
$$
L[M] := L\otimes_K K[M], \ \ (a\otimes \mathbf{e}_{m}) \cdot (b\otimes
  \mathbf{e}_{n}) := (a\lsb{v(m)}{b})\otimes \mathbf{e}_{mn}
$$
for elements $a,b\in L$ and $m,n\in M$.  It is straightforward to
verify this multiplication is $K$-linear and associative with
$1\otimes \mathbf{1_N}$ as an identity element.  

\mni
Define $I$ to be the $2$-sided ideal in $L[M]$ defined by the
generators,
$$
I := \langle \lambda\otimes \mathbf{e}_{1_M} - 1\otimes
\mathbf{e}_{u(\lambda)} | \lambda \in L^* \rangle.
$$
Denote the quotient algebra by $A_M := L[M]/I$.  As a (left)
$L$-vector space, it is straightforward to check that $A_M$ has
dimension equal to $\# \G(L/K) = n$.  This algebra is usually denoted
as the \emph{cross-product algebra} associated to the extension $M$.  

\mni
It is a somewhat involved exercise to check that $A_M$ is a central
simple algebra and that $\Delta_n(A_M)$ equals $[M]$.  This is usually
done by choosing a set map splitting $s:\G(L/K)\rightarrow M$ of $v$
and
using this to translate everything into cocycles.  Since the details
do not seem very illuminating, they are left to the interested reader,
cf. \cite[Exercises 1 and 2, p.159]{LocalFields}.   
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Division algebras
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Division algebras} 
\label{ssec-Div}
\marpar{ssec-Div}

\begin{defn} \label{defn-Div}
\marpar{defn-Div}
A \emph{division algebra} over $L/K$ is a central simple algebra $A$
over $L/K$ such that every nonzero element of $A$ is invertible.
Equivalently, it is a central simple algebra whose only right ideals
are $\{0\}$ and $A$ (and the same also holds with left ideals in place
of right ideals).
When $L$ is a separable closure $K^s$ of $K$, these are referred to
simply as \emph{division algebras} over $K$.    
\end{defn}

\begin{lem} \label{lem-Div}
\marpar{lem-Div}
Let $A$ be a division algebra over $K$
Every finitely generated left (or right) $A$-module is free.
\end{lem}

\begin{proof}
Let $M$ be a left $A$-module.  If $M$ equals $\{0\}$ then 
the assertion is vacuous,
$M = A^{\oplus 0}$.  Thus assume $M$ is nonzero.
Consider a surjective homomorphism of $D$-modules,
$$
q:A^{\oplus n} \twoheadrightarrow M
$$
for which the integer $n$ is minimal.  Denote by $N$ the kernel of
$q$.  If $N$ is nonempty, then for some integer $1\leq i \leq n$, 
the projection
$$
\pi_i:K \rightarrow A
$$
is nonzero.  Thus $\pi_i(N)$ is a nonzero left $A$-submodule of $A$,
i.e., a left ideal in $A$.  Since $A$ is a division algebra,
$\pi_i(N)$ equals $A$.  But this means that, modulo $N$, the
$i^\text{th}$ basis element of $A^{\oplus n}$ is congruent to a linear
combination of the other basis elements.  So the restriction of $q$ to
the free submodules generated by the other basis elements is still
surjective, contradicting the minimality of $n$.  Therefore $N$ equals
$\{0\}$, i.e., $q$ is an isomorphism of $A$-modules.
\end{proof}

\mni
In order to prove the next result, it is useful to describe the right
ideals in $\Mat{n\times n,K}$.  The proof is left as an exercise for
the reader.

\begin{lem} \label{lem-right}
\marpar{lem-right}
For a nonzero, finite dimensional $K$-vector space $V$, the right
ideals of $\Mat{V,k}$ are in $1$-to-$1$ correspondence with the
$K$-linear subspaces $W$ of $V$ via the rule,
$$
W\mapsto \text{Hom}_{K-\text{v. space}}(V,W).
$$
\end{lem}

\begin{lem}[Schur's Lemma] \label{lem-Schur}
\marpar{lem-Schur}
Let $A$ be a central simple algebra over $K$.  Assume that $A$ is not
a division algebra, i.e., $A$ has a proper, nonzero right ideal $I$.  
Let $I$ be
a proper, nonzero right ideal which is minimal, i.e., has minimal
dimension as a $K$-vector space.  Then every nonzero left $A$-module
homomorphism,
$$
\phi:I \rightarrow I
$$
is an isomorphism and the $K$-algebra of $A$-module
homomorphisms,
$$
D := \text{End}_{A-\lmod}(I,I)
$$
is a division algebra.
\end{lem}

\begin{proof}
The image $\phi(I)$ is a left ideal in $A$.  Since $I$ is minimal, if
$\phi$ is nonzero then $\phi(I)$ equals $I$, i.e., $\phi$ is
surjective.  A surjective $K$-linear endomorphism of a finite
dimension $K$-vector space is injective, thus $\phi$ is an
isomorphism.  Thus every nonzero element of $D$ is nonzero.  After
base change from $K$ to $L$, $A\otimes_K L$ is isomorphic to $\Mat{V,L}$ for
some nonzero, finite dimensional $L$-vector space $V$.  By Lemma
~\ref{lem-right},  $I\otimes_K L$ is isomorphic to
$\text{Hom}_{L-\text{v. space}}(V,W)$ for some nonzero, proper
$L$-subspace $W$ of $V$.  It is straightforward to check that 
$$
\text{End}_{\Mat{V,L}}(\text{Hom}_{L-\text{v. space}}(V,W)) = \Mat{W,L}.
$$ 
Thus $D\otimes_K L$ is isomorphic to $\Mat{W,L}$, i.e., $D$ is a
central simple algebra over $K$.  Therefore $D$ is a division algebra.
\end{proof}

\mni
There is an induced left action of $D$ on $I$ which commutes with the
right action of $A$ on $I$.  Thus there is an induced $K$-algebra
homomorphism,
$$
A\rightarrow \text{End}_{D}(I).
$$

\begin{thm}[Artin-Wedderburn Theorem] \label{thm-AW}
\marpar{thm-AW}
Every central simple algebra $A$ over $L/K$ is isomorphic to
$D\otimes_K \Mat{n\times n,K}$ for a unique division algebra $D$ over
$L/K$.  
\end{thm}

\begin{proof}
By base change from $K$ to $L$, one can check that the $K$-linear map
$$
A\rightarrow \text{End}_{D}(I)
$$
above is an isomorphism.  By Lemma ~\ref{lem-Div}, $I$ is a free
$D$-module, say $I\cong D^{\oplus n}$.  Therefore $A$ is isomorphic to
$D\otimes_K \Mat{n\times n,K}$.

\mni
Let $D$ be a division algebra and let $A$ be $D\otimes_K \Mat{n\times
  n,K}$.  Then, up to isomorphism, the unique (nonzero) simple
$A$-module is $M := D^{\oplus n}$ with its natural $A$-action.  And
the natural map
$$
D\rightarrow \text{End}_{A-\lmod}(M)
$$
is an isomorphism of algebras.  Therefore $D$ is uniquely determined
by $A$.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: The period-index problem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The period-index problem} 
\label{ssec-pip}
\marpar{ssec-pip}

\mni
\begin{defn} \label{defn-periodindex}
\marpar{defn-periodindex}
Let $A$ be a central simple algebra over $K$.  The \emph{index} of
$A$, denote $\text{index}(A)$,
is the unique integer $n$ such that $\text{dim}_K(D)=n^2$ where $D$ is
a division algebra Morita equivalent to $A$.  The \emph{period} or
\emph{exponent} of $A$, denoted $\text{period}(A)$, 
is the order of $[A]$ in $\text{Br}(K)$.  
A \emph{splitting field} for $A$ is a separable field extension $L/K$
such that $A\otimes_K L$ is isomorphic to $\Mat{m\times m,L}$ for some
integer $m$.
\end{defn}

\mni
It turns out that $\text{index}(A)$ equals both the minimal degree
$[L:K]$ of a splitting field of $A$ as well as the greatest common
divisor of the degrees $[L:K]$ of all splitting fields.  

\begin{prop}[R. Brauer] \label{prop-periodindex}
\marpar{prop-periodindex}
For every central simple algebra $A$ over $K$, $\text{period}(A)$
divides $\text{index}(A)$, and both integers have the same prime
factors (but often with different multiplicities).
\end{prop}

\begin{proof}
See \cite[Exercise 3, p. 159]{LocalFields}.
\end{proof}

\mni
In light of this proposition, it is natural to ask about the precise
relation between the period and the index.

\begin{prob}[The Period-Index Problem] \label{prob-PI}
\marpar{prob-PI}
Let $K$ be a field.  If there exists a nonnegative 
integer $r$ such that
$\text{index}(A)$ divides $\text{period}(A)^r$ for every central
simple algebra $A$, the least such integer is the \emph{power} of
$K$, denoted $r(K)$; otherwise $r(K)$ is defined to be $\infty$.  What
is the power of $K$?  For a given integer $r$, what ``conditions'' on $K$
guarantee that $r(K) \leq r$?  
\end{prob}

\mni
Of course this problem is ill-posed since the meaning of ``condition''
is not specified.  In Chapter \ref{pt-PI} 
we will explain some answers which
hopefully give a clearer meaning to this problem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: The universal cover sequence  
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The universal cover sequence} 
\label{sec-cover}
\marpar{sec-cover}

\mni
There is a generalization of the map $\Delta_n$ from Theorem
~\ref{thm-BrBrprime} to other linear algebraic group schemes.  The
generalization uses some of the structure theory for linear algebraic
group schemes.  Unipotent groups were defined in Definition
~\ref{defn-unipot}.

\begin{defn} \label{defn-multss}
\marpar{defn-multss}
A connected, quasi-compact, smooth group scheme $T$ is
\emph{multiplicative} or 
\emph{of
  multiplicative type} if for some integer $n\geq 0$, 
$T\otimes_K K^s$ is isomorphic to
$\Gm{m,K^s}^n$ as a group scheme over $K^s$.  

\mni
For a
quasi-compact, smooth group scheme $G$ over $K$, the \emph{unipotent
  radical} of $G$, $R_u(G)$, is the maximal connected, normal subgroup
scheme 
of $G$ which is unipotent.  If $R_u(G)$ is trivial, then $G$ is
\emph{reductive}.  

\mni
For a connected, 
smooth, linear algebraic group scheme $G$ over $K$, the
\emph{multiplicative quotient} of $G$, $T(G)$, is the maximal
quotient group of $G$ (by a closed, normal subgroup scheme) which
is multiplicative.  If $R_u(G)$ and $T(G)$ are trivial, then $G$ is
\emph{semisimple}.  
If $T(G)$ is the quotient of $G$ by $R_u(G)$, then
$G$ is \emph{solvable}. 
The maximal connected, normal, solvable
subgroup of $G$ is the \emph{solvable radical} of $G$, or simply the
\emph{radical} of $G$, denoted $R(G)$.  
\end{defn}

\mni
Let $G$ be a connected, smooth, linear algebraic group scheme over
$K$.  
Of course $R_u(G)$ is the unipotent radical of $R(G)$.  And $R_u(G)$
is contained in the kernel of $G\rightarrow T(G)$.  Thus there is an
induced homomorphism of group schemes over $K$,
$$
R(G)/R_u(G) \rightarrow T(G).
$$
This is a finite and faithfully flat morphism, i.e., an \emph{isogeny}
of group schemes.  But it is not necessarily an isomorphism. (Notice
however that $T(G)$ is trivial if and only if $R(G)$ equals $R_u(G)$,
so that $G$ is semisimple if and only if $R(G)$ is trivial.)
Because $T(G)$ and $R(G)/R_u(G)$ are different,
there are $2$ different decompositions that arise naturally in
the study of group schemes.  

\begin{notat} \label{notat-L}
\marpar{notat-L}
Let $G$ be a connected, smooth, linear algebraic group scheme over
$K$.  The \emph{first associated semisimple group} of $G$ is the group
$$
L(G) = \text{Ker}(G\rightarrow T(G))/R_u(G).
$$
The \emph{second associated semisimple group} of $G$ is the group
$$
L'(G) = G/R(G).
$$
\end{notat}

\mni
As the definition of $L'(G)$ is simpler, it is the one that arises
most often.  But $L(G)$ is slightly better behaved with respect to
universal covers and fundamental groups.  Both of these definitions
should be considered as decompositions of $G$ into a unipotent group,
$R_u(G)$, a multiplicative group, $T(G)$, resp. $R(G)/R_u(G)$, and a
semisimple group, $L(G)$, resp., $L'(G)$.  

\begin{defn} \label{defn-pi}
\marpar{defn-pi}
A connected, semisimple algebraic group $L$ is \emph{simply connected}
if for every connected, semisimple algebraic group $M$, every isogeny
$$
M\rightarrow L
$$
is an isomorphism.  For every connected, semisimple algebraic group
$L$, there exists a connected, simply connected, semisimple algebraic
group $\widetilde{L}$ and an isogeny
$$
\widetilde{L}\rightarrow L,
$$
called the \emph{universal cover} of $L$.
The kernel of the universal cover, denote $\pi_1L$, is the
\emph{fundamental group scheme} of $L$.
\end{defn}

\mni
There is a central extension of group schemes,
$$
\begin{CD}
0 @>>> \pi_1L @>>> \widetilde{L} @>>> L @>>> 1
\end{CD}
$$
Note that $\pi_1L$ is sometimes not smooth, e.g.,
$\pi_1\PGL{n,K}$ is the non-smooth group scheme
$\mun{n,K}$ when $\text{char}(K)$ divides $n$.  
There is a generalization of
Galois cohomology, cohomology for the fppf topology,
$H^1_{\text{fppf}}(K;-)$.  This agrees with Galois cohomology for
smooth group schemes.
This cohomology
theory also has a long exact sequence, leading to a sequence
$$
0 \rightarrow H^0_{\text{fppf}}(K;\pi_1 G) \rightarrow
H^0(\G(K^s/K),\widetilde{G}(K^s)) \rightarrow
H^0(\G(K^s/K),G(K^s)) \xrightarrow{\delta} 
$$
$$
H^1_{\text{fppf}}(K,\pi_1 G) \xrightarrow{u_*}
H^1(\G(K^s/K),\widetilde{G}(K^s)) \xrightarrow{v_*} H^1(\G(K^s/K),G(K^s))
\xrightarrow{\Delta} 
H^2_{\text{fppf}}(K,\pi_1 G).
$$
This sequence satisfies the same conditions as in Proposition
~\ref{prop-les6}.  

\mni
Serre posed two conjectures about this sequence.  The first conjecture
was proved by R. Steinberg.

\begin{thm}[Steinberg's Theorem = Serre's ``Conjecture I'']
\cite{Steinberg} \label{thm-St}
\marpar{thm-St}
If $K$ has cohomological dimension $\leq 1$, then for every connected,
semisimple group scheme $G$ over $K$, $H^1(\G(K^s/K),G(K^s))$ is
$\{*\}$, and the converse holds.
More generally, if $K$ has cohomological dimension $\leq 1$, then
$H^1(\G(K^s/K),G(K^s))$ equals $\{*\}$ for every connected, reductive
group scheme $G$ over $K$.  If $K$ is perfect, it also holds for every
connected, linear algebraic group scheme $G$ over $K$.
\end{thm}

\mni
If $H^1(\G(K^s/K),\PGL{n,K}(K^s))$ equals $\{*\}$ for every
$n$, then $\text{Br}'(K)$ equals $\{[K]\}$ by Theorem
~\ref{thm-BrBrprime}.  Using Proposition ~\ref{prop-mu} and similar
arguments, this implies $\text{cd}(K)\leq 1$, cf. also Theorem
\ref{thm-Galois}.  
For a field of cohomological dimension $\leq 1$,
$H^1(\G(K^s/K),T)$ equals $\{*\}$ for every multiplicative group $T$,
cf. \cite[Application, p. 162]{LocalFields}.  Thus Steinberg's theorem
reduces to the case that $G$ is connected and semisimple.  There is a
further reduction to the case that $G$ is simply connected and
\emph{quasi-split}, i.e., there exists a closed subgroup $B$ of $G$
such that $B\otimes_K K^s$ is a maximal solvable subgroup (i.e., a
\emph{Borel subgroup}) of $G\otimes_K K^s$.  
Steinberg
proves that for every field $K$, for every 
simply connected, semisimple group $G$ with a Borel
subgroup defined over $K$, every
$G$-torsor arises as $i_*E$ where $i:T\rightarrow G$ is the inclusion
of a multiplicative subgroup and where $E$ is a $T$-torsor.  
When $K$ has
cohomological dimension $\leq 1$, $E$ is a trivial $T$-torsor, and
thus also $i_* E$ is a trivial $G$-torsor.      

\begin{conj}[Serre's ``Conjecture II''] \cite[\S 3]{GalCoh} 
\label{conj-S2}
\marpar{conj-S2}
If $K$ has cohomological dimension $2$ and if $K$ is perfect, 
then for every connected,
simply connected, semisimple group scheme $\widetilde{G}$ over $K$,
$H^1(\G(K^s/K),\widetilde{G}(K^s))$ is $\{*\}$.  Equivalently, for
every connected, semisimple group scheme $G$ over $K$, the connecting
map
$$
\Delta:H^1(\G(K^s/K),G(K^s)) \rightarrow H^2_{\text{fppf}}(K;\pi_1 G)
$$
is injective.  
\end{conj}

\begin{rmk} \label{rmk-S}
\marpar{rmk-S}
In \cite{SerrePP}, Serre explains that the hypothesis that $K$ is
perfect is too strong in these conjectures.  They should also hold if
the perfect hypothesis is replaced by the hypothesis that $[K:K^p]
\leq p^2$ and $H^3_p(K')$ is $0$ for all finite, separable extensions
$K'$ of $K$.  In particular, this new hypothesis holds when $K$ is a
function field of a surface over an algebraically closed field $k$,
i.e., $K$ is a finitely generated $k$-extension and
$\text{tr.deg.}(K/k)$ equals $2$.  
\end{rmk}

\mni
This conjecture remains open in general, although much is known,
cf. \cite[\S 3]{GalCoh}, \cite{SerrePP}, \cite{CTGP}.  At least when
$K$ is perfect, a converse theorem is known, cf. Theorem
~\ref{thm-Galois}(ii).  
We will return
to this conjecture in Chapter \ref{pt-S2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Part: The Chevalley-Warning and Tsen-Lang theorems
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{The Chevalley-Warning and Tsen-Lang theorems} 
\label{pt-TL}
\marpar{pt-TL}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: The Chevalley-Warning Theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Chevalley-Warning Theorem} 
\label{sec-CW}
\marpar{sec-CW}

\mni
A motivating problem in both arithmetic and geometry is the
following.

\begin{problem} \label{prob-vague}
\marpar{prob-vague}
Given a field $K$ and a $K$-variety $X$ find
sufficient, resp. necessary, conditions for existence of a $K$-point
of $X$.
\end{problem}

\mni
The problem depends dramatically
on the type of $K$: number
field, finite field, $p$-adic field, function field over a finite field,
or function field over an algebraically closed field.  
In arithmetic the number field case is most exciting.
However the geometric case, i.e., the case of a function field over
an algebraically closed field, is typically easier and may suggest
approaches and conjectures in the arithmetic case.  

\mni
Two results, the Chevalley-Warning theorem and Tsen's
theorem, deduce 
a sufficient condition for existence of $K$-points by
``counting''.  More generally, counting
leads to a relative result:
the Tsen-Lang theorem that a strong property about existence of
$k$-points for a field $k$ propagates to a weaker property
about $K$-points for certain field extensions $K/k$.
The prototype result, both historically and logically, 
is a theorem of Chevalley and its generalization
by Warning.  The counting result at the heart of the proof is
Lagrange's theorem together with the observation that a
nonzero single-variable polynomial of degree $\leq q-1$ cannot have
$q$ distinct zeroes.

\begin{lem} \label{lem-Ch}
\marpar{lem-Ch}
For a finite field $K$ with $q$ elements, the polynomial
$1-x^{q-1}$ vanishes on $K^*$ and $x^q-x$ vanishes on all of $K$.
For every integer $n\geq 0$, for
the $K$-algebra homomorphism
$$
\text{ev}_n:K[X_0,\dots,X_n] \rightarrow
\text{Hom}_{\text{Sets}}(K^{n+1},K), 
$$
$$ 
\text{ev}_n(p(X_0,\dots,X_n)) =
((a_0,\dots,a_n) \mapsto p(a_0,\dots,a_n)),
$$
the kernel equals the ideal
$$
I_n = \langle X_0^q - X_0,\dots, X_n^q-X_n \rangle.
$$
Finally, the collection $(X_i^q - X_i)_{i=0,\dots,n}$ is a Gr\"obner
basis 
with respect to every monomial order refining the grading of monomials
by total order.
In particular, for every $p$ in $I_n$ some term of $p$ of highest
degree is in the
ideal $\langle X_0^q,\dots,X_n^q \rangle$.
\end{lem}

\begin{proof}
Because $K^*$ is a group of order $q-1$, Lagrange's theorem implies
$a^{q-1}=1$ for every element $a$ of $K^*$, i.e., $1-x^{q-1}$ vanishes
on $K^*$.  Multiplying by $x$ shows that $x^q-x$ vanishes on $K$.
Thus the ideal $I_n$ is at least contained in the kernel of $\text{ev}_n$.

\mni
Modulo
$X_n^q-X_n$, every element of $K[X_0,\dots,X_n]$ 
is congruent to one of the form
$$
p(X_0,\dots,X_n) = p_{q-1}X_n^{q-1} + \dots  + p_0 X_n^0, \ \
p_0,\dots,p_{q-1} \in K[X_0,\dots,X_{n-1}].
$$
(Of course $K^n$ is defined to be $\{0\}$ and 
$K[X_0,\dots,X_{n-1}]$ is defined to be $K$ if $n$ equals $0$.)
Since $K$ has $q$ elements and since a nonzero polynomial of degree
$\leq q-1$ can have at most $q-1$ distinct zeroes, for every
$(a_0,\dots,a_{n-1})\in K^n$ the polynomial $p(a_0,\dots,a_{n-1},X_n)$
is zero on $K$ if and only if 
$$
p_0(a_0,\dots,a_{n-1})=\dots=p_{q-1}(a_0,\dots,a_{n-1}) = 0.
$$
Thus $\text{ev}_n(p)$ equals $0$ if and only if
each $\text{ev}_{n-1}(p_i)$ equals $0$.
In that case, by the induction hypothesis, 
each $p_i$ is in $I_{n-1}$ (in case $n=0$, each
$p_i$ equals $0$).  Then, since $I_{n-1}K[X_0,\dots,X_n]$ is
in $I_n$, $p$ is in $I_n$.
Therefore, by induction on $n$, the kernel of $\text{ev}_n$ is
precisely $I_n$.  

\mni
Finally, Buchberger's algorithm applied to the set
$(X_0^q-X_0,\dots,X_n^q-X_n)$ produces $S$-polynomials
$$
S_{i,j} = X_j^q(X_i^q-X_i) - X_i^q(X_j^q-X_j) = X_j(X_i^q-X_i) -
X_i(X_j^q-X_j)
$$
which have remainder $0$.
Therefore this set is a Gr\"obner basis by Buchberger's criterion.
\end{proof}


\begin{thm}\cite{Chevalley35},\cite{Warning} \label{thm-Ch}
\marpar{thm-Ch}
Let $K$ be a finite field.  Let $n$ and $r$ be positive integers.  Let
$F_1,\dots,F_r$ be nonconstant, homogeneous polynomials in
$K[X_0,\dots,X_n]$.  If
$$
\text{deg}(F_1) + \dots + \text{deg}(F_r) \leq n
$$
then there exists $(a_0,\dots,a_n) \in K^{n+1} - \{0\}$ such that for
every $i=1,\dots,r$, $F_i(a_0,\dots,a_n)$ equals $0$.
Stated differently, the projective scheme
$\mathbb{V}(F_1,\dots,F_r) \subset \PP^n_K$ 
has a $K$-point. 
\end{thm}

\begin{proof}
Denote by $q$ the number of elements in $K$.  
The polynomial
$$
G(X_0,\dots,X_n) = 1 - \prod_{i=0}^n(1-X_i^{q-1})
$$
equals $0$ on $\{0\}$ and equals $1$ on $K^{n+1}-\{0\}$.
For the same
reason, the polynomial
$$
H(X_0,\dots,X_n) = 1 - \prod_{j=1}^r(1-F_j(X_0,\dots,X_n)^{q-1})
$$
equals $0$ on 
$$
\{(a_0,\dots,a_n)\in K^{n+1} | F_1(a_0,\dots,a_n) = \dots =
F_r(a_0,\dots,a_n) = 0 \}
$$
and equals $1$ on the complement of this set in $K^{n+1}$.
Since each $F_i$ is
homogeneous, $0$ is a common zero of $F_1,\dots,F_r$.  Thus the
difference $G-H$ equals $1$ on
$$
\{(a_0,\dots,a_n) \in K^{n+1} -\{0\} | F_1(a_0,\dots,a_n) = \dots =
F_r(a_0,\dots,a_n) = 0\}
$$
and equals $0$ on the complement of this set in $K^{n+1}$.  Thus, to
prove that $F_1,\dots,F_r$ have a nontrivial common zero, it suffices
to prove the polynomial $G-H$ does not lie in the ideal $I_n$.

\mni
Since
$$
\text{deg}(F_1) + \dots + \text{deg}(F_r) \leq n,
$$
$H$ has strictly smaller degree than $G$.  Thus
the leading term of $G-H$ equals the leading term of $G$.  There is
only one term of $G$ of degree $\text{deg}(G)$.  Thus, for every
monomial ordering refining the grading by total degree, the leading
term of $G$ equals
$$
(-1)^{n+1}X_0^{q-1}X_1^{q-1}\cdots X_n^{q-1}.
$$
This is clearly divisible by none of $X_i^q$ for $i=0,\dots,n$, i.e.,
the leading term of $G-H$
is not in the ideal $\langle X_0^q,\dots,X_n^q \rangle$.  
Because $(X_0^q-X_0,\dots,X_n^q-X_n)$ is a Gr\"obner basis for $I_n$
with respect to the monomial order, $G-H$ is not in $I_n$.
\end{proof}

%% \begin{exer} \label{exer-Ch}
%% \marpar{exer-Ch}
%% Of course the previous exercise proves Chevalley's theorem, but it
%% might be useful to see a slightly different perspective.  This
%% exercise solves the case $n=d=2$ using the method of incidence
%% correspondences.  

%% \mni
%% Let $q$ denote the number of elements of $K$. 
%% Observe that the number of $K$-rational points of the incidence
%% correspondence 
%% $$
%% I := \{([X,Y,Z],[A,B,C,D,E,F]) \in \PP^2_K\times \PP^5_K | AX^2 + BXY + CXZ
%% + DY^2 + EYZ + FZ^2 = 0\}
%% $$
%% equals
%% $$
%% |I(K)| = \sum_{p\in \PP^2_K(K)} |\pi_1^{-1}(p)| = \sum_{p\in
%%   \PP^2_K(K)}|\PP^4_K(K)| = |\PP^2_K(K)|\cdot |\PP^4_K(K)| =
%% \frac{q^3-1}{q-1} \times \frac{q^5-1}{q-1}.
%% $$
%% On the other hand, considering the projection $\pi_2:I\rightarrow
%% \PP^5_K$ and counting points on singular conics, $|I(K)|$ also equals 
%% $$
%% M_{\text{double}}\cdot N_{\text{double}} +
%% M_{\text{reducible}}\cdot N_{\text{reducible}} +
%% M_{\text{geom. reducible}}N_{\text{geom. reducible}} +
%% M_{\text{smooth, split}} N_{\text{smooth, split}}
%% $$
%% where
%% $$
%% M_{\text{double}} = \#\{[F]\in \PP^5_K(K)| \mathbb{V}(F)\text{ is a
%%   double line}\},
%% $$
%% $$
%% N_{\text{double}} = \#\{K-\text{points on a double line}\} =
%% |\PP^1_K(K)| = \frac{q^2-1}{q-1},
%% $$
%% $$
%% M_{\text{reducible}} = \#\{[F] \in \PP^5_K(K)| \mathbb{V}(F)\text{ is
%%   reducible}\}, 
%% $$
%% $$
%% N_{\text{reducible}} = \#\{K-\text{points on a reducible conic}\} =
%% 2\frac{q^2-1}{q-1} - 1, 
%% $$
%% $$
%% M_{\text{geom. reducible}} = \#\{[F]\in \PP^5_K(K)|
%% \mathbb{V}(F)\text{ is irreducible and geometrically reducible}\},
%% $$
%% $$
%% N_{\text{geom. reducible}} = \#\{K-\text{points on an irreducible,
%%   geometrically reducible plane conic}\} = 1,
%% $$
%% $$
%% M_{\text{smooth, split}} = \#\{[F]\in \PP^5_K(K)| \mathbb{V}(F)\text{
%%   is smooth and has a }K-\text{point}\},
%% $$
%% and
%% $$
%% N_{\text{smooth, split}} = \#\{K-\text{points on a smooth conic
%%   with a }K-\text{point}\} = \frac{q^2-1}{q-1}.
%% $$
%% Compute $M_{\text{double}}$, $M_{\text{reducible}}$, and
%% $M_{\text{geom. reducible}}$ and use this to solve for
%% $N_{\text{smooth, split}}$.  
%% Deduce that
%% every smooth conic has a $K$-point.
%% \end{exer}

%% \begin{exer} \label{exer-Ch2}
%% \marpar{exer-Ch2}
%% A \emph{normic form} is a nonconstant, homogeneous, degree $d$
%% polynomial in $K[X_0,\dots,X_{d-1}]$ having only the trivial
%% $K$-solution, cf. \cite[p. 230, Definition IV.6.4.2]{K}.  
%% Let $K$ be a field which is not algebraically closed.  Prove there
%% exists an integer $d\geq 2$ and a degree $d$ normic form over $K$.

%% \mni
%% If $K$ is a finite field prove $K$ has a normic form of \emph{every}
%% degree $d\geq 2$.  In this sense Chevalley's theorem is sharp: the
%% inequality is the weakest for which the theorem holds.  
%% (Also prove that $K$ has a normic form of every degree $d\geq 2$ if
%% $K$ is a number field, resp. $p$-adic field, function field over a
%% finite field, and function field over an algebraically closed field.)

%% \mni
%% \textbf{Hint.}  Given a finite field extension $L/K$ of degree $d$,
%% view $L$ as a
%% $K$-vector space and (via multiplication in $L$) also 
%% as a subspace of the space of $K$-linear maps of $L$,
%% $$
%% L\subset \text{Hom}_{K-\text{linear}}(L,L).
%% $$
%% What are the ``characteristic'' polynomial maps $L\rightarrow K$?  Is
%% there one of degree $d$?  What would it mean if this map had a
%% nontrivial zero?  
%% \end{exer}

%% \begin{exer} \label{exer-Ch2.5}
%% \marpar{exer-Ch2.5}
%% Does a finite field $K$ have a \emph{smooth} normic form?  In other words,
%% does there exist a normic form $F$ such that the associated projective
%% scheme
%% $$
%% \mathbb{V}(F) = \text{Proj} \ K[X_0,\dots,X_n]/\langle F \rangle
%% $$ 
%% is smooth over $\SP
%% k$?  Does it have a smooth normic form of every degree, resp. of
%% infinitely many different degrees?  You might want to start by
%% considering Fermat hypersurfaces, 
%% $$
%% F = X_0^d + \dots + X_{d-1}^d.
%% $$
%% \end{exer}

%% \begin{exer} \label{exer-Ch3}
%% \marpar{exer-Ch3}
%% Define a \emph{normic sequence} to be a sequence of nonconstant,
%% homogeneous polynomials
%% $$
%% F_1,\dots,F_r \in K[X_0,\dots,X_n]
%% $$
%% of respective degrees $d_1,\dots,d_r$ satisfying $d_1 + \dots + d_r = n+1$
%% and 
%% whose only common solution $F_1(p)= \dots = F_r(p)=0$
%% is $p=(0,\dots,0)$.  

%% \mni
%% Let $K$ be a finite field.
%% For every sequence of
%% positive integers $(d_1,\dots,d_r,n)$ with $d_1 + \dots + d_r = n+1$,
%% does there exist a normic sequence of that degree?  What if $K$ is a
%% function field (of transcendence degree $\geq 1$)
%% over an algebraically closed field?
%% \end{exer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: The Tsen-Lang Theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Tsen-Lang Theorem} 
\label{sec-TL}
\marpar{sec-TL}

\mni
On the geometric side, an analogue of Chevalley's theorem was proved
by Tsen, cf. \cite{Tsen}.
This was later generalized independently by Tsen and Lang,
cf. \cite{Tsen36}, \cite{Lang52}.
Lang introduced a definition which simplifies the argument.

\begin{defn} \cite{Lang52} \label{defn-Cm}
\marpar{defn-Cm}
Let $m$ be a nonnegative integer.  A field $K$ is called $C_m$, or
said to have \emph{property $C_m$}, if it satisfies the following.
For every positive integer $n$ and every sequence of positive
integers $(d_1,\dots,d_r)$ satisfying
$$
d_1^m + \dots + d_r^m \leq n,
$$
every sequence $(F_1,\dots,F_r)$ of
homogeneous polynomials $F_i \in K[X_0,\dots,X_n]$ with
$\text{deg}(F_i)= d_i$ has a common zero in $K^{n+1}-\{0\}$.  
\end{defn}

\begin{rmk} \label{rmk-Cm}
In fact the definition in \cite{Lang52} is a little bit different than
this.  For fields having normic forms, Lang proves the definition
above is equivalent to his definition.  And the definition above works
best with the following results.
\end{rmk}

\mni
With this definition, the statement of the Chevalley-Warning theorem
is quite simple: every finite field has property $C_1$.  The next
result proves that property $C_m$ is preserved by algebraic extension.

\begin{lem} \label{lem-TsenLang}
\marpar{lem-TsenLang}
For every nonnegative integer $m$, every algebraic extension of a
field with property $C_m$ has property $C_m$.
\end{lem}

\begin{proof}
Let $K$ be a field with property $C_m$ and let $L'/K$ be an algebraic
extension.  For every sequence of polynomials $(F_1,\dots,F_r)$ as in
the definition, the coefficients generate a finitely generated
subextension $L/K$ of $L'/K$.  Thus clearly it suffices to prove the
lemma for finitely generated, algebraic extensions $L/K$.

\mni
Denote by $e$ the finite dimension $\text{dim}_K(L)$.
Because multiplication on $L$ is $K$-bilinear, each homogeneous,
degree $d_i$, polynomial map of $L$-vector spaces,
$$
F_i:L^{\oplus (n+1)} \rightarrow L,
$$
is also a homogeneous, degree $d_i$, polynomial map of $K$-vector spaces.
Choosing a $K$-basis for $L$ and decomposing $F_i$ accordingly, $F_i$
is equivalent to $e$ distinct homogeneous, degree $d_i$, polynomial
maps of $K$-vector spaces,
$$
F_{i,j}:L^{\oplus (n+1)} \rightarrow K, \ \ j=1,\dots,e.
$$
The set of common zeroes of the collection of 
homogeneous polynomial maps $(F_i|i=1,\dots,r)$ equals the
set of common zeroes of the collection of homogeneous polynomial
functions $(F_{i,j}| i =1,\dots, r, j=1,\dots,e)$.   
Thus it suffices to prove there is a nontrivial common zero of all the
functions $F_{i,j}$.

\mni
By hypothesis,
$$
\sum_{i=1}^r \text{deg}(F_i)^m \text{ is no greater than } n.
$$
Thus, also
$$
\sum_{i=1}^r \sum_{j=1}^e \text{deg}(F_{i,j})^m = e \sum_{i=1}^r
\text{deg}(F_i)^m \text{ is no greater than } en.
$$
Since $K$ has property $C_m$ and since
$$
\text{dim}_K(L^{\oplus (n+1)}), \text{ i.e., }  (n+1)\text{dim}_K(L) =
e(n+1),
$$
is larger than $en$, the collection of homogeneous polynomials
$F_{i,j}$ has a common zero in $L^{\oplus (n+1)}-\{0\}$.  
\end{proof}

\mni
The heart of the Tsen-Lang theorem is the following proposition.  

\begin{prop} \label{prop-TsenLang}
\marpar{prop-TsenLang}
Let $K/k$ be a function field
of a curve, i.e., a
finitely
generated, separable field extension of transcendence degree $1$.  
If $k$ has property $C_m$ then $K$ has property $C_{m+1}$.
\end{prop}

\mni
This is proved in a series of steps.
Let $n$, $r$ and $d_1,\dots,d_r$ be positive integers such that
$$
d_1^{m+1} + \dots + d_r^{m+1} \leq n.
$$
For every collection of homogeneous polynomials
$$
F_1,\dots,F_r \in K[X_0,\dots,X_n], \ \ \text{deg}(F_i) = d_i,
$$
the goal is to prove that the collection of homogeneous, degree $d_i$, 
polynomial
maps of $K$-vector spaces
$$
F_1,\dots,F_r: K^{\oplus(n+1)} \rightarrow K
$$
has a common zero.  Of course, as in the proof of Lemma
~\ref{lem-TsenLang}, 
this 
is also a collection of homogeneous polynomial maps of $k$-vector spaces.
Unfortunately both of these $k$-vector spaces are 
infinite dimensional.  However, using geometry, these polynomial maps
can be realized as the colimits of polynomial
maps of finite dimensional $k$-vector spaces.  For these maps there is
an analogue of the Chevalley-Warning argument replacing the 
counting argument by a \emph{parameter counting argument} which
ultimately follows from the Riemann-Roch theorem for curves. 
The first
step is to give a \emph{projective model} of $K/k$.

\begin{lem} \label{lem-projmodel1}
\marpar{lem-projmodel1}
For every separable, finitely generated field extension $K/k$ of
transcendence degree $1$, there exists a smooth, projective, connected
curve $C$ over $k$ and an isomorphism of $k$-extensions $K\cong
k(C)$.  Moreover the pair $(C,K\cong k(C))$ is unique up to unique
isomorphism. 
\end{lem}

\begin{proof}
This is essentially the \emph{Zariski-Riemann surface} of the
extension $K/k$.  For a proof in the case that $k$ is algebraically
closed, see \cite[Theorem I.6.9]{H}.  The proof in the general case is
similar.  
\end{proof}

\mni
The isomorphism $K \cong k(C)$ is useful because the infinite
dimensional
$k$-vector space
$k(C)$ has a plethora of naturally-defined finite dimensional
subspaces.  For every Cartier divisor $D$ on $C$, denote by $V_D$ the
subspace
$$
V_D := H^0(C,\OO_C(D)) = \{ f\in k(C) | \text{div}(f) + D \geq 0\}.
$$
The collection of all Cartier divisors $D$ on $C$ is a partially
ordered set where
$$
D' \geq D \text{ if and only if } D'-D \text{ is effective.}
$$
The system of subspaces $V_D$ of $k(C)$ is compatible for this partial
order, i.e., if $D'\geq D$ then $V_{D'} \supset V_D$.  
And $K$ is the union of all the subspaces $V_D$, i.e., it is the
colimit of this compatible system of finite dimensional $k$-vector spaces.
Thus for all
$k$-multilinear algebra operations which commute with colimits, the
operation on $k(C)$ can be understood in terms of its restrictions to
the finite dimensional subspaces $k(C)$.  The next lemma makes this
more concrete for the polynomial map $F$.

\begin{lem} \label{lem-projmodel2}
\marpar{lem-projmodel2}
Let $C$ be a smooth, projective, connected curve over a field $k$ and
let 
$$
F_i \in k(C)[X_0,\dots,X_n]_{d_i}, \ \ i=1,\dots,r
$$
be a collection of polynomials in the spaces
$k(C)[X_0,\dots,X_n]_{d_i}$ 
of homogeneous, degree $d_i$ polynomials.
There exists an effective, Cartier divisor $P$ on $C$ and for every
$i=1,\dots,r$ there exists a global
section $F_{C,i}$ of the coherent sheaf
$\OO_C(P)[X_0,\dots,X_n]_{d_i}$
such that for every $i=1,\dots,r$ 
the germ of $F_{C,i}$ at the generic point of $C$ equals $F_i$.
\end{lem}  

\begin{rmk} \label{rmk-projmodel2}
\marpar{rmk-projmodel2}
In particular, for every Cartier divisor $D$ on $C$ and for every
$i=1,\dots,r$ 
there is a
homogeneous, degree $d$, polynomial map of $k$-vector spaces
$$
F_{C,D,i}:V_D^{\oplus (n+1)} \rightarrow W_{d_i,P,D}, \ \ W_{d_i,P,D} :=
V_{d_iD+P},
$$
such that for every $i=1,\dots,r$
the restriction of $F_i$ to $V_D^{\oplus(n+1)}$ equals
$F_{C,D,i}$ considered as a map with target $K$ (rather than the
subspace $V_{dD+P}$).
\end{rmk}

\begin{proof}
The coefficients of each $F_i$ are rational
functions on $C$.  Each such function has a polar divisor.  Since
there are only finitely many coefficients of the finitely many
polynomials $F_1,\dots,F_r$, there exists a single
effective, Cartier divisor $P$ on $C$ such that every coefficient 
is a global section of $\OO_C(P)$. 
\end{proof}

\mni
Because of Lemma ~\ref{lem-projmodel2}, the original polynomial maps
$F_1,\dots,F_r$ can be understood in terms of their 
restrictions to the subspaces
$V_D$.  The dimensions of these subspaces are determined by the
Riemann-Roch theorem.

\begin{thm} [Riemann-Roch for smooth, projective curves] \label{thm-RR}
\marpar{thm-RR}
Let $k$ be a field.  
Let $C$ be a smooth, projective, connected curve over $k$.  
Denote by $\omega_{C/k}$ the sheaf of
relative differentials of $C$ over $k$ and denote by $g(C) =
\text{genus}(C)$ 
the unique
integer such that $\text{deg}(\omega_{C/k}) = 2g(C) - 2$.  
For every invertible sheaf $\mc{L}$ on $C$, 
$$
h^0(C,\mc{L}) - h^0(C,\omega_C\otimes_{\OO_C} \mc{L}^\vee) =
\text{deg}(\mc{L}) + 1 - g(C).
$$
\end{thm}

\begin{rmk} \label{rmk-RR}
In particular, if
$\text{deg}(\mc{L}) > \text{deg}(\omega_C) = 2g(C)-2$
so that $\omega_C\otimes_{\OO_C}\mc{L}^\vee$ has negative degree, then
$h^0(C,\omega_C\otimes_{\OO_C}\mc{L}^\vee)$ equals zero.  And then
$$
h^0(C,\mc{L}) = \text{deg}(\mc{L}) + 1 - g(C).
$$
\end{rmk}

\mni
For a Cartier divisor $D$ satisfying 
$$
\text{deg}(D) > 2g(C) - 2 \text{  and for each  } i = 1,\dots,r,
d_i\text{deg}(D) + \text{deg}(P) > 2g(C)-2,
$$ 
the Riemann-Roch theorem
gives that $V_D^{\oplus(n+1)}$ and $W_{d_i,P,D}$ are finite dimensional
$k$-vector spaces of respective dimensions,
$$
\text{dim}_k(V_D^{\oplus (n+1)}) = (n+1)h^0(C,\OO_C(D)) = 
(n+1)(\text{deg}(D) + 1-g)
$$
and
$$
\text{dim}_k(W_{d_i,P,D}) = \text{dim}(V_{d_iD+P}) =
d_i\text{deg}(D) + \text{deg}(P) + 1 - g.
$$
In this case, choosing a basis for $W_{d_i,P,D}$ and decomposing 
$$
F_{C,D,i}:V_D^{\oplus(n+1)} \rightarrow W_{d_i,P,D}
$$
into its associated components, there exist $\text{dim}_k(W_{d_i,P,D})$
homogeneous, degree $d$, polynomial functions 
$$
(F_{C,D,i})_j:V_D^{\oplus (n+1)} \rightarrow k, \ \ j=1,\dots,
\text{dim}_k(W_{d_i,P,D})
$$
such that a zero of $F_{C,D,i}$ is precisely the same as a
common zero of all the functions $(F_{C,D,i})_j$.  

\begin{proof}[Proof of Proposition ~\ref{prop-TsenLang}]
By hypothesis, each $d_i$ and $n+1 - \sum_{i=1}^r d_i^{m+1}$ are nonzero
so that the fractions
$$
\frac{2g(C)-2 - \text{deg}(P)}{d_i} \text{ for each } i=1,\dots,r, \ 
\frac{(n+1 - \sum_{i=1}^r d_i^{m})(g-1) +
  \sum_{i=1}^rd_i^{m}\text{deg}(P)}{n+1-\sum_{i=1}^r d_i^{m+1}}
$$
are all defined.  Let $D$ be an effective, Cartier divisor on $C$ such
that
$$
\text{deg}(D) > 2g(C)-2, \ \text{deg}(D) > \frac{2g(C)-2 -
  \text{deg}(P)}{d_i}, i=1,\dots,r, \text{ and}
$$
$$
\text{deg}(D) > \frac{(n+1 - \sum_{i=1}^r d_i^{m})(g-1) +
  \sum_{i=1}^rd_i^{m}\text{deg}(P)}{n+1-\sum_{i=1}^r d_i^{m+1}}.
$$

\mni
Because $\text{deg}(D) > 2g(C)-2$, the Riemann-Roch theorem states
that
$$
\text{dim}_k(V_D^{\oplus(n+1)}) = (n+1)\text{dim}_k(V_D) =
(n+1)(\text{deg}(D)+1-g).
$$
For every $i=1,\dots,r$, 
because $d_i$ is positive and because $\text{deg}(D) >
(2g(C)-2-\text{deg}(P))/d_i$, also
$$
\text{deg}(d_iD+P) = d_i\text{deg}(D) + \text{deg}(P) 
\text{ is greater than } 2g(C)-2.
$$
Thus the Riemann-Roch theorem states that
$$
\text{dim}_k(W_{d_i,P,D}) = \text{dim}_k(V_{d_iD+P}) = d_i\text{deg}(D) +
\text{deg}(P) + 1-g(C).
$$
Thus for the collection of polynomial functions $(F_{C,D,i})_j$,
$$
\text{dim}_k(V_D^{\oplus (n+1)}) - \sum_{i=1}^r \sum_j
\text{deg}((F_{C,D,i})_j)^{m}
$$
equals
$$
(n+1)(\text{deg}(D)+1-g) - \sum_{i=1}^r (d_i\text{deg}(D) +
\text{deg}(P)+1-g(C)) d_i^{m} =
$$
$$
(n+1-\sum_{i=1}^r d_i^{m+1}) \text{deg}(D) - [(n+1-\sum_{i=1}^r
d_i^{m})(g-1) + \sum_{i=1}^r d_i^{m}\text{deg}(P)].
$$
Because
$$
\text{deg}(D) > \frac{(n+1 - \sum_{i=1}^r d_i^{m})(g-1) +
  \sum_{i=1}^rd_i^{m}\text{deg}(P)}{n+1-\sum_{i=1}^r d_i^{m+1}}
$$
and because $n+1 - \sum_{i=1}^r d_i^{m+1}$ is positive, also
$$
(n+1-\sum_{i=1}^r d_i^{m+1}) \text{deg}(D) > [(n+1-\sum_{i=1}^r
d_i^{m})(g-1) + \sum_{i=1}^r d_i^{m}\text{deg}(P)].
$$
Therefore
$$
\text{dim}_k(V_D^{\oplus(n+1)}) \text{ is greater than }
\sum_{i=1}^r \sum_j
\text{deg}((F_{i,C,D})_j)^{m}.
$$

\mni
Because of the inequality above, and because $k$ has property $C_m$,
there is a nontrivial
common zero of the collection of
homogeneous polynomial functions 
$(F_{C,D,i})_j$, $i=1,\dots,r$, $j=1,\dots,
\text{dim}_k(W_{d_i,P_i,D})$.  
Therefore there is a nontrivial common zero of the collection of
homogeneous polynomial maps $F_{C,D,i}$, $i=1,\dots,r$.  By Lemma
~\ref{lem-projmodel2}, 
the image
of this nonzero element in $K^{\oplus (n+1)}$ is a nonzero element
which is a common zero of the polynomials $F_1,\dots,F_r$.  
\end{proof}

\mni
Proposition ~\ref{prop-TsenLang} is the main step in the proof of the
Tsen-Lang theorem.

\begin{thm} [The Tsen-Lang Theorem] \cite{Lang52} \label{thm-TsenLang}
\marpar{thm-TsenLang}
Let $K/k$ be a field extension with finite transcendence degree,
$\text{tr.deg.}(K/k) = t$.  If $k$ has property $C_m$ then $K$ has
property $C_{m+t}$.  
\end{thm}

\begin{proof}
The proof of the theorem is by induction on $t$.
When $t=0$, i.e., when $K/k$ is algebraic, the result follows from
Lemma ~\ref{lem-TsenLang}.  Thus assume $t>0$ and the result is known
for $t-1$.  Let $(b_1,\dots,b_t)$ be a transcendence
basis for $K/k$.  Let $E_t$, resp. $E_{t-1}$, 
denote the subfield of $K$ generated by
$k$ and $b_1,\dots,b_t$, resp. generated by $k$ and
$b_1,\dots,b_{t-1}$.  Since $E_{t-1}/k$ has transcendence degree
$t-1$, by the induction hypothesis $E_{t-1}$ has property
$C_{m+t-1}$.  
Now $E_t/E_{t-1}$ is a purely
transcendental extension of transcendence degree $1$.  In particular,
it is finitely generated and separable.  Since $E_{t-1}$ has property
$C_{m+t-1}$, by Proposition
~\ref{prop-TsenLang} $E_t$ has property $C_{m+t}$.
Finally by Lemma ~\ref{lem-TsenLang} again, since $K/E_t$ is
algebraic and $E_t$ has property $C_{m+t}$, also $K$ has property
$C_{m+t}$.
\end{proof}

\mni
The homogeneous version of the Nullstellensatz implies a field $k$ has
property $C_0$ if and only if $k$ is algebraically closed.  Thus one
corollary of Theorem ~\ref{thm-TsenLang} is the following.

\begin{cor} \label{cor-TsenLang}
\marpar{cor-TsenLang}
Let $k$ be an algebraically closed field and let $K/k$ be a field
extension of finite transcendence degree $t$.  The field $K$ has
property $C_t$.
\end{cor}

\mni
In particular, the case $t=1$ is historically the first result in
this direction.   

\begin{cor}[Tsen's theorem] \cite{Tsen36} \label{cor-Tsen}
\marpar{cor-Tsen}
The function field of a curve over an algebraically closed field has
property $C_1$.
\end{cor}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Applications to Brauer groups
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Applications to Brauer groups} 
\label{sec-Br}
\marpar{sec-Br}

\mni
Chevalley and Tsen recognized that property $C_1$, which they called
\emph{quasi-algebraic closure}, has an important consequence for
division algebras.
Lang recognized that property $C_2$ also has an important consequence
for division algebras, cf. \cite[Theorem 13]{Lang52}.

\mni
Let $A$ be a central simple algebra over $K$ with $\text{dim}_K(A) =
n^2$, and let $E$ be the corresponding $\PGL{n,K}$-torsor.
Because the determinant polynomial $\text{det}$ on
$\Mat{n\times n, K}$ is invariant for the inner action of
$\PGL{n,K}$, $\text{det}$ determines a well-defined
polynomial map on $A$.

\begin{defn} \label{defn-Nrm}
\marpar{defn-Nrm}
For a central simple algebra $A$ over $K$ with $\text{dim}_K(A) =
n^2$, the \emph{reduced norm} is the unique, degree $n$ polynomial map
$$
\text{Nrm}_{A/k}:A\rightarrow K
$$
such that the induced degree $n$ polynomial map
$$
\text{Nrm}_{A/K}\otimes \text{Id}:A\otimes_K K^s \rightarrow K^s
$$
agrees with the determinant map
$$
\text{det}:\Mat{n\times n,K^s} \rightarrow K^s
$$
via one, and hence every, $K^s$-algebra isomorphism of $A$ and
$\Mat{n\times n,K^s}$.  
\end{defn}

\mni
Since $\text{det}$ is multiplicative, so is $\text{Nrm}_{A/K}$, i.e., 
$$
\forall a,b\in A, \ \ 
\text{Nrm}_{A/K}(ab) = \text{Nrm}_{A/K}(a)\text{Nrm}_{A/K}(b).
$$
And the restriction to the center $K$ is the polynomial map $\lambda
\mapsto \lambda^n$.  These properties characterize the reduced norm.
By the same type of Galois invariance argument as above, and using
Cramer's rule, an element $a$ of $A$ has a (left and right) inverse if
and only if $\text{Nrm}_{A/K}(a)$ is nonzero.  In particular, if $D$
is a division algebra the only zero of $\text{Nrm}_{A/K}$ is $a=0$.

\begin{prop} \label{prop-division}
\marpar{prop-division}
Let $K$ be a field
\begin{enumerate}
\item[(i)]
If $K$ has property $C_1$, then the only division algebra with
center $K$ is $K$ itself.  Thus $\text{Br}(K)$ equals $\{[K]\}$.  
\item[(ii)]
If $K$ has property $C_2$ then for
every division algebra $D$ with center $K$ the reduced norm map
$$
\text{Nrm}_{D/K}:D \rightarrow K
$$
is surjective.
\end{enumerate}
\end{prop}

\begin{proof}
\textbf{(i).}
Let $D$ be a division algebra with center $K$.  Denote by $n$ the
index of $D$.
Because $\Mat{n\times n}(\overline{K})$ has dimension $n^2$ as
a $\overline{K}$-vector space, also $D$ has dimension $n^2$ as a
$K$-vector space.  If $K$ has property $C_1$, then since the
homogeneous polynomial map
$\text{Nrm}_{D/K}$ has only the trivial zero,
$$
n = \text{deg}(\text{Nrm}_{D/K}) \geq \text{dim}_K(D) = n^2,
$$
i.e., $n=1$.  Thus for a field $K$ with property $C_1$, the only
finite dimensional, division algebra with center $K$ has dimension
$1$, i.e., $D$ equals $K$.

\mni
\textbf{(ii).}
Next suppose that $K$ has property $C_2$.  Clearly
$\text{Nrm}_{D/k}(0)$ equals $0$.  Thus to prove that
$$
\text{Nrm}_{D/K}:D \rightarrow K
$$
is surjective, it suffices to prove that for every nonzero $c\in K$ 
there exists $b$ in $D$ with $\text{Nrm}_{D/K}(b) = c$.
Consider the homogeneous,
degree $n$, polynomial map
$$
F_c:D\oplus K \rightarrow K, \ \ (a,\lambda) \mapsto
\text{Nrm}_{D/K}(a) - c\lambda^n.
$$
Since
$$
\text{dim}_K(D\oplus K) = n^2 + 1 > \text{deg}(F_c)^2,
$$
by property $C_2$ the map $F_c$ has a zero $(a,\lambda) \neq (0,0)$,
i.e., $\text{Nrm}_{D/k}(a) = c\lambda^n$.
In particular, $\lambda$ must be nonzero since otherwise $a$ is a
nonzero element of $D$ with $\text{Nrm}_{D/K}(a) = 0$.  But then $b=
(1/\lambda)a$ is an element of $D$ with $\text{Nrm}_{D/k}(b) = c$.
\end{proof}

\mni
It was later recognized, particularly through the work of Merkurjev and
Suslin, 
that these properties of division algebras are
equivalent to properties of Galois cohomology.  

\begin{thm}\cite[Proposition 5, \S I.3.1]{GalCoh}, \cite[Corollary
  24.9]{Suslin84}  
\label{thm-Galois}
\marpar{thm-Galois}
Let $K$ be a field.
\begin{enumerate}
\item[(i)]  
The cohomological dimension of $K$ is $\leq 1$ if and only
  if for every finite extension $L/K$, the only division algebra with
  center $L$ is $L$ itself.
\item[(ii)]
If $K$ is perfect, the cohomological dimension of $K$ is $\leq 2$ if
and only if for every finite extension $L/K$, for every division
algebra $D$ with center $L$, the reduced norm map $\text{Nrm}_{D/L}$
is surjective.
\end{enumerate}
\end{thm}

%% \begin{exer} \label{exer-Tsen1}
%% \marpar{exer-Tsen1}
%% Use the same approach as above to prove the following
%% generalized form of Theorem ~\ref{thm-Tsen}.  
%% Let $n$ be a positive integer and let $d_1,\dots,d_r$ be a sequence of
%% positive integers. 
%% Every sequence $(F_1,\dots,F_r)$ of homogeneous
%% polynomials $F_i \in K[X_0,\dots,X_n]$ with $\text{deg}(F_i) = d_i$
%% has a nontrivial common solution if
%% $$
%% d_1 + \dots + d_r \leq n.
%% $$
%% Stated differently, the projective variety
%% $\mathbb{V}(F_1,\dots,F_r)\in \PP^n_K$ has a $K$-point.
%% \end{exer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Part: Rationally connected fibrations over curves
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Rationally connected fibrations over curves} \label{pt-RC}
\marpar{pt-RC}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Rationally connected varieties
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rationally connected varieties} 
\label{sec-RC}
\marpar{sec-RC}

\mni
The theorems of Chevalley-Warning and Tsen-Lang are positive answers
to Problem ~\ref{prob-vague} for certain classes of fields.  It is
natural to ask whether these theorems can be generalized for such fields.

\begin{problem} \label{prob-vague2}
\marpar{prob-vague2}
Let $r$ be a nonnegative integer.  Give sufficient geometric
conditions on a variety such that for every $C_r$ field $K$ 
(or perhaps every $C_r$ field satisfying some additional hypotheses)
and for every $K$-variety satisfying the conditions, $X$ has a $K$-point.
\end{problem}

\mni
As with Problem ~\ref{prob-vague}, this problem is quite vague.
Nonetheless there are important partial answers.  One such 
answer is the following.

\begin{thm}\cite{CubicForms} \cite{CT87} \label{thm-CTM}
\marpar{thm-CTM}
Let $K$ be a $C_1$ field and let $X$ be a projective $K$-variety.  If
$X\otimes_K \overline{K}$ is birational to
$\PP^2_{\overline{K}}$ then $X$ has a $K$-point.
\end{thm}

\mni
This begs the question: What (if anything) is the common feature of
rational surfaces and of the varieties occurring in the
Chevalley-Warning and Tsen-Lang theorems, i.e., 
complete intersections in $\PP^n$ of
hypersurfaces of degrees $d_1,\dots,d_r$ with $d_1 + \dots + d_r \leq
n$?  One answer is \emph{rational connectedness}.  This is a property
that was studied by Koll\'ar-Miyaoka-Mori and Campana, cf. \cite{K}.
%% The theory of rational connectedness 
%% is quite well developed with a
%% number of important results.  To be concise we will only describe
%% those results which directly bear on Problem ~\ref{prob-vague2}.

\begin{defn} \label{defn-RC}
\marpar{defn-RC}
Let $k$ be an algebraically closed field.  An integral (thus
nonempty), 
separated,
finite type, $k$-scheme $X$ is \emph{rationally connected},
resp. \emph{separably rationally connected}, if there
exists an integral, finite type $k$-scheme $M$ and a morphism of
$k$-schemes
$$
u:M\times_k \PP^1_k \rightarrow X, \ \ (m,t) \mapsto u(m,t)
$$
such that the induced morphism of $k$-schemes
$$
u^{(2)}:M\times_k \PP^1_k \times_k \PP^1_k \rightarrow X\times_k X, \
\ (m,t_1,t_1) \mapsto (u(m,t_1),u(m,t_2))
$$
is surjective, resp. surjective and generically smooth.

\mni
In a similar way, $X$ is \emph{rationally chain connected},
resp. \emph{separably rationally chain connected}, if for some integer
$m\geq 1$, the analogous property holds after replacing $\PP^1_k$ by the
proper, connected,
nodal, reducible curve $C_m$ which is a chain of $m$ smooth rational
curves. 
\end{defn}

\mni
In other references, the condition on $u^{(2)}$ is that it be dominant
rather than surjective.  In the following arguments, it is preferrable to
demand that $u^{(2)}$ is surjective.  When $X$ is proper and
$\text{char}(k)=0$, the definitions turn out to be equivalent.

\mni
Figure ~\ref{fig-RC1a} shows a rationally connected variety, where
every pair of points is contained in an image $u(\PP^1)$ of the
projective line.

%FIGURE RC1a
\begin{figure}
\begin{picture}(330,270)
\put(10,10){\makebox(330,240){\includegraphics{RC1a_small.eps}}}
\end{picture}
\caption{Every pair of points in a rationally connected variety lie in
an image of the projective line}
\label{fig-RC1a}
\end{figure}

\

\mni
The definition of rational connectedness, resp. rational
chain connectedness, mentions a particular parameter space $M$. 
However, using
the general theory of Hilbert schemes, it suffices to check that every
pair $(x_1,x_2)$ of $K$-points of $X\otimes_k K$ is contained in some
rational $K$-curve, resp. a chain of rational $K$-curves, 
(not necessarily from a fixed parameter space) 
for one 
\emph{sufficiently large}, algebraically closed, field extension $K/k$,
i.e., for an algebraically closed extension $K/k$ such that for
every countable
collection of proper closed subvarieties $Y_i\subsetneq X$, there
exists a $K$-point of $X$ contained in none of the sets $Y_i$.  For
instance, $K/k$ is sufficiently large if $K$ is uncountable or if
$K/k$ contains the fraction field $k(X)/k$ as a subextension.

\mni
A very closely related property is the existence of a  
very free rational curve.  For a $d$-dimensional variety $X$, a
\emph{very free rational curve} is a morphism
$$
f:\PP^1_k \rightarrow X_{\text{smooth}}
$$
into the smooth locus of $X$ such that $f^* T_X$ is ample, i.e.,
$$
f^* T_X \cong \OO_{\PP^1_k}(a_1) \oplus \dots \oplus
\OO_{\PP^1_k}(a_d), \ a_1,\dots,a_d > 0.
$$

\begin{defn} \label{defn-vflocus}
\marpar{defn-vflocus}
Let $k$ be a field and let $X$ be a quasi-projective $k$-scheme.
Denote by $X_{\text{smooth}}$ the smooth locus of $X$.
The \emph{very free locus} $X_\text{v.f.}$ 
of $X$ is the union of the images in $X_{\text{smooth}}$ 
of all very free rational curves to $X_{\text{smooth}}\otimes_k K$ as 
$K/k$ varies over all algebraically closed extensions.  
More generally, for a flat, quasi-projective morphism of schemes,
$$
\pi:X\rightarrow B,
$$
denoting by $X_{\pi,\text{smooth}}$ the smooth locus of the morphism
$\pi$, 
the \emph{very free locus} $X_{\pi,\text{v.f.}}$
is the union in $X_{\pi,\text{smooth}}$ of the images of every very
free rational curve in every geometric fiber of
$X_{\pi,\text{smooth}}$ over $B$.  
\end{defn}

\mni
The next theorem explains the relation of these different properties.

\begin{thm}\cite[\S IV.3]{K}, \cite{HT06}, \label{thm-equiv}
\marpar{thm-equiv}
Unless stated otherwise, all varieties below are $d$-dimensional,
reduced, irreducible,
quasi-projective schemes over
an algebraically closed field $k$.
\begin{enumerate}
\item[(0)]
In characteristic $0$, every rationally connected variety is separably
rationally connected.
\item[(i)]
For every flat, proper morphism $\pi:X\rightarrow B$ (not necessarily
of quasi-projective varieties over a field),
the subset of $B$
parameterizing points whose geometric fiber is rationally chain
connected is stable under specialization. (If one bounds the degree of
the chains with respect to a relatively ample invertible
$\OO_X$-module, then it is a closed subset.)
\item[(ii)]
The very free locus of a quasi-projective variety is open.  More
generally, 
for every flat, quasi-projective 
morphism, $\pi:X\rightarrow B$, 
the subset $X_{\pi,\text{v.f.}}$ of $X_{\pi,\text{smooth}}$ is an open
subset.  
\item[(iii)]
The very free locus $X_\text{v.f.}$ of a quasi-projective variety
is
(separably) rationally connected in the following strong sense.  For
every positive integer $N$, for every positive integer $m$, and for 
every positive integer $a$, for every
collection of distinct closed points $t_1,\dots,t_N\in \PP^1_k$, for
every collection of closed points $x_1,\dots,x_N \in X_{\text{v.f.}}$, 
and for every specification of an $m$-jet of a smooth curve
in $X$ at each point $x_i$, there exists a morphism
$$
f:\PP^1_k \rightarrow X_{\text{v.f.}}
$$
such that for every $i=1,\dots,N$, $f$ is unramified at $t_i$, 
$f(t_i)$ equals $x_i$ and the $m$-jet of
$t_i$ in $\PP^1_k$ maps isomorphically to the specified $m$-jet at
$x_i$, and
$$
f^* T_X \cong \OO_{\PP^1_k}(a_1) \oplus \dots \oplus
\OO_{\PP^1_k}(a_d), \ a_1,\dots,a_d \geq a.
$$
\item[(iv)]
Every rational curve in $X_{\text{smooth}}$ intersecting
$X_{\text{v.f.}}$ is contained in $X_{\text{v.f.}}$.  
Thus for every smooth, rationally chain connected 
variety, 
if $X$ contains a very free rational curve then $X_{\text{v.f.}}$ 
equals all of $X$.  
\item[(v)]
A proper variety $X$ is rationally chain connected if it
is \emph{generically} rationally chain connected, i.e., if there
exists a morphism $u$ as in the definition such that $u^{(2)}$ is
dominant (but not necessarily surjective).
\item[(vi)]
For the morphism $u:M\times_k \PP^1_k \rightarrow X$, let $l$ be a
closed point
of $M$ such that $u_l:\PP^1 \rightarrow X$ has image in
$X_\text{smooth}$ and such that $u^{(2)}$ is smooth at $(l,t_1,t_2)$
for some $t_1,t_2\in \PP^1_k$.  Then the morphism $u_l$ is very free.  
Thus an irreducible, quasi-projective variety $X$ contains a very free
curve if and only if there is a separably rationally connected open
subset of $X_\text{smooth}$.  Also, a smooth, quasi-projective variety
$X$ in characteristic $0$ which is generically rationally connected
contains a very free morphism.  
\item[(vii)]
For a surjective morphism $f:X\rightarrow Y$ of varieties over an
algebraically closed field, if $X$ is rationally connected,
resp. rationally chain connected, then also $Y$ is rationally
connected, resp. rationally chain connected.
\item[(viii)]
For a birational morphism $f:X\rightarrow Y$ of proper varieties over
an algebraically closed field, if $Y$ is rationally connected then $X$
is rationally chain connected.  If the characteristic is zero, then
$X$ is rationally connected.
\end{enumerate}
\end{thm}

\begin{rmk} \label{rmk-equiv}
Item (ii) is proved in Proposition ~\ref{prop-vfRC}.  The
\emph{generic case} of Item (iii), which is all we will need, 
is proved in Proposition
~\ref{prop-genWA}.  The complete result was proved by Hassett and
Tschinkel, \cite{HT06}.  Item (iv) follows from Corollary
~\ref{cor-genWA}.  The remaining items are not proved, nor are they
used in the proof of the main theorem.
For the most
part they are proved by similar arguments; complete proofs are in
\cite[\S IV.3]{K}.
\end{rmk}

%% \mni
%% Let $X$ be a smooth, projective variety over an algebraically closed
%% field $k$.  An algebraically closed field extension $K/k$ is
%% \emph{sufficiently large} if for every countable collection
%% $(Y_i)_{i=1,2,\dots}$ of proper closed subsets $Y_i\subsetneq X$ there exists
%% a $K$-point of $X$ contained in no $Y_i$.  For instance, $K$ is
%% sufficiently large if it is
%% uncountable or if $K$ contains $k(X)$.

%% \begin{prop}[REFERENCES] \label{prop-equiv}
%% \marpar{prop-equiv}
%% The smooth, projective variety $X$ is rationally connected
%% if and only if it satisfies one of the following conditions.  
%% \begin{enumerate}
%% \item[(i)]
%% For every algebraically closed extension $K/k$, 
%% resp. for at least one sufficiently large extension $K/k$, and
%% for every pair $(x_1,x_2)$ of $K$-points of $X$, resp. for every pair
%% in some Zariski dense open subset of $(X\times_k X)\otimes_k K$, 
%% there exists a
%% $K$-morphism 
%% $$
%% f:\PP^1_K \rightarrow X\otimes_k K
%% $$ 
%% whose image contains
%% $x_1$ and $x_2$.
%% \item[(ii)]
%% With hypotheses on $K$, $x_1$ and $x_2$ as above, there exists a curve
%% $C$ in $X\otimes_k K$ containing $x_1$ and $x_2$ whose irreducible
%% components all have geometric genus $0$, i.e., 
%% the pair is contained in a \emph{chain} of rational curves.
%% \item[(iii)]
%% For every algebraically closed extension $K/k$,
%% for every positive integer $r$, for every
%% $r$-tuple of $K$-points $p_1,\dots,p_r$ of $X$ and for every collection of
%% tangent vectors $(v_i \in T_{p_i}(X\otimes_k K))_{i=1,\dots,r}$, there exists a
%% $K$-morphism 
%% $$
%% f:\PP^1_K \rightarrow X\otimes_k K
%% $$ 
%% and $K$-points
%% $(t_i\in \PP^1_K)_{i=1,\dots,r}$ such that 
%% $$
%% \forall i=1,\dots,r, \ \ f(t_i) = p_i \text{ and } v_i \in
%% df_{t_i}(T_{t_i}\PP^1_K).
%% $$
%% \item[(iv)]  
%% The scheme $X$ is connected and there exists a \emph{very free}
%% rational curve, i.e., there exists a 
%% $k$-morphism
%% $$
%% f:\PP^1_k \rightarrow X
%% $$ 
%% such that $f^* T_X$ is an ample sheaf on
%% $\PP^1_k$, 
%% $$
%% f^* T_X \cong \OO_{\PP^1}(a_1) \oplus \dots \oplus \OO_{\PP^1}(a_n), \
%% \ a_1,\dots,a_n > 0.
%% $$
%% \end{enumerate}
%% \end{prop}

%% \mni
%% Rational connectedness is analogous to path connectedness in topology,
%% where continuous maps from the unit interval are replaced by morphisms
%% from the projective line.  Many properties of path connected
%% topological spaces have analogues among rationally connected varieties.

%% \begin{prop}[REFERENCES] \label{prop-RCrobust}
%% \marpar{prop-RCrobust}
%% \begin{enumerate}
%% \item[(i)]
%% In a flat family of
%% smooth, proper varieties over a connected base, one geometric fiber
%% is rationally connected if and only if every geometric fiber is
%% rationally connected.  
%% \item[(ii)]
%% Among proper varieties, rational connectedness is a birational
%% invariant.
%% \item[(iii)]
%% A proper variety is rationally connected if it is the image of a
%% rationally connected variety.  In particular, every unirational
%% variety is rationally connected.
%% \end{enumerate}
%% \end{prop}

\mni
Rational connectedness is analogous to path connectedness in topology,
and satisfies the analogues of many properties of path connectedness.
One property of path connectedness is this:
for a fibration of CW complexes, if
the base space and the fibers are path connected, then also the total
space is path connected.  This led to two conjectures 
by Koll\'ar, Miyaoka and Mori.

\begin{conj}\cite[Conjecture IV.5.6]{K} \label{conj-KMM1}
\marpar{conj-KMM1}
Let $\pi:X\rightarrow B$ be a surjective morphism of smooth, projective
schemes over an algebraically closed field of characteristic $0$.  
If both $B$ and a general fiber of $\pi$ are rationally
connected, then $X$ is also rationally connected.
\end{conj}

\mni
Conjecture ~\ref{conj-KMM1} is implied by the following conjecture
about rationally connected fibrations over curves.

\begin{conj}\cite[Conjecture IV.6.1.1]{K} \label{conj-KMM}
\marpar{conj-KMM}
Let $\pi:X\rightarrow B$ be a surjective morphism of projective
schemes over an algebraically closed field of characteristic $0$.  
If $B$ is a smooth curve and if a general fiber of $f$
is rationally connected, then there exists a morphism $s:B\rightarrow
X$ such that $\pi\circ s$ equals $\text{Id}_B$, i.e., $s$ is a section
of $\pi$.
\end{conj}

%% \mni
%% Conjecture ~\ref{conj-KMM} implies a positive answer to Question
%% ~\ref{ques-RC}.  For two general points $x_1$ and $x_2$ of $X$, each
%% point $x_i$ is rationally connected to every point in the fiber
%% $f^{-1}(f(x_i))$ by Proposition ~\ref{prop-RCrobust} (i).
%% Thus, if there exists a rational curve in $X$
%% intersecting each fiber $f^{-1}(f(x_1))$ and $f^{-1}(f(x_2))$, then
%% $x_1$ and $x_2$ are contained in a chain of rational curves.  This
%% implies $X$ is rationally connected by Proposition ~\ref{prop-equiv} (ii).
%% Let $U$ be the open subset of $X$ on which $f$ is a
%% Since $B$
%% is rationally connected, there exists a morphism $g:\PP^1 \rightarrow
%% B$ whose image contains $f(x_1)$ and $f(x_2)$.  The base change
%% morphism
%% $$
%% \PP^1 \times_B X \rightarrow \PP^1
%% $$ 
%% satisfies the hypotheses of Conjecture ~\ref{conj-KMM}.  Thus the
%% conjecture implies there exists a morphism $s:\PP^1\rightarrow X$
%% whose image intersects both fibers $f^{-1}(f(x_1))$ and $f^{-1}(f(x_2))$.

\mni
Our next goal is to prove the following result.

\begin{thm} \cite{GHS} \label{thm-GHS}
\marpar{thm-GHS}
Conjecture ~\ref{conj-KMM} of Koll\'ar-Miyaoka-Mori is true.
Precisely, let $k$ be an algebraically closed field of characteristic
$0$ and let $\pi:X\rightarrow B$ be a surjective morphism from a
normal, projective $k$-scheme $X$ to a smooth, projective, connected
$k$-curve $B$.  If the geometric generic fiber
$X_{\overline{\eta}_B}$ is a normal, integral scheme whose smooth
locus contains a very free curve, then there exists a morphism
$s:B\rightarrow X$ such that $\pi\circ s$ equals $\text{Id}_B$.  
\end{thm}

\mni
This was generalized by A. J. de Jong to the case that $k$ is
algebraically closed of arbitrary characteristic, \cite{dJS}.  The key
difference has to do with extensions of valuation rings in
characteristic $0$ and in positive characteristic.  Given a flat
morphism of smooth schemes in characteristic $0$, $\pi:U\rightarrow
B$, and given codimension $1$ points $\eta_D$ of $U$ and $\eta_\Delta$
of $B$ with $\pi(\eta_D)=\eta_\Delta$, the induced local homomorphism
of stalks 
$\pi_U^*:\widehat{\OO}_{B,\eta_\Delta} \rightarrow \widehat{\OO}_{U,\eta_D}$, 
is equivalent to 
$$
k(\Delta)\Sem{t} \rightarrow k(D)\Sem{r}, \ \ t\mapsto u r^m
$$ 
for a unit $u$ and a positive
integer $m$, cf. the proof of Lemma ~\ref{lem-U} below.  In
particular, it is \emph{rigid} in the sense that $t\mapsto ur^m +
vr^{m+1} + \dots$ is equivalent to $t\mapsto ur^m$. However,
extensions of positive characteristic valuation rings are not rigid, 
e.g., $t\mapsto r^p + v_1r^{p+1}$ is equivalent to $t\mapsto
r^p + v_2r^{p+1}$ only if $v_1= v_2$.  
But there is a weak rigidity of local homomorphisms,
Krasner's lemma in the theory of non-Archimedean valuations.  This is a
key step in the generalization to positive characteristic.

\mni
Of course when $k$ has characteristic $0$, then 
since $X$ is normal
the fiber $X_{\overline{\eta}_B}$ is automatically normal.  If $X$ is
also smooth (which can be achieved thanks to resolution of
singularities in characteristic $0$),
then also $X_{\overline{\eta}_B}$ is smooth.  Then the hypothesis
on $X_{\overline{\eta}_B}$ is equivalent to rational connectedness.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Outline of the proof
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Outline of the proof} \label{sec-out}
\marpar{sec-out}

\mni
The proof that follows is based on a proof by T. Graber, J. Harris and
myself (not quite the version we chose to publish) together with
several major simplifications due to A. J. de Jong.
The basic idea is to choose a smooth curve
$C\subset X$ such that $\pi|_C:C\rightarrow B$ is finite, and then
try to deform $C$ as a curve in $X$ until it specializes to a
reducible curve in $X$, one component of which is the image of a
section $s$ of $\pi$.  Here are some definitions that make this
precise.

\begin{defn} \label{defn-defo}
\marpar{defn-defo}
Let $\pi_C:C \rightarrow B$ be a finite morphism of smooth,
projective $k$-curves.  A \emph{linked curve with handle $C$} is a
reduced, connected, projective curve $C_\text{link}$ with irreducible
components  
$$
C_{\text{link}} = C \cup L_1 \cup \dots \cup L_m
$$
together with a morphism
$$
\pi_{C,\text{link}}:C_{\text{link}} \rightarrow B
$$
such that 
\begin{enumerate}
\item[(i)]
$\pi_{C,\text{link}}$ restricts to $\pi_C$ on the component $C$,
\item[(ii)]
the restriction of $\pi_{C,\text{link}}$ to each \emph{link}
component $L_i$ is a
constant morphism with image $b_i$, where $b_1,\dots,b_m$ are distinct
closed points of $B$,
\item[(iii)]
and each link $L_i$ is a smooth, rational curve intersecting
$C$ in a finite number of nodes of $C_\text{link}$.
\end{enumerate}
If every link $L_i$ intersects $C$ in a single node of
$C_{\text{link}}$, then $(C_{\text{link}},\pi_{C,\text{link}})$ is
called a \emph{comb} and the links $L_i$ are called \emph{teeth}.  For
combs we will use the notation $C_{\text{comb}}$ rather than
$C_{\text{link}}$.  

\mni
A \emph{one-parameter deformation} of a linked curve
$(C_\text{link},\pi_{C,\text{link}})$ 
is a datum of a smooth, connected, pointed curve $(\Pi,0)$ and a
projective morphism
$$
(\rho,\pi_{\mc{C}}):\mc{C} \rightarrow \Pi\times_k B
$$
such that $\rho$ is flat and such that $\mc{C}_0 := \rho^{-1}(0)$ 
together
with the restriction of $\pi_{\mc{C}}$ equals
the linked curve $(C_{\text{link}},\pi_{C,\text{link}})$. 

\mni
A one-parameter
deformation \emph{specializes to a section curve} if there exists a
closed point $\infty \in \Pi$ and an irreducible component $B_i$ of
$\mc{C}_\infty := \rho^{-1}(\infty)$
such that
\begin{enumerate}
\item[(i)]
$\mc{C}_\infty$ is reduced at the generic point of $B_i$ 
\item[(ii)]
and the
restriction of $\pi_{\mc{C}}$ to $B_i$ is an isomorphism 
$$
\pi_{\mc{C}}|_{B_i}:
B_i \xrightarrow{\cong} B.
$$
\end{enumerate}

\mni
Given a linked curve, a one-parameter deformation of the linked curve
and a
$B$-morphism $j:C_{\text{link}} \rightarrow X$, an
\emph{extension} of $j$ is an open neighborhood of $0$, $0\in N
\subset B$ and a $B$-morphism 
$$
j_N:\mc{C}_N \rightarrow X, \ \ \mc{C}_N :=
\rho^{-1}(N)
$$
restricting
to $j$ on $\mc{C}_0 = C_\text{link}$. 
\end{defn}

\mni
Figure ~\ref{fig-comb1a} shows a linked curve with some links
intersecting the handle in more than $1$ point.  And Figure
~\ref{fig-comb1b} shows a comb, where every tooth intersects the
handle precisely once.  

%FIGURE comb1a
\begin{figure}
\begin{picture}(330,270)
\put(0,10){\makebox(330,240){\includegraphics{comb1a_small.eps}}}
\end{picture}
\caption{A linked curve with handle $C$ and some links intersecting
  $C$ in more than 1 point}
\label{fig-comb1a}
\end{figure}


%FIGURE comb1b
\begin{figure}
\begin{picture}(330,270)
\put(0,10){\makebox(330,240){\includegraphics{comb1b_small.eps}}}
\end{picture}
\caption{A comb is a linked curve where every link, or tooth,
  intersects the handle precisely once}
\label{fig-comb1b}
\end{figure}


\mni
For the purposes of producing a section, the particular parameter
space $(\Pi,0)$ of the one-parameter deformation is irrelevant.  Thus,
it is allowed to replace the one-parameter deformation by the new
one-parameter deformation obtained from a finite base change
$(\Pi',0')\rightarrow (\Pi,0)$.  The following lemma is
straightforward.  

\begin{lem} \label{lem-bc}
\marpar{lem-bc}
Let $(\Pi,0,\infty)$ together with $(\rho,\pi_{\mc{C}}):\mc{C}
\rightarrow \Pi\times_k B$ be a one-parameter deformation of
$(C_\text{link},\pi_{C,\text{link}})$ specializing to a section curve
$B_i$.  For every morphism of $2$-pointed, smooth, connected curves
$$
(\Pi',0',\infty') \rightarrow (\Pi,0,\infty),
$$
the base change morphism
$$
\Pi' \times_{\Pi} \mc{C} \rightarrow \Pi'\times_k B
$$
is also a one-parameter deformation of
$(C_\text{link},\pi_{C,\text{link}})$ specializing to the section
curve $B_i$.
\end{lem}

\mni
The usefulness of these definitions is the following simple
consequence of the valuative criterion of properness.

\begin{lem} \label{lem-val}
\marpar{lem-val}
Let $(C_{\text{link}},\pi_{\text{link}})$ be a linked curve together
with a $B$-morphism $j:C_{\text{link}} \rightarrow X$.  If there
exists a one-parameter deformation of the linked curve specializing
to a section curve and if there exists an extension of $j$, then there
exists a section $s:B\rightarrow X$ of $\pi$.
\end{lem}

\begin{proof}
Let $R$ denote the stalk $\OO_{\mc{C},\eta_{B_i}}$ of $\OO_{\mc{C}}$ at
the generic point $\eta_{B_i}$ of $B_i$.  By the hypotheses on
$\mc{C}$ and $B_i$, $R$ is a discrete valuation ring with residue
field $\kappa = k(B_i)$ and fraction field $K = k(\mc{C})$.  
The restriction
of $j_N$ to the generic point of $\mc{C}$ is a $B$-morphism 
$$
j_K:\SP K \rightarrow X.
$$
Because $\pi:X\rightarrow B$ is proper, by the valuative criterion
of properness the $B$-morphism $j_K$ extends to a $B$-morphism
$$
j_R:\SP R \rightarrow X,
$$
which in turn gives a $B$-morphism from the residue field $\SP \kappa$
to $X$, i.e., a rational $B$-map
$$
j_{B_i}:B_i \supset U \rightarrow X, \ \ U \subset B_i \text{ a dense, Zariski
  open}.
$$
Finally, because $B_i$ is a smooth curve, the valuative
criterion applies once more and this rational transformation extends to a
$B$-morphism
$$
j_{B_i}:B_i \rightarrow X.
$$
Because $\pi_{\mc{C}}|_{B_i}:B_i \rightarrow B$ is an isomorphism,
there exists a unique $B$-morphism
$$
s:B\rightarrow X
$$
such that $j_{B_i}= s\circ \pi_{\mc{C}}|_{B_i}$.  
The morphism $s$ is a section of $\pi$.
\end{proof}

\mni
Thus the proof of the theorem breaks into three parts:
\begin{enumerate}
\item[(i)]
find a ``good'' linked curve $j:C_{\text{link}} \rightarrow X$,
\item[(ii)]
find a one-parameter deformation of the linked curve specializing to a
section curve,
\item[(iii)]
and find an extension of $j$ to the one-parameter deformation.
\end{enumerate}

\mni
The first step in finding $j:C_{\text{link}} \rightarrow X$ is to form
a curve $C_{\text{init}}$ 
which is an intersection of $X$ with $\text{dim}(X)-1$
general hyperplanes in projective space.  By Bertini's theorem, if the
hyperplanes are sufficiently
general, then $C_{\text{init}}$ will satisfy any
reasonable transversality property.  Moreover, there is a technique
due to Koll\'ar-Miyaoka-Mori -- the smoothing combs technique --
for improving $C_{\text{init}}$ to another curve $C\subset X$ still
satisfying the transversality property and also satisfying a positivity
property with respect to the vertical tangent bundle of
$\pi:X\rightarrow B$.  

\mni
Unfortunately, even after such an improvement,
there may be no one-parameter deformation of $\pi|_C:C\rightarrow B$
specializing to a section curve.  However, after attaching
sufficiently many link 
components over general closed points of $B$, there does exist a
one-parameter deformation of $C_\text{link}$
specializing to a section curve.  
This is one aspect of the well-known theorem that for a fixed base
curve $B$ and for a fixed degree $d$, if the number $\beta$ of branch points
is sufficiently large the Hurwitz scheme of degree $d$ covers of $B$
with $\beta$ branch points is irreducible.  (This was proved by
Hurwitz when $g(B) = 0$, \cite{Hurw}, proved by Richard Hamilton for
arbitrary genus
in
his thesis, and periodically reproved ever since, cf. \cite{GHS2}.)
Because
the general fibers of $\pi:X\rightarrow B$ are rationally connected,
the inclusion $C\subset X$ extends to a $B$-morphism $j:C_\text{link}
\rightarrow X$.  

\mni
Finally the positivity property mentioned above implies $j$ extends to
the one-parameter deformation, at least after base change by a
morphism $\Pi' \rightarrow \Pi$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Hilbert schemes and smoothing combs
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Hilbert schemes and smoothing combs} \label{sec-Hilb}
\marpar{sec-Hilb}

\mni
The smoothing combs technique of Koll\'ar-Miyaoka-Mori depends on a
result from the deformation theory of Hilbert schemes.  Here is the
setup for this result.
Let $Y \rightarrow S$ be a flat, quasi-projective morphism and
let 
$$
(\rho_{\text{Hilb}}:\text{Hilb}(Y/S) \rightarrow S, \text{Univ}(Y/S)
\subset \text{Hilb}(Y/S)\times_S T)
$$
be universal among pairs $(\rho:T\rightarrow S, Z\subset T\times_S
Y)$ of an $S$-scheme $T$ and a closed subscheme $Z\subset
T\times_S Y$ such that $Z\rightarrow T$ is proper, flat and
finitely presented.  In other words, $\text{Hilb}(Y/S)$ is the
\emph{relative Hilbert scheme} of $Y$ over $S$.

\mni
In particular, for every field $K$ the $K$-valued points of
$\text{Hilb}(Y/S)$ are naturally in bijection with pairs
$(s,Z)$ of a $K$-valued point $s$ of $S$ and a closed subscheme $Z$
of $Y_s := \{s\} \times_S Y$.  The closed immersion $Z\rightarrow Y_s$
is a \emph{regular embedding} if at every point of $Z$ the stalk of
the ideal sheaf $\mc{I}_{Z/Y_s}$ is generated by a regular sequence of
elements in the stalk of $\OO_{Y_s}$.  In this case the \emph{conormal
  sheaf} $\mc{I}_{Z/Y_s}/\mc{I}_{Z/Y_s}^2$ is a locally free
$\OO_Z$-module, and hence also the \emph{normal sheaf} 
$$
\mc{N}_{Z/Y_s} :=
\textit{Hom}_{\OO_Z}(\mc{I}_{Z/Y_s}/\mc{I}_{Z/Y_s}^2,\OO_S)
$$
is a locally free $\OO_Z$-module.  The regular embeddings which arise
in the proof of Theorem ~\ref{thm-GHS} are precisely closed immersions
of at-worst-nodal curves in a smooth variety.  

\begin{prop}\cite[Theorem I.2.10, Lemma I.2.12.1, Proposition
  I.2.14.2]{K} \label{prop-Hilb}
\marpar{prop-Hilb}
If $Z\subset Y_s$ is a regular embedding and if
$h^1(Z,\mc{N}_{Z/Y_s})$ equals $0$, then $\text{Hilb}(Y/S)$ is smooth
over $S$ at $(s,Z)$.
\end{prop}

\mni
There is a variation of this proposition 
which is also useful.  There is a
\emph{flag Hilbert scheme} of $Y$ over $S$, i.e., a universal pair
$$
(\rho_{\text{fHilb}}:\text{fHilb}(Y/S) \rightarrow S,
\text{Univ}_1(Y/S) \subset \text{Univ}_2(Y/S) \subset
\text{Hilb}(Y/S)\times_S T)
$$
among all pairs $(\rho:T\rightarrow S, Z_1\subset Z_2 \subset
T\times_S Y)$ of an $S$-scheme $T$ and a nested pair of closed
subschemes $Z_1 \subset Z_2 \subset T\times_S Y$ such that for
$i=1,2$, the projection $Z_i \rightarrow T$ is proper, flat and
finitely presented.  There are obvious forgetful morphisms
$$
F_i: \text{fHilb}(Y/S) \rightarrow \text{Hilb}(Y/S), \ \ F_i(s,Z_1,Z_2)
= (s,Z_i).
$$

\begin{prop} \label{prop-Hilb2}
\marpar{prop-Hilb2}
Let $K$ be a field and let $(s,Z_1,Z_2)$ be a $K$-point of
$\text{fHilb}(Y/S)$.  
If each closed immersion
$Z_1 \subset Z_2$ and 
$Z_2\subset Y_s$ is a regular embedding and if
$$
h^1(Z_2,\mc{N}_{Z_2/Y_s}) = 0, h^1(Z_1,\mc{N}_{Z_1/Z_2}) = 0, \text{
  and if } h^i(Z_2,\mc{I}_{Z_1/Z_2}\cdot \mc{N}_{Z_2/Y_2}) = 0 \text{
  for } i=1,2,
$$
then $\text{fHilb}(Y/S)$ is smooth over $S$ at $(s,Z_1,Z_2)$, for each
$i=1,2$, $\text{Hilb}(Y/S)$ is smooth over $S$ at $(s,Z_i)$, and each
forgetful morphism $F_i:\text{fHilb}(Y/S) \rightarrow
\text{Hilb}(Y/S)$ is smooth at $(s,Z_1,Z_2)$.  
\end{prop}

\begin{proof}
Since $h^1(Z_2,\mc{N}_{Z_2/Y_s})$ equals $0$, $\text{Hilb}(Y/S)$ is
smooth at $(s,Z_2)$ by Proposition ~\ref{prop-Hilb}.  It is easy to
see that the forgetful morphism $F_2$ is equivalent to the relative
Hilbert scheme $\text{Hilb}(\text{Univ}(Y/S)/\text{Hilb}(Y/S))$ over
$\text{Hilb}(Y/S)$.  Thus, applying Proposition ~\ref{prop-Hilb} to
this Hilbert scheme, the vanishing of $h^1(Z_1,\mc{N}_{Z_1/Z_2})$
implies $F_2$ is smooth at $(s,Z_1,Z_2)$.  Since a composition of
smooth morphisms is smooth, also $\text{fHilb}(Y/S)$ is smooth over
$S$ at $(s,Z_1,Z_2)$.  The long exact sequence of cohomology
associated to the short exact sequence
$$
\begin{CD}
0 @>>> \mc{I}_{Z_1/Z_2}\cdot \mc{N}_{Z_2/Y_s} @>>> \mc{N}_{Z_2/Y_s}
@>>> \mc{N}_{Z_2/Y_s}|_{Z_1} @>>> 0
\end{CD}
$$
implies that $h^1(Z_1,\mc{N}_{Z_2/Y_s}|_{Z_1})$ equals
$h^1(Z_2,\mc{N}_{Z_2/Y_s})$, which is $0$.  Thus, the long exact
sequence of cohomology associated to
$$
\begin{CD}
0 @>>> \mc{N}_{Z_1/Z_2} @>>> \mc{N}_{Z_1/Y_s} @>>>
\mc{N}_{Z_2/Y_s}|_{Z_1} @>>> 0
\end{CD}
$$
implies that $h^1(Z_2,\mc{N}_{Z_1/Y_2})$ equals $0$.  So again by
Proposition \ref{prop-Hilb}, $\text{Hilb}(Y/S)$ is smooth over $S$ at
$(s,Z_1)$.  Finally, $F_1$ is a morphism of smooth $S$-schemes at
$(s,Z_1,Z_2)$.  Thus, to prove $F_1$ is smooth, it suffices to prove
it is surjective on Zariski tangent vector spaces.  This follows from
the vanishing of $h^1(Z_2,\mc{I}_{Z_1/Z_2}\cdot \mc{N}_{Z_2/Y_s})$.  
\end{proof}

\mni
Another ingredient in the smoothing combs technique is a 
a simple result about \emph{elementary transforms} of locally free
sheaves on a curve: the higher cohomology of the sheaf becomes zero
after applying elementary transforms at sufficiently many points.

\begin{lem} \label{lem-elemtrans}
\marpar{lem-elemtrans}
Let $C$ be a projective, at-worst-nodal, connected curve over a field $k$ and
let $\mc{E}$ be a locally free $\OO_C$-module. 
\begin{enumerate}
\item[(i)]
There exists a short exact sequence of coherent sheaves,
$$
0 \rightarrow \mc{F}^\vee \rightarrow \mc{E}^\vee \rightarrow \mc{T}
\rightarrow 0
$$
such that $\mc{T}$ is a torsion sheaf with support in $C_\text{smooth}$
and such that $h^1(C,\mc{F})$
equals $0$.
\item[(ii)]
Inside the parameter space of 
torsion quotients $q:\mc{E}^\vee \twoheadrightarrow \mc{T}$ with
support in $C_{\text{smooth}}$, denoting
$$
\mc{F}^\vee := \text{Ker}(\mc{E}^\vee \rightarrow \mc{T}) \text{ and }
\mc{F} := \textit{Hom}_{\OO_C}(\mc{F}^\vee, \OO_C),
$$
the subset parameterizing quotients for which $h^1(C,\mc{F}) = 0$ is an
open subset.  
\item[(iii)]
If $h^1(C,\mc{F})$ equals $0$, then for every short exact sequence of
coherent sheaves
$$
0 \rightarrow \mc{G}^\vee \rightarrow \mc{E}^\vee \xrightarrow{q'}
\mc{S} \rightarrow 0
$$
admitting a morphism $r:\mc{S}\rightarrow \mc{T}$ of torsion sheaves
with support in $C_\text{smooth}$
for which $q=r\circ q'$, $h^1(C,\mc{G})$ equals $0$.
\end{enumerate}
\end{lem}  

\begin{proof}
\textbf{(i)}
By Serre's vanishing theorem, there exists an effective, ample divisor
$D$ in the smooth locus of $C$
such that $h^1(C,\mc{E}(D))$ equals $0$.  Define $\mc{F} =
\mc{E}(D)$, define $\mc{E} \rightarrow \mc{F}$ to be the obvious
morphism $\mc{E} \rightarrow \mc{E}(D)$, and define $\mc{T}$ to be the
cokernel of $\mc{F}^\vee \rightarrow \mc{E}^\vee$.

\mni
\textbf{(ii)}
This follows immediately from the semicontinuity theorem, cf. \cite[\S
III.12]{H}.  

\mni
\textbf{(iii)}
There exists an injective morphism of coherent sheaves
$\mc{F}\rightarrow \mc{G}$ with torsion cokernel.  
Because
$h^1(C,\mc{F})$ equals $0$ and because $h^1$ of every torsion sheaf
is zero, the long exact sequence of cohomology implies that also
$h^1(C,\mc{G})$ equals $0$.  
\end{proof}

\mni
It is worth noting one interpretation of the sheaf $\mc{F}$ associated
to a torsion quotient $\mc{T}$.  
Assume that $\mc{T}$ is isomorphic to
a direct sum of skyscraper sheaves at $n$ distinct points
$c_1,\dots,c_n$ of $C_\text{smooth}$.  (Inside the parameter space of torsion
quotients, those with this property form a dense, open subset.)  For
each point $c_i$, the linear functional 
$\mc{E}^\vee|_{c_i} \twoheadrightarrow \mc{T}|_{c_i}$ gives a one-dimensional
subspace $\text{Hom}_k(\mc{T}|_{c_i},k) \hookrightarrow
\mc{E}|_{c_i}$.  The sheaf $\mc{F}$ is precisely the sheaf of rational
sections of $\mc{E}$ having at worst a simple pole at each point $c_i$
in the direction of this one-dimensional subspace of $\mc{E}|_{c_i}$.
This is often called an \emph{elementary transform up} of $\mc{E}$ at
the point $c_i$ in the specified direction.  
So Lemma ~\ref{lem-elemtrans} says that $h^1$
becomes zero after sufficiently many elementary transforms up at
general points in general directions.

%% \begin{defn} \label{defn-combs}
%% \marpar{defn-combs}
%% Let $\pi_C:C\rightarrow B$ be a finite morphism of smooth, projective
%% $k$-curves. 
%% A \emph{comb with handle $C$} is 
%% is a linked curve $(C_{\text{comb}},\pi_{C,\text{comb}})$
%% such that each link $L_i$ intersects the
%% handle $C$ in a single node of $C_{\text{comb}}$. For a comb, the
%% links $L_i$ are called \emph{teeth}.
%% A \emph{comb in
%%   $X$} is a comb $(C_{\text{comb}},\pi_{C,\text{comb}})$ together with
%% a $B$-morphism $j:C_{\text{comb}} \rightarrow X$.  
%% \end{defn}

\mni
This interpretation is useful because the normal sheaf of a reducible
curve can be understood in terms of elementary transforms up.  
To be precise, let $Y$ be a $k$-scheme, let $C$ be a proper, nodal
curve, let $C_0$ be a closed subcurve (i.e., a union of irreducible
components of $C$), and
let 
$j:C\rightarrow Y$ be a regular embedding such that $Y$ is smooth at
every node $p_1,\dots,p_n$ of $C$ which is
contained in $C_0$ and which is not a node of $C_0$.
Then $j_0:C_0\rightarrow Y$ is also a regular embedding and
both
$\mc{N}_{C/Y}|_{C_0}$ and $\mc{N}_{C_0/Y}$ are locally free sheaves on
$C_0$. 
For each $i$, there is a branch $C_i$ of $C$
at $p_i$ other than $C_0$.  
Denote by $T_{C_i,p_i}$ the tangent direction of this branch in
$T_{Y,p_i}$.  

\begin{lem} \cite[Lemma 2.6]{GHS} \label{lem-N}
\marpar{lem-N}
The restriction
$\mc{N}_{C/Y}|_{C_0}$ equals the sheaf of rational sections of
$\mc{N}_{C_0/Y}$ 
having at most a simple pole at each point $p_i$ in the normal
direction determined by $T_{C_i,p_i}$.  
\end{lem}

\begin{proof}
The restrictions of $\mc{N}_{C/Y}|_{C_0}$ and $\mc{N}_{C_0/Y}$ to
the complement of $\{p_1,\dots,p_n\}$ are canonically isomorphic.  The
lemma states that this canonical isomorphism is the restriction of an
injection
$\mc{N}_{C_0/Y} \hookrightarrow \mc{N}_{C/Y}|_{C_0}$ which identifies
$\mc{N}_{C/Y}|_{C_0}$ with the sheaf of rational sections, etc.  This
local assertion can be verified in a formal neighborhood of
each node $p_i$.

\mni
Locally near $p_i$, $C \rightarrow Y$ is
formally isomorphic to the union of the two axes inside a $2$-plane
inside an $n$-plane, i.e., the subscheme of $\AAA^n_k$ with ideal
$I_{C/Y} = \langle x_1x_2, x_3,\dots,x_n \rangle$.  The
branch $C_0$ corresponds to just one of the axes, e.g., the subscheme
of $\AAA^n_k$ with ideal $I_{C_0/Y}=\langle x_2,x_3,\dots,x_n \rangle$.  The
tangent direction of the other branch $C_i$ is spanned by
$(0,1,0,\dots,0)$.  
Thus
it is clear that $I_{C/Y}/I_{C_0/Y}\cdot I_{C/Y}$ is the
submodule of $I_{C_0/Y}/I_{C_0/Y}^2$ 
of elements whose fiber at $0$ is contained in
the annihilator of $T_{C_i,p_i}$.
Dualizing gives the lemma.
\end{proof}

\mni
The final bit of deformation theory needed has to do with deforming
nodes.  
Let
$C$ be a proper, nodal curve and let $j:C\rightarrow Y$ be a regular
embedding.  Let $p$ be a node of $C$ and assume that $Y$ is smooth at
$p$.  There are two branches $C_1$ and $C_2$ of $C$ at $p$ (possibly
contained in the same irreducible component of $C$).  The sheaf
$$
\mc{T} := \textit{Ext}^1_{\OO_C}(\Omega_C,\OO_C)
$$
is a skyscraper sheaf supported at $p$ and with fiber canonically
identified to
$$
\mc{T}|_p = T_{C_1,p}\otimes_k T_{C_2,p}.
$$
The following lemma is as much definition as lemma.  

\begin{lem} \label{lem-node}
\marpar{lem-node}
There exists a quotient of coherent sheaves 
$$
\mc{N}_{C/Y} \twoheadrightarrow \mc{T}
$$
such that 
for both $i=1,2$ the
quotient $\mc{N}_{C/Y}|_{C_i}/\mc{N}_{C_i/Y}$ equals $\mc{T}$.
A first-order deformation of $C\subset Y$, i.e., a  
global
section of $\mc{N}_{C/Y}$ is said to \emph{smooth the node $p$
to first-order} if 
the image of the section in $T_{C_1,p}\otimes_k T_{C_2,p}$
is nonzero.  For a deformation 
$$
\mc{C} \subset \Pi \times_k Y
$$
of $C\subset Y$ over a smooth
pointed curve $(\Pi,0)$ (i.e., $\mc{C}_0 = C$), 
if the associated first-order deformation of
$C\subset Y$ smooths the node $p$ to first-order, then $p$ is not
contained in the closure of the singular locus of the projection,
$$
(\Pi-\{0\})\times_\Pi \mc{C} \rightarrow (\Pi-\{0\})
$$
i.e., a general fiber $\mc{C}_t$ of the deformation \emph{smooths the
  node}.  
\end{lem}

\mni
This is a well-known result.  A good reference for this result, and
many other results about deformations of singularities, is
\cite{ArtinDef}, particularly \S I.6.  Here is a brief remark on the
proof. 
Because $C\subset Y$
is a regular embedding, the conormal sequence is exact on the left,
i.e.,
$$
\begin{CD}
0 @>>> \mc{I}_{C/Y}/\mc{I}_{C/Y}^2 @>>> \Omega_Y|_C @>>> \Omega_C @>>>
0
\end{CD}
$$
is a short exact sequence.  Applying global Ext, there is a connecting
map
$$
\delta: H^0(C,\mc{N}_{C/Y}) \rightarrow
\text{Ext}^1_{\OO_C}(\Omega_C,\OO_C).
$$
There is also a local-to-global sequence for global Ext inducing a map
$$
\text{Ext}^1_{\OO_C}(\Omega_C,\OO_C) \rightarrow
H^0(C,\textit{Ext}^1_{\OO_C}(\Omega_C,\OO_C)) = H^0(C,\mc{T}) =
T_{C_1,p}\otimes_k T_{C_2,p}.
$$
The composition of these two maps is precisely the map on global
sections associated to $\mc{N}_{C/Y} \rightarrow \mc{T}$.
%% The induced map to $H^0(C,\textit{Ext}^1_{\OO_C}(\Omega_C,\OO_C)) =
%% \mc{T}$ is precisely the map above.  
The global Ext group
is identified with the first-order
deformations of $C$ as an abstract scheme, and the Ext term
is identified with the first-order deformations of the node.  
It is worth noting that even if the first-order
deformation does not smooth the node, the full deformation $\mc{C}
\subset \Pi \times_k Y$
may smooth the node if the total space $\mc{C}$ is singular
at $(0,p)$.    

\mni
The first result using the smoothing combs technique is the following.

\begin{prop} \label{prop-vfRC}
\marpar{prop-vfRC}
Let $Y$ be a quasi-projective scheme over an
algebraically closed field $k$.  The very free locus $Y_\text{v.f.}$
is an open subset of $Y$.  More generally, for a flat,
quasi-projective morphism $\pi:Y\rightarrow B$, the relative very free
locus $Y_{\pi,\text{v.f.}}$ is an open subset of $Y$.

\mni
Let $Y$ be an irreducible, quasi-projective scheme over an
algebraically closed field $k$.
Denote by $t_1$, resp. $t_2$, the closed
point of $\PP^1_k$, $t_1 = 0$, resp. $t_2 = \infty$. 
Let $y_1$ and $y_2$ be closed points of $Y_\text{v.f.}$, let $a$ and
$k$ be nonnegative integers, and let there be given curvilinear
$k$-jets in $Y$ at each of $y_1$ and $y_2$.  If the given $k$-jets are
general among all curvilinear $k$-jets at $y_1$ and $y_2$, then there
exists a morphism
$$
f:(\PP^1_k,t_1,t_2) \rightarrow (Y_\text{v.f.},y_1,y_2)
$$ 
mapping the $k$-jet of $\PP^1$ at $t_i$ isomorphically to the given
$k$-jet at $y_i$ for $i=1,2$ and such that
$$
f^*T_Y
\cong \OO_{\PP^1}(a_1) \oplus \dots \oplus \OO_{\PP^1}(a_n), \
a_1,\dots,a_n \geq a.
$$
\end{prop}

\begin{proof}
In the absolute case, resp. relative case, the very free locus $Y_{\text{v.f.}}$,
resp. $Y_{\pi,\text{v.f.}}$,  
is defined to be the same as the very free locus of the smooth locus
$Y_{\text{smooth}}$, resp. $Y_{\pi,\text{smooth}}$.  Since the smooth
locus is open in $Y$, and since an open subset of an open subset is an
open subset, it suffices to prove the very free locus is open under
the additional hypothesis that $Y$ is smooth, resp. that $\pi$ is smooth.

\mni
By the
definition of $Y_\text{v.f.}$, for each $i=1,2$ there exists 
a very free morphism 
$$
f_i:(\PP^1,0) \rightarrow (Y_\text{v.f.},y_i), \ \ f_i^* T_Y \cong
\OO_{\PP^1}(a_1) \oplus \dots \oplus \OO_{\PP^1}(a_n), a_1,\dots,a_n
\geq 1.
$$
In particular, 
for each $i=1,2$, $h^1(\PP^1,f_i^*T_Y(-\ul{0}-\ul{\infty}))$ equals
$0$, where $\ul{0}$, resp. $\ul{\infty}$, is the Cartier divisor of
the point $0$, resp. $\infty$, in $\PP^1$.
Since the normal sheaf of $f_i$ is a quotient of $f_i^*T_Y$, also
$h^1(\PP^1, \mc{N}_{f_i}(-\ul{0}-\ul{\infty}))$ equals $0$.  
Thus, applying
Proposition ~\ref{prop-Hilb2} where $Z_1 = \{0,\infty\}$ and $Z_2 = \PP^1$,
there exist deformations of the morphism $f_i$ such that $f_i(0)$
equals $y_i$ and $f_i(\infty)$ is any point in a nonempty Zariski open
subset of $Y$.  The same argument holds in the relative case.

\mni
Next assume that $Y$ is irreducible and quasi-projective.  Then the
smooth locus $Y_{\text{smooth}}$ is also irreducible (or empty).
Thus, by the same argument as above, a proof of the second result for
smooth varieties implies the second result in general.  Thus assume
$Y$ is also smooth.

\mni
Since $Y$ is
irreducible, the open for $i=1$ intersects the open for $i=2$.  Thus
there exist very free morphisms $f_1$ and $f_2$ such that $f_1(\infty)
= f_2(\infty)$.  Let $C$ be the nodal curve with two irreducible
components $C_1$ and $C_2$ each isomorphic to $\PP^1$ and with a
single node which, when
considered as a point in either $C_1$ or $C_2$, corresponds to
$\infty$ in $\PP^1$.  
Let $f:C\rightarrow Y$ be the unique morphism whose
restriction to each component $C_i$ equals $f_i$.  Denote by
$$
\begin{CD}
0 @>>> \mc{N}'_{C/Y} @>>> \mc{N}_{C/Y} @>>> \mc{T} @>>> 0
\end{CD}
$$
the short exact sequence coming from Lemma ~\ref{lem-node}.
Using Lemma ~\ref{lem-N}, there is an exact
sequence
$$
\begin{CD}
0 @>>> \mc{N}_{C/Y}|_{C_1}(-\ul{0}-\ul{\infty}) @>>> \mc{N}'_{C/Y}(-y_1-y_2) 
@>>> \mc{N}_{C_2/Y}(-\ul{0}) @>>> 0
\end{CD}
$$
and an exact sequence
$$
\begin{CD}
0 @>>> \mc{N}_{C_1/Y}(-\ul{0}-\ul{\infty}) @>>>
\mc{N}_{C/Y}|_{C_1}(-\ul{0}-\ul{\infty}) 
@>>> \kappa_\infty @>>> 0
\end{CD}
$$
where $\kappa_\infty$ is the skyscraper sheaf on $C_1$ supported at
$\infty$.  Applying the long exact sequence of cohomology, using that
$h^1(C_i,\mc{N}_{C_i/Y}(-\ul{0}-\ul{\infty}))$ 
equals $0$ for $i=1,2$, and chasing
diagrams, this finally gives that $h^1(C,\mc{N}'_{C/Y}(-y_1-y_2))$
also equals $0$.  

\mni
This has two consequences.
First, this implies $h^1(C,\mc{N}_{C/Y}(-y_1-y_2))$ 
equals $0$, and thus the space of deformations of $C$ containing $y_1$
and $y_2$ is smooth by Proposition ~\ref{prop-Hilb2}.  And second, 
the map 
$$
H^0(C,\mc{N}_{C/Y}(-y_1-y_2)) \rightarrow T_{C_1,\infty}\otimes
T_{C_2,\infty}
$$
is surjective.
Thus there exist first-order deformations of $C$
containing $y_1$ and $y_2$ and smoothing the node at $\infty$.  Since
the space of deformations containing $y_1$ and $y_2$ is smooth, this
first-order deformation is the one associated to a one-parameter
deformation
$$
\mc{C} \subset \Pi \times_k Y
$$
of
$[C]$ over a smooth, pointed curve $(\Pi,0)$ (e.g., choose $\Pi$ to be
a general complete intersection curve in the smooth deformation space
containing the given Zariski tangent vector).
By Lemma ~\ref{lem-node}, for a general point $t$ of $\Pi$, $\mc{C}_t$
is a smooth, connected curve containing $y_1$ and $y_2$.  Since the
arithmetic genus of $C$ is $0$, the arithmetic genus of $\mc{C}_t$ is
also $0$, i.e., $\mc{C}_t \cong \PP^1_k$.  Let
$$
f_1:\PP^1_k \rightarrow \mc{C}_t
$$
be an isomorphism with $f_1(t_i) = y_i$ for $i=1,2$.
Because
$h^1(C,\mc{N}_{C/Y}(-y_1-y_2))$ equals $0$, by the semicontinuity
theorem also $h^1(\mc{C}_t,\mc{N}_{\mc{C}_t/Y}(-y_1-y_2))$ equals
$0$.  This implies that
$$
f_1^*T_Y \cong \OO_{\PP^1}(a_1)\oplus \dots \oplus
\OO_{\PP^1}(a_n), \text{ for integers } a_1,\dots,a_n \geq 1.
$$

\mni
Next, for every integer $a$, let $g_a:(\PP^1 \rightarrow \PP^1$ be the
morphism $z\mapsto z^a$.  Then the composition $f_a = f_1\circ g_1$ is
a morphism
$$
f_a:(\PP^1_k,t_1,t_2) \rightarrow (Y_\text{v.f.},y_1,y_2)
$$
with
$$
f_a^* T_Y = g_a^*(f_1^* T_Y) \cong \OO_{\PP^1}(a_1)\oplus \dots \oplus
\OO_{\PP^1}(a_n) \text{ for integers } a_1,\dots,a_n \geq a,
$$
namely the new integer $a_i(f_a)$ equals $a\cdot a_i(f_1)$.  Next,
choosing $a\geq 2k+1$, this implies that
$$
h^1(\PP^1,f_a^* T_Y(-(k+1)(t_1+t_2))) = 0.
$$
Applying Proposition ~\ref{prop-Hilb2} with $\PP^1\times_k Y$ in the
place of $Y$, with the graph of $f_a$ in the place of $Z_2$ and with
$Z_1=(k+1)(t_1+t_2)$ in the place of $Z$, deformations of $f_a$ map
the $k$-jet of $\PP^1$ at $t_1$, resp. at $t_2$, isomorphically to a
general $k$-jet at $y_1$, resp. at $y_2$.
\end{proof}

\mni
The following proposition is the strongest 
generalization of Proposition ~\ref{prop-vfRC} we will need.
It is
stated as a theorem about finding new
sections of a rationally connected fibration under the hypothesis that
one such section exists.
In this sense it may seem premature (and
dangerously close to circular logic), 
since
Theorem ~\ref{thm-GHS} is not yet proved.  In fact the proposition is
used in the proof of Theorem ~\ref{thm-GHS} not for the original
fibration, but only for a \emph{constant} fibration 
$$
\text{pr}_{\PP^1}: \PP^1_k \times_k Y \rightarrow \PP^1_k
$$
which obviously admits sections (constant sections).  So there is
nothing circular in the application of the proposition to the proof of
Theorem ~\ref{thm-GHS}.

\begin{prop}[Generic weak approximation] \cite{KMM}, \cite{HT06}
\label{prop-genWA}
\marpar{prop-genWA}
Let $B$ be a smooth, connected, projective curve over an algebraically
closed field $k$.  Let $\pi:U \rightarrow B$ be a smooth,
quasi-projective morphism having irreducible geometric fibers.
Assume there exists a section $s:B\rightarrow U$ mapping the generic
point of $B$ into the very free locus of the generic fiber of $\pi$.  Let
$(b_1,\dots,b_M,b'_1,\dots,b'_{M'})$ 
be distinct closed points of $B$ such that $s(b_i)$ is in
the very free locus $U_{b_i,\text{v.f.}}$ of the fiber $U_{b_i}$ for
each $i=1,\dots,M$.
Let
$k$ and $a$ be nonnegative integers.
For
each $i$, let $x_i$ be a closed point of $U_{b_i,\text{v.f.}}$ and let
there be given a curvilinear $k$-jet in $U$ at $x_i$. 
Assuming each
of these $k$-jets is a general $k$-jet at $x_i$, 
there exists a section $\sigma:B\rightarrow U$
such that
\begin{enumerate}
\item[(i)]
for each $i=1,\dots,M$, $\sigma(b_i)$ equals $x_i$,
\item[(ii)]
for each $i=1,\dots,M'$, $\sigma(b'_i)$ equals $s(b'_i)$, 
\item[(iii)]
for every invertible $\OO_B$-module $\mc{L}$ of degree $\leq a$,
$h^1(B,\mc{N}_{\sigma(B)/U}\otimes_{\OO_B}\mc{L}^\vee)$ equals $0$,
\item[(iv)]
and $\sigma$ maps the $k$-jet of $b_i$ in $B$ isomorphically to the given
$k$-jet at $x_i$ for each $i$.
\end{enumerate}
\end{prop}

\mni
In fact Hassett and Tschinkel proved much more: the result holds 
for \emph{arbitrary} $k$-jets transverse to the fibers of
$\pi$ (i.e., for $k$-jets whose associated Zariski tangent vector is
not contained in a fiber of $\pi$).  In what follows we only need the
``generic'' result, which is all we prove.  

\begin{proof}
Denote by $\Omega_\pi$ the locally free sheaf of relative
differentials of $\pi$, 
and denote by $T_\pi$ the dual locally free sheaf.
Choose a large integer $N$ and enlarge the set of pairs
$((b_i,x_i))|_{i=1,\dots,M}$ to a set $((b_i,x_i))_{i=1,\dots,N}$
having the same properties above and 
such that the collection
$(b_i)_{i=M+1,\dots,N}$ is a general collection of $N-M$ points in $B$
(this is possible because for all but
finitely many closed points of $B$, $s(b)$ is contained in
$U_{b,\text{v.f.}}$).  
By Proposition ~\ref{prop-vfRC}, applied with $k=1$, i.e., in the case
that 
$k$-jets are simply tangent directions, 
for every $i=1,\dots,N$ there exists a morphism
$$
f_i:(\PP^1,0,\infty) \rightarrow (U_{b_i,\text{v.f.}},s(b_i),x_i)
$$
such that 
$$
f_i^* T_{\pi} \cong \OO_{\PP^1}(a_1)\oplus \dots \oplus
\OO_{\PP^1}(a_n), \ \ a_1,\dots,a_n \geq 1
$$
and the tangent direction of $f_i(\PP^1)$ at $s(b_i)$ is a general
tangent direction in $T_{U_{b_i},s(b_i)}$.  But of course the tangent
space $T_{U_{b_i},s(b_i)}$ equals the normal space
$N_{s(B)/U}|_{s(b_i)}$.  Thus the tangent direction of $f_i(\PP^1)$ at
$s(b_i)$ gives a general normal direction to $s(B)$ in $U$ at
$s(b_i)$.  

\mni
Form the comb $j:C_{\text{comb}} \rightarrow U$
with handle $s(B)$ and with each
morphism $f_i$ being a tooth $L_i$ attached at $s(b_i)$. 
By Lemma ~\ref{lem-N}, $\mc{N}_{C_\text{comb}/U}|_{s(B)}$ equals the
sheaf of rational sections of $\mc{N}_{s(B)/U}$ having at most a
simple pole at each point $s(b_i)$ in a general normal direction at
$s(b_i)$.  
Assuming the integer $N$ is sufficiently large, Lemma
~\ref{lem-elemtrans} then implies that $h^1(B,
s^*\mc{N}_{C_\text{comb}/U})$ equals $0$.  Moreover, fixing an
auxiliary invertible sheaf $\mc{M}$ on $B$ of degree $g(B)+1$ and
applying Lemma ~\ref{lem-elemtrans} to
$s^*\mc{N}_{s(B)/U}(-(b'_1+\dots+b'_{M'}))\otimes_{\OO_B}
\mc{M}^\vee$, for $N$ sufficiently large also
$h^1(B,s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+\dots+b'_{M'}))\otimes_{\OO_B}
\mc{M}^\vee)$  equals $0$.

\mni
For every $i$, there is a short exact sequence
$$
\begin{CD}
0 @>>> f_i^* \mc{N}_{L_i/U_{b_i}} @>>> f_i^* \mc{N}_{L_i/U} @>>> f_i^*
\mc{N}_{U_{b_i}/U} @>>> 0.
\end{CD}
$$
Of course the normal sheaf $\mc{N}_{U_{b_i}/U}$
is just $\OO_{U_{b_i}}$ since
$U_{b_i}$ is a smooth fiber of a morphism to a curve.  Also the tangent
direction of $s(B)$ at $s(b_i)$ surjects onto the fiber of
$\mc{N}_{U_{b_i}/U}$ at $s(b_i)$.  Thus the elementary transform up of
$\mc{N}_{L_i/U}$ at $s(b_i)$ in this direction surjects onto
the elementary transform up of $\OO_\PP^1$ at $\infty$, i.e., it
surjects onto $\OO_{\PP^1}(1)$.  Thus, by Lemma ~\ref{lem-N}, there
is a short exact sequence
$$
\begin{CD}
0 @>>> f_i^* \mc{N}_{L_i/U_{b_i}} @>>> f_i^*
\mc{N}_{C_{\text{comb}}/U} @>>> \OO_{\PP^1}(1) @>>> 0.
\end{CD}
$$
Twisting 
by $\OO_{\PP^1}(-2)$ and applying the long exact
sequence of cohomology associated to the short exact sequence,
$h^1(\PP^1,f_i^*\mc{N}_{C_{\text{comb}}/U}(-\ul{0}-\ul{\infty}))$ equals $0$.
Combined with the result of the previous paragraph and joining
the two types of normal sheaf via the short exact sequence
$$
0 \rightarrow 
\oplus_{i=1}^N \mc{N}_{C_\text{comb}/U}|_{L_i}(-x_i-s(b_i))
\rightarrow 
\mc{N}_{C_\text{comb}/U}(-(x_1 + \dots + x_N)-(b'_1+\dots+b'_{M'})) 
$$
$$
\rightarrow
\mc{N}_{C_\text{comb}/U}|_{s(B)}(-(b'_1+\dots+b'_{M'}))
\rightarrow 0,
$$ 
the long exact sequence of cohomology implies both that
$$
h^1(C_{\text{comb}},\mc{N}_{C_\text{comb}/U}(-(x_1+\dots+x_N) -(b'_1+
\dots + b'_{M'}))) \text{ equals } 0,
$$
and that the map
$$
H^0(C_{\text{comb}},
\mc{N}_{C_\text{comb}/U}(-(x_1+\dots+x_N)-(b'_1+\dots+b'_{M'})))  
\rightarrow H^0(B,s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+ \dots + b'_{M'})))
$$
is surjective.  

\mni
Thus, by Proposition ~\ref{prop-Hilb2}, the space of
deformations of $C_{\text{comb}}$ containing $x_1,\dots,x_N$ and
$b'_1,\dots,b'_{M'}$ 
is smooth.  
And, by Lemma ~\ref{lem-node},
to prove there exists a deformation smoothing every node
of $C_{\text{comb}}$, it suffices to prove for every $i$ there exists
a section of $s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+\dots+b'_{M'}))$ 
whose image in 
$T_{s(B),s(b_i)}\otimes_k T_{L_i,s(b_i)}$ is
nonzero.  Of course this skyscraper sheaf $\mc{T}_{s(b_i)}$
is a quotient of the fiber
of $s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+\dots+b'_{M'}))$ 
at $b_i$.  Thus it suffices to prove for every $i$ that
$h^1(B,s^*\mc{N}_{C_\text{comb}/U}(-b_i - (b'_1 + \dots + b'_{M'})))$
equals $0$.  Recall 
the auxiliary invertible sheaf $\mc{M}$ of degree $g(B)+1$.  Because the
invertible sheaf $\mc{M}(-b_i)$ has degree $g(B)$, it is effective,
say $\OO_B(\Delta_i)$.  Thus there exists an injective $\OO_B$-module
homomorphism
$$
s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+\dots+b'_{M'}))\otimes_{\OO_B}\mc{M}^\vee
\hookrightarrow 
$$
$$
s^*\mc{N}_{C_\text{comb}/U}(
-(b'_1+\dots b'_{M'}))\otimes_{\OO_B}\mc{M}^\vee(\Delta_i) 
= 
s^*\mc{N}_{C_\text{comb}/U}(-b_i-(b'_1+\dots+b'_{M'}))
$$
with torsion cokernel.
Since
$h^1(B, s^*\mc{N}_{C_\text{comb}/U}(-(b'_1+\dots+b'_{M'}))
\otimes_{\OO_B}\mc{M}^\vee)$   
equals $0$, and since every torsion sheaf has $h^1$ equal to $0$, also
$h^1(B,s^*\mc{N}_{C_\text{comb}/U}(-b_i-(b'_1+\dots+ b'_{M'})))$
equals $0$ for every $i$. 
Therefore there exist a one-parameter family of
deformations $(\mc{C}_t)_{t\in \Pi}$
of $C_\text{comb}$ containing each of $x_1,\dots,x_M$, containing each 
of $s(b'_1),\dots,s(b'_{M'})$  
and smoothing
every node of $C_\text{comb}$, i.e., for $t$ general, $\mc{C}_t$ is smooth.

\mni
Because $\pi_U$ maps $s(B)$ to $B$ with degree $1$, also $\pi_U$ maps
$\mc{C}_t$ to $B$ with degree $1$.  Because $\mc{C}_t$ is smooth, this
means the projection $\mc{C}_t \rightarrow B$ is an isomorphism.
Therefore there exists a section $\sigma_t:B\rightarrow U$ of $\pi_U$
with image $\mc{C}_t$.  In particular, $\sigma_t(b_i) = x_i$ for every
$i=1,\dots,M$ and $\sigma_t(b'_i) = s(b'_i)$ for every $i=1,\dots,M'$.
Because
$h^1(C_\text{comb},\mc{N}_{C_\text{comb}/U}(-(x_1+\dots+x_N)))$ equals
$0$, by semicontinuity also
$h^1(B,\sigma_t^*\mc{N}_{\sigma_t(B)/U}(-(x_1+\dots+x_N)))$ equals $0$ for
$t$ general.  In particular, if $N\geq a + g(B)$, then for every
invertible sheaf $\mc{L}$ of degree $\leq a$,
$\mc{L}^\vee(x_1+\dots+x_N)$ has degree $\geq g(B)$ and thus is
effective, say $\OO_B(\Delta)$.  Therefore there exists an injective
sheaf homomorphism
$$
\sigma_t^* \mc{N}_{\sigma_t(B)/U}(-(x_1+\dots+x_N)) \hookrightarrow
\sigma_t^* \mc{N}_{\sigma_t(B)/U}(-(x_1+\dots+x_N) + \Delta) =
\sigma_t^* \mc{N}_{\sigma_t(B)/U}\otimes_{\OO_B}\mc{L}^\vee
$$
with torsion cokernel.  So, by the same type of argument as above,
$h^1(B,\sigma_t^* \mc{N}_{\sigma_t(B)/U}\otimes_{\OO_B}\mc{L}^\vee)$
equals $0$ for every invertible sheaf $\mc{L}$ of degree $\leq a$.

\mni
Finally, applying the last result when $a=(k+1)(M+M')$ and
$\mc{L}=\OO_B((k+1)(b_1+\dots + b_M + b'_1 + \dots + b'_{M'}))$, 
there exists a section
$\sigma:B\rightarrow U$ of $\pi_U$ as above and satisfying
$h^1(B,\sigma^*
\mc{N}_{\sigma(B)/U}(-(k+1)(b_1 + \dots + b_M + b'_1 + \dots +
b'_{M'})))$  equals $0$.  Therefore,
by Proposition \ref{prop-Hilb2} once more, for a general deformation
of $\sigma(B)$ containing $x_1,\dots,x_M$ and $s(b'_1), \dots,
s(b'_{M'})$, 
the $k$-jet of the curve at
each point $x_i$ and $s(b'_i)$ 
is a general curvilinear $k$-jet in $U$ at that point.
\end{proof}

\mni
The main application is to the case when $U$ equals $\PP^1\times_k Y$ where
$Y$ is a smooth, irreducible, quasi-projective $k$-scheme whose very
free locus $Y_\text{v.f.}$ is nonempty.

\begin{cor} \label{cor-genWA} 
\marpar{cor-genWA}
Every rational curve in $Y$ intersecting $Y_\text{v.f.}$ is contained
in $Y_\text{v.f.}$.  
For every integer
$k$, for every integer $a$, for every collection of distinct, closed points
$b_1,\dots,b_M$ of $\PP^1$, 
for every collection of closed points
$y_1,\dots,y_M$ of $Y_\text{v.f.}$ (not necessarily distinct), 
and for every choice of a curvilinear $k$-jet in
$Y$ at each point $y_i$, if each $k$-jet is general among curvilinear
$k$-jets at $y_i$, then there exists a morphism
$$
f:(\PP^1,b_1,\dots,b_M) \rightarrow (Y,y_1,\dots,y_M)
$$
mapping the $k$-jet of $\PP^1$ at $b_i$ isomorphically onto the
given $k$-jet at $y_i$ and such that
$$
f^*T_Y \cong \OO_{\PP^1}(a_1) \oplus \dots \oplus \OO_{\PP^1}(a_n), \
a_1,\dots,a_n \geq a.
$$
\end{cor}

\begin{proof}
Let $B=\PP^1$, let $U=B\times_k Y$ and let $\pi_B$ be the obvious
projection.  The sections of $\pi_B$ are precisely the graphs of
morphisms $f:\PP^1 \rightarrow Y$.  In particular, if $f$ is a
morphism whose image intersects $Y_\text{v.f.}$, then the section
$s=(\text{Id}_B,f)$ satisfies the hypotheses of Proposition
~\ref{prop-genWA}.  Thus, for every point $b'=b'_1$ of $\PP^1$, there
exists a section $\sigma=(\text{Id}_{\PP^1},\phi)$ with $\sigma(b') =
s(b')$ and with $h^1(B,\sigma^*\mc{N}_{\sigma(B)/U}(-2))$ equal to $0$.  In
other words, $\phi:\PP^1_k \rightarrow Y$ is a morphism with $\phi(b')
= f(b')$ and with $h^1(\PP^1,\phi^*T_Y(-2))$ equal to $0$.  Thus $\phi$
is a very free morphism whose image contains $f(b')$.  Therefore every
point in the image of $f$ is contained in the very free locus, i.e.,
every rational curve in $Y$ intersecting $Y_{\text{v.f.}}$ is
contained in $Y_{\text{v.f.}}$.  

\mni
The rest of the corollary is just a straightforward translation of
Proposition ~\ref{prop-genWA} to this context.
\end{proof}

\mni
There is one more result in this direction which is useful. The proof
is similar to the arguments above.

\begin{lem} \cite[Lemma II.7.10.1]{K} \label{lem-vanish}
\marpar{lem-vanish}
Let $C_\text{comb}$ be a comb with handle $C$ and teeth
$L_1,\dots,L_n$.  
Let $\rho:\mc{C} \rightarrow \Pi$ be a one-parameter deformation 
of $C_\text{comb}$ over a pointed curve $(\Pi,0)$ whose general fiber
$\mc{C}_t$ is smooth.
Let
$\mc{E}$ be a locally free sheaf on $\mc{C}$.  If $\mc{E}|_{L_i}$ is
ample for every $i$ and if $h^1(C,(\mc{E}|_C)\otimes_{\OO_C} \mc{M})$
equals $0$ for every invertible $\OO_C$-module $\mc{M}$ of degree
$\geq n$, then $h^1(\mc{C}_t,\mc{E}|_{\mc{C}_t})$ equals $0$ for
general $t$ in $\Pi$.
\end{lem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Ramification issues
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Ramification issues} \label{sec-ram}
\marpar{sec-ram}

\mni
The argument sketched in Section ~\ref{sec-out} and the powerful
smoothing combs technique from Section ~\ref{sec-Hilb} form 
the core of the proof of Theorem ~\ref{thm-GHS}.  
However there is a technical issue
complicating matters.   There may be codimension $1$ points
of $X$ at which the morphism $\pi:X\rightarrow B$ is not smooth.  
In other words,
finitely many scheme-theoretic fibers of $\pi$ may have irreducible
components occurring with multiplicity $\geq 1$.  
This is a well-known
issue when working with fibrations.  Although there are sophisticated
ways to deal with this (using log structures or Deligne-Mumford
stacks), for the purposes of this proof it suffices to deal with this
in a more naive manner.

\mni
In fact there may be codimension $0$ points of $X$ at which $\pi$ is
not smooth, at least if $k$ has positive characteristic.  The
hypotheses in Theorem ~\ref{thm-GHS} prevent this, but something
slightly weaker suffices.  Let $B$ be a smooth $k$-curve, let $X$ be a
reduced, finite type $k$-scheme and let $\pi:X\rightarrow B$ be a flat
morphism.  From here on, we assume the following hypothesis.

\begin{hyp} \label{hyp-pi}
\marpar{hyp-pi}
The geometric generic fiber of $\pi$ is reduced.  Equivalently, 
$\pi$ is smooth at every generic point of $X$,
cf. ~\cite[Proposition 4.6.1]{EGA4}.
This hypothesis is automatic if $\text{char}(k)$
equals $0$.  
\end{hyp}

\begin{defn} \label{defn-U}
\marpar{defn-U}
The \emph{good locus} of $\pi$ is the maximal open subscheme $U$ of
$X$ such that
$U$ is smooth and such that
for every point $b$ of $B$ the reduced scheme 
of the fiber
$\pi^{-1}(b)\cap U$ is smooth.  Denote the restriction of $\pi$ to
$U$ by $\pi_U$.  The morphism $\pi$ is \emph{good} if the good locus
equals all of $X$.
The \emph{log divisor} of $\pi$ is the Cartier divisor
$D_{\pi,\text{log}}$ of $U$ given by
$$
D_{\pi,\text{log}} := \sum_{b\in B(k)} \pi_U^*(b) - \pi_U^*(b)_{\text{red}},
$$
where $\pi_U^*(b)_{\text{red}}$ is the reduced Cartier divisor.

\mni
Since the geometric generic fiber of $\pi$ is reduced, so is the
geometric generic fiber of $\pi_U$ (or else it is empty if $U$ is
empty).  Thus the sum in the definition of the log divisor reduces 
to a sum over those finitely many closed points $b$ of
$B$ for 
which $\pi_U^*(b)$ is nonreduced. 
\end{defn}

\begin{lem} \label{lem-U}
\marpar{lem-U}
The complement of $U$ in $X$ has codimension $\geq 2$.  
If $\text{char}(k)$ equals $0$, then
the pullback
map on relative differentials
$$
\pi_U^*:\pi_U^*\Omega_{B/k} \rightarrow \Omega_{U/k}
$$
factors uniquely through the inclusion
$$
\pi_U^*\Omega_{B/k} \hookrightarrow
\pi_U^*\Omega_{B/k}(D_{\pi,\text{log}})
$$
and the cokernel
$$
\Omega_{\pi,\text{log}} :=
\text{Coker}(\pi_U^*\Omega_{B/k}(D_{\pi,\text{log}}) \rightarrow
\Omega_{U/k}) 
$$
is locally free.
\end{lem}

\begin{proof}
To construct $U$, first remove the closure of the singular locus of
the geometric generic fiber of $\pi$ and next remove the singular
locus from the reduced scheme of the finitely many singular fibers.
Both of these sets have codimension $2$ in $X$ (the first by
Hypothesis ~\ref{hyp-pi}).

\mni
The proof of the second part uses that
$\text{char}(k)=0$.  It can be checked formally locally
near every closed point $x$ of $U$.  Denote by $b$ the image
$\pi(x)$ in $B$ and denote by $D$ the
reduced structure on the irreducible component of $\pi^{-1}(b)$
containing $x$.  Since $x$ is in $U$, $D$ is a smooth Cartier
divisor in $U$.  Let $r$ be a defining equation for $D$ in $U$ and
let $t$ be a defining equation for $b$ in $B$.  Near $x$, $\pi^*(b) =
m D +$ other terms.  Thus, in $\widehat{\OO}_{U,x}$, 
$$
\pi^* t = a_m r^m + a_{m+1} r^{m+1} + \dots
$$
where $a_m$ is a unit.  Because $\text{char}(k)=0$, the power series
$$
u = \sqrt[m]{a_m + a_{m+1} r + \dots}
$$
is a well-defined unit in $\widehat{\OO}_{U,x}$.  Thus, after
replacing $r$ by $ur$, there exists a regular system of parameters
$r,r_2,\dots,r_n$ for $\widehat{\OO}_{U,x}$ with respect to which 
the pullback homomorphism
$\pi^*$ is the unique local homomorphism 
with 
$$
\pi^* t = r^m \text{ and } \pi^*(dt) = mr^{m-1} dr.
$$ 
In particular, the pullback homomorphism
$\pi^*$ on relative differentials 
locally factors through $\pi^*\Omega_B((e-1)D) =
\pi^*\Omega_B(D_{\pi,\text{log}})$.  Moreover 
the stalk of the cokernel of the induced homomorphism
is the free module generated by $dr_2,\dots,dr_n$.  
\end{proof}

\mni
The locally free quotient $\Omega_{\pi,\text{log}}$ of $\Omega_\pi$ is
called the sheaf of 
\emph{log relative differentials}.  Of course it equals the
torsion-free quotient of $\Omega_\pi$.  But its true importance comes
from the following lemma:
given a base change $V\rightarrow B$ for which the
normalized 
fiber product $\widetilde{U\times_B V}$ is smooth over $V$, the sheaf
$\Omega_{\widetilde{U\times_B V}/V}$
of relative differentials of $\widetilde{U\times_B V}$ over $V$ equals the
pullback of $\Omega_{\pi,\text{log}}$.  Thus the relative
deformation theory of
$\widetilde{U\times_B V}$ over $V$ is already captured by the sheaf
$\Omega_{\pi,\text{log}}$ on $U$.  
Before stating the lemma precisely, there
is some setup.

\mni
Let 
$$
\pi:U \rightarrow B, \ \ \varpi:V \rightarrow B
$$
be two good morphisms with respective log divisors
$D_{\pi,\text{log}}$ and $E_{\varpi,\text{log}}$.  
Let $b$ be a closed point of $B$.  
Let $D$ be a prime divisor of $U$ in
$\text{Supp}(D_{\pi,\text{log}}) \cap \pi^{-1}(b)$, and let $E$ be a
prime divisor of $V$ in $\text{Supp}(E_{\varpi,\text{log}})\cap
\varpi^{-1}(b)$.  
Denote by $m_D - 1$, resp. $m_E - 1$, the coefficient of $D$ in 
$D_{\pi,\text{log}}$, resp. the coefficient of $E$ in
$E_{\varpi,\text{log}}$.   
The \emph{normalized fiber product} of $U$ and $V$ along $D$ and $E$
is the normalization
$\widetilde{U\times_B V}$ of $U\times_B V$ along $D\times_{\{b\}} E$.
Denote by 
$$
\text{pr}_U: U\times_B V \rightarrow U, \ \ \text{pr}_V: U\times_B V
\rightarrow V
$$
the two projections, and denote by
$$
\widetilde{\text{pr}}_U: \widetilde{U\times_B V} \rightarrow U, \ \
\widetilde{\text{pr}}_V: \widetilde{U\times_B V} \rightarrow V
$$
the compositions with the normalization morphism.  Denote by $\text{Exc}$
the exceptional locus of the morphism, i.e.,
$$
\text{Exc} := (\widetilde{\text{pr}}_U^{-1}(D) \cap
\widetilde{\text{pr}}_V^{-1}(E))_{\text{reduced}}.
$$  

\mni
From this point forward we explicitly assume that $\text{char}(k)$ equals
$0$.

\begin{hyp} \label{hyp-char0}
\marpar{hyp-char0}
The algebraically closed ground field $k$ has characteristic $0$.  In
particular, this implies Hypothesis ~\ref{hyp-pi}.
\end{hyp}

\mni
The sheaves $\Omega_\pi$ and $\Omega_{\pi,\text{log}}$ agree on a
dense open subset of $U$, namely
$U-\text{Supp}(D_{\pi,\text{log}})$.  
Because $\widetilde{\text{pr}}_V$ and
$\text{pr}_V$ are isomorphic over a dense open subset of $V$ (namely
$V-E$) 
also
$\Omega_{\widetilde{\text{pr}}_V}$ agrees with
$\widetilde{\text{pr}}_V^* \Omega_\pi$ on a dense open subset of
$\widetilde{U\times_B V}$.  Therefore also $\Omega_\varpi$ agrees with
$\widetilde{\text{pr}}_V^* \Omega_{\pi,\text{log}}$ on a dense open
subset of $\widetilde{U\times_B V}$. 

\begin{lem} \label{lem-etale}
\marpar{lem-etale}
The morphism
$$
\widetilde{\text{pr}}_V:\widetilde{U\times_B V} \rightarrow V
$$
is smooth at every point of $\text{Exc}$ if
and only if $m_D$ divides $m_E$.  In this case the reduced
normalization equals the blowing up of $U\times_B V$ along the closed
subscheme $\text{pr}_U^{-1}(D) \times \text{pr}_V^{-1}((m_E/m_D) E)$ and 
$\text{Exc}$ is contained in the
maximal open neighborhood of $\widetilde{U\times_B V}$ on
which $\Omega_{\widetilde{\pi}}$ agrees with $(\widetilde{\text{pr}_V})^*
\Omega_{\pi,\text{log}}$.
\end{lem}

\begin{proof}
This is proved in much the same way as the second part of Lemma
~\ref{lem-U}.  For every closed point $x$ of $U$ and $y$ of $V$ with
common image point $b=\pi(x) = \varpi(y)$, there exist a regular
system of parameters
$(r,r_2,\dots,r_n)$ for $\widehat{\OO}_{U,x}$,
resp. $(s,s_2,\dots,s_p)$ for $\widehat{\OO}_{V,y}$, 
and a regular parameter $t$ for $\widehat{\OO}_{B,b}$ such
that
$$
\pi^* t = r^{m_D} \text{ and } \varpi^* t = s^{m_E},
$$
and thus,
$$
\widehat{\OO}_{U\times_B V, (x,y)} =
k\Sem{r,r_2,\dots,r_n,s,s_2,\dots,s_p}/\langle r^{m_D} - s^{m_E}
\rangle.
$$
Denoting by $m$ the greatest common factor of $m_D$ and $m_E$,
the stalk of the normalization equals
$$
k\Sem{u,r,r_2,\dots,r_n,s,s_2,\dots,s_p}/\langle
r-u^{m_E/m},s-u^{m_D/m} \rangle.
$$
Thus it is formally smooth as a $k\Sem{s,s_2,\dots,s_p}$-algebra if
and only if $m_D/m$ equals $1$, i.e., if and only if $m_D$ divides
$m_E$.  In this case it is easy to see that the normalization is the
blowing up at the ideal $\langle s, t^{m_E/m_D} \rangle$ and it is
easy to see that the module of relative differentials is the free
module generated by $dr_2,\dots,dr_n$, i.e., it is the pullback of
$\Omega_{\pi,\text{log}}$.  
\end{proof}

\mni
Having introduced the ideas need to deal with the ramification issues,
we now resume the proof of Theorem ~\ref{thm-GHS}.  So from this point on
we assume the following.

\begin{hyp} \label{hyp-GHS}
\marpar{hyp-GHS}
The following hypotheses of Theorem ~\ref{thm-GHS} hold.
\begin{enumerate}
\item[(i)]
The algebraically closed ground field $k$ has characteristic $0$,
i.e., Hypothesis ~\ref{hyp-char0} holds.
\item[(ii)]
The
smooth $k$-curve $B$ is projective and connected.
\item[(iii)] 
The reduced, finite
type $k$-scheme
$X$ is normal and projective.
\item[(iv)]
And the geometric generic fiber of the
flat morphism $\pi:X\rightarrow B$ is a normal, integral scheme whose
smooth locus contains a very free curve.
\end{enumerate}
\end{hyp}

\begin{defn} \label{defn-preflex}
\marpar{defn-preflex}
A \emph{log preflexible curve} is a connected, smooth, proper curve
$C\subset U$ such that 
\begin{enumerate}
\item[(i)]
the generic fiber of $C$ over $B$ is contained in the very free locus
of the generic fiber of $U$ over $B$,
\item[(ii)]
$\pi_U(C)$ equals $B$, 
\item[(iii)]
and every intersection point of $C$ with
$\text{supp}(D_{\pi,\text{log}})$ is 
transverse, i.e., the tangent direction of $C$ at the intersection
point is not contained in the tangent space of
$\text{supp}(D_{\pi,\text{log}})$.  
\end{enumerate}
A \emph{linked log preflexible
  curve} is a $B$-morphism from a linked curve $j:C_{\text{link}}
\rightarrow U$ such that the handle $C$ is log preflexible and for
every link $L_i$
the image in $B$ of $L_i$ is disjoint from the image
in $B$ of $D_{\pi,\text{log}}$.

\mni
A log preflexible curve $C$ is a \emph{log flexible curve} if 
$$
h^1(C, T_{\pi,\text{log}}|_C) \text{ equals } 0, \text{ where } 
T_{\pi,\text{log}} :=
\textit{Hom}_{\OO_U}(\Omega_{\pi,\text{log}},\OO_U).
$$
A linked log preflexible curve is a \emph{linked log flexible curve}
if
$$
h^1(C_{\text{link}},j^*T_{\pi,\text{log}}) \text{ equals } 0.
$$
\end{defn}

\mni
Figure ~\ref{fig-fiber1d} shows a log preflexible curve intersecting a
singular, but reduced fiber.  Because the curve is transverse to the
fiber, the morphism to $B$ is unramified.  On the other hand, Figure
~\ref{fig-fiber1e} shows a log preflexible curve intersecting a
nonreduced fiber -- the middle component has multiplicity $2$.
Necessarily the map from the curve to $B$ is ramified.

%FIGURE fiber1d
\begin{figure}
\begin{picture}(330,270)
\put(0,10){\makebox(330,240){\includegraphics{fiber1d_small.eps}}}
\end{picture}
\caption{Where a log preflexible curve intersects a reduced, possibly singular,
  fiber the map from the curve to $B$ is unramified.}
\label{fig-fiber1d}
\end{figure}

%FIGURE fiber1e
\begin{figure}
\begin{picture}(330,270)
\put(0,10){\makebox(330,240){\includegraphics{fiber1e_small.eps}}}
\end{picture}
\caption{Where a log preflexible curve intersects a nonreduced fiber
  -- e.g., the middle component has multiplicity 2 -- the map from the
  curve to $B$ is necessarily ramified.}
\label{fig-fiber1e}
\end{figure}


\begin{lem} \label{lem-Bertini}
\marpar{lem-Bertini}
There exists a log preflexible curve $C$.  In fact, every intersection of
$X$ with $\text{dim}(X)-1$ general hyperplanes is a log preflexible curve.
\end{lem}

\begin{proof}
Because $X-U$ has codimension $2$ in $X$, a general complete
intersection curve in $X$ is disjoint from $X-U$, i.e., it is
contained in $U$.  By hypothesis, $U_{\pi,\text{v.f.}}$ is a dense
open subset of 
$U$ and thus a general complete intersection curve intersects this
open.  Finally, by Bertini's theorem a general complete intersection
curve in $U$ is smooth and intersects
$\text{supp}(D_{\pi,\text{log}})$ 
transversally.
\end{proof}

% \begin{prop}\cite[\S II.7]{K} \cite[Lemma 2.6]{GHS} \label{prop-combs}
% \marpar{prop-combs}
% \begin{enumerate}
% \item[(i)]
% The restriction
% $\mc{N}_j|_C$ equals the sheaf of rational sections of $\mc{N}_{j_0}$
% having at most a simple pole at each point $p_i$ in the normal
% direction determined by $dj(T_{L_i,p_i})$.  
% \item[(ii)]
% For given $j_0:C\rightarrow U$, there exists an integer $n_0$ such that
% if the number $n$ of teeth of the comb
% $(C_{\text{comb}},\pi_{C,\text{comb}})$ is $\geq n_0$, if the
% attachment points $p_i$ are general points of $C$, if $j$ maps each
% tooth $L_i$ to a very free curve (or even just a ``free curve'')
% in a fiber of $\pi_X$, and if the
% normal directions $dj(T_{L_i,p_i})$ are general one-dimensional
% subspaces of $\mc{N}_{j_0}|_{p_i}$, 
% then there exists a one-parameter deformation 
% $(\rho,\pi_{\mc{C}}):\mc{C} \rightarrow \Pi\times_k B$
% of
% $(C_{\text{comb}},\pi_{C,\text{comb}})$ and an extension of $j$ to the
% deformation such that $\mc{C}_t$ is smooth for all but finitely many
% closed points of $\Pi$.  
% \item[(iii)]
% Moreover, given a locally free sheaf $\mc{E}$ on $\mc{C}$, if each
% $\mc{E}|_{L_i}$ is ample and if $h^1(C,\mc{E}|_{C}\otimes_{\OO_C} M)$
% equals $0$ for a general invertible sheaf $M$ of degree $\geq n$ on
% $C$, then for all but finitely many closed points $t$ of $\Pi$,
% $h^1(\mc{C}_t, \mc{E}|_{\mc{C}_t})$ equals $0$.
% \end{enumerate} 
% \end{prop}

% \begin{proof}
% Assertion (i) can be checked locally at
% each point $p_i$.  Locally near $p_i$, $C_{\text{comb}} \cong X$ is
% formally isomorphic to the union of the two axes inside a $2$-plane
% inside an $n$-plane, i.e., the subscheme of $\AAA^n_k$ with ideal
% $I_{\text{comb}} = \langle x_1x_2, x_3,\dots,x_n \rangle$.  The
% component $C$ corresponds to just one of the axes, e.g., the subscheme
% of $\AAA^n_k$ with ideal $I=\langle x_2,x_3,\dots,x_n \rangle$.  Thus
% it is clear that $I_{\text{comb}}/I\cdot I_{\text{comb}}$ is the
% submodule of $I/I^2$ of elements whose fiber at $0$ is contained in
% the annihilator of the tangent direction $(0,1,0,\dots,0)$, i.e., the
% annihilator of $dj(T_{L_i,p_i})$.  Dualizing gives (i).

% \mni
% For any locally free sheaf $\mc{N}$ on $C$, e.g., $\mc{N} =
% \mc{N}_{j_0}$, there exists an integer $n_0$ such that for every
% $n\geq n_0$,
% if $\mc{N}'$ is the sheaf of rational sections of $\mc{N}$ having at
% worst a pole at each of $n$ general points $p_i$ of $C$ in the direction of
% a general one-dimensional subspace of $\mc{N}|_{p_i}$, then
% $h^1(C,\mc{N}')$ equals $0$.  Indeed, 
% there exists 
% an effective, reduced, ample divisor $D$ such that $h^1(C,\mc{N}(D))$ equals
% $0$.  Now take $n = \text{rank}(\mc{N})\cdot \text{deg}(D)$.  
% The sheaf $\mc{N}(D)$ can be thought of as iteratively forming the
% sheaf of rational sections with a simple pole at each point $p$ of
% $D$ in the direction of a one-dimensional subspace of the fiber at
% $p$, as the one-dimensional subspaces vary over the spans of a basis
% of the fiber.  Since vanishing of $h^1$ is open in a family of
% sheaves, the conclusion also holds for a general choice of $n$ points
% general one-dimensional subspaces of the fiber at each point.

% \mni
% In
% particular, applying this in the case that $\mc{N} = \mc{N}_{j_0}$,
% $h^1(C,\mc{N}_j|_{C_0})$ equals $0$.  Because each tooth $L_i$ is very
% free, also $h^1(L_i,\mc{N}_j|_{L_i}(-p_i))$ equals $0$ (this also
% holds if the tooth is free).  Thus, applying the long exact sequence
% of cohomology to the short exact sequence
% $$
% \begin{CD}
% 0 @>>> \oplus_{i=1}^n \mc{N}_j|_{L_i}(-p_i) @>>> \mc{N}_j @>>>
% \mc{N}_j|_C @>>> 0
% \end{CD}
% $$
% gives that $h^1(C_{\text{comb}},\mc{N}_j)$ equals $0$.  Thus, by the
% deformation theory of the Hilbert scheme, $[C_{\text{comb}}]$ is a
% smooth point of the Hilbert scheme (or Kontsevich space, etc.).
% Moreover, after replacing $n_0$ by $n_0+1$, if $n\geq n_0$, then for
% every tooth $L_i$, denoting by $C_i$ the union of all components of
% $C_{\text{comb}}$ other than $L_i$, then $h^1(C_i,\mc{N}_{C_i/X})$
% equals $0$.  Thus, applying the long exact sequence of cohomology to
% the short exact sequence
% $$
% \begin{CD}
% 0 @>>> \mc{N}_j|_{L_i}(-p_i) \oplus \mc{N}_{C_i/X} @>>> \mc{N}_j @>>>
% T_{C_i,p_i} \otimes_k T_{L_i,p_i} @>>> 0
% \end{CD}
% $$
% a general first-order deformation of $[C_{\text{comb}}]$ 
% deforms the node $p_i$.
% Since there are only finitely many nodes $p_i$, a general first-order
% deformation of $C$ deforms every node $p_1,\dots,p_n$.  Since the
% Hilbert scheme is smooth at $[C_{\text{comb}}]$, this first-order
% deformation is 
% contained in the tangent space at $0=[C]$ for a smooth curve $(\Pi,0)$
% in the Hilbert scheme.  Letting $\mc{C}$ be the pullback to $(\Pi,0)$
% of the universal family of curves in $X$, $\mc{C} \rightarrow
% \Pi\times_k X$
% is a one-parameter
% deformation of $C_{\text{comb}}$ whose general member is smooth.

% \mni
% The third part is similar to the second, cf. \cite[Lemma II.7.10.1]{K}.
% \end{proof}

\mni
An important consequence of the smoothing combs technique is the
following result.  

\begin{prop} \label{prop-logflex}
\marpar{prop-logflex}
There exists a log flexible curve in $X$.  In fact,
for every comb in $X$ with log preflexible
handle $C$ and with sufficiently many very free teeth in fibers of $\pi_U$
attached at
general points of $C$ and with general tangent directions, there exists a
one-parameter deformation of the comb whose general member is a log
flexible curve.
\end{prop}

\begin{proof}
By hypothesis, $C$ intersects the very free locus
$U_{\pi,\text{v.f.}}$ of the morphism $\pi_U$.  By the same argument
as in the proof of Proposition ~\ref{prop-vfRC}, $U_{\pi,\text{v.f.}}$
is open.  Therefore all but finitely many points of $C$ are contained
in $U_{\pi,\text{v.f.}}$.  By Proposition ~\ref{prop-vfRC} applied to
$1$-jets, i.e., to tangent directions, for each such point $c$ there
exists a very free rational curve in $U_{\pi_U(c)}$ containing $c$ and
whose tangent direction at $c$ is a 
general tangent direction in $U_{\pi_U(c)}$.

\mni
Let $C_\text{comb}$ be a comb obtained by attaching to $C$ a number
of teeth $L_1,\dots,L_N$ as in the previous paragraph at general
points of $C$ (in particular, points where $C\rightarrow B$ is unramified)
and with general tangent directions in $U_{\pi_U(c)}$.  These tangent
directions are the same as normal directions to $C$ in $U$.  By the
same argument as in the proof of Proposition ~\ref{prop-genWA}, if $N$
is sufficiently large there is a one-parameter deformation
$$
\mc{C} \subset \Pi \times_k U
$$
of $C_\text{comb}$ such that $\mc{C}_t$ is smooth for general $t$ in
$\Pi$.  The properties (i), (ii) and (iii) of Definition
~\ref{defn-preflex} are all open properties and hold for $\mc{C}_0 =
C_\text{comb}$, thus also hold for $\mc{C}_t$
so long as $t$ is general.   

\mni
For each tooth $L_i$ in a fiber $U_{b_i}$, 
$T_{\pi,\text{log}}|_{L_i}$ equals $T_{U_{b_i}}|_{L_i}$.  Since $L_i$
is very free, this is an ample locally free sheaf.  Thus, by Lemma
~\ref{lem-vanish} with the pullback of $T_{\pi,\text{log}}$ in the
place of $\mc{E}$, we have that 
$h^1(\mc{C}_t,T_{\pi,\text{log}}|_{\mc{C}_t})$ equals $0$ for $t$ a
general point of $\Pi$.  Therefore, for $t$ a general point of $\Pi$,
$\mc{C}_t$ is a log flexible curve.
\end{proof}

% [HERE, Make the comb smoothing argument precise, then do generic weak
% approximation, then use that to deduce existence of very free teeth
% with given, general tangent directions.]

% \begin{cor}[Generic weak approximation] \cite{KMM}, \cite{HT06}
% \label{cor-genWA}
% \marpar{cor-genWA}
% Assume there exists a section $s:B\rightarrow U$.  Let $b_1,\dots,b_M$
% be distinct closed points of $B$ such that each $s(b_i)$ is in
% the very free locus $U_{b_i,\text{v.f.}}$ of the fiber $U_{b_i}$.  Let
% $e\geq 1$ be an integer.
% For
% each $i$, let $x_i$ be a closed point of $U_{b_i,\text{v.f.}}$ and let
% there be given an $(e-1)$-jet of a smooth curve at $x_i$.
% There exists a section $\sigma:B\rightarrow U$
% such that
% \begin{enumerate}
% \item[(i)]
% for each $i$, $\sigma(b_i)$ equals $x_i$, 
% \item[(ii)]
% assuming the $(e-1)$-jet is a general $(e-1)$-jet at $x_i$, $\sigma$
% maps the $(e-1)$-jet of $b_i$ in $B$ isomorphically to the given
% $(e-1)$-jet.
% \end{enumerate}
% Moreover, given an integer $a$, the section can be chosen so that for
% every invertible sheaf $\mc{L}$ on $B$ of degree $\leq a$
% $$
% h^1(B,\sigma^*T_{\pi,\text{log}}\otimes_{\OO_B}\mc{L}^\vee) \text{
%   equals } 0.
% $$
% \end{cor}

% \begin{proof}
% Here is the point.  By the definition of the very free locus, 
% for each $i$ there exists a morphism
% $$
% f_i:(\PP^1,0,\infty) \rightarrow (U_{b_i,\text{v.f.}},s(b_i),x_i)
% $$
% such that 
% $$
% f_i^* T_{\pi,\text{log}} \cong \OO_{\PP^1}(a_1)\oplus \dots \oplus
% \OO_{\PP^1}(a_n), \ \ a_1,\dots,a_n \geq 1.
% $$
% Then, for every positive integer $e \geq 1$, the composition of $f_i$ with
% the morphism
% $$
% (\PP^1,0,\infty) \rightarrow (\PP^1,0,\infty), \ [T_0,T_1] \mapsto
% [T_0^e,T_1^e]
% $$
% is a new morphism $f_i$ such that
% $$
% f_i^* T_{\pi,\text{log}} \cong \OO_{\PP^1}(a_1)\oplus
% \OO_{\PP^1}(a_n), \ \ a_1,\dots,a_n \geq e.
% $$
% Thus, the pullback of $T_{\pi,\text{log}}$ can be made arbitrarily
% large.  
% Now choose a large number $f_1,\dots,f_M,f_{M+1},\dots,f_N$ of
% such morphisms including the original morphisms connecting $s(b_i)$
% and $x_i$ for each $i=1,\dots,M$, i.e., increase $M$ by choosing some
% additional points $b_i$ and points $x_i$.
% Form the comb $j:C_{\text{comb}} \rightarrow U$
% with handle $\text{Image}(s)$ and with each
% morphism $f_i$ being a tooth $L_i$ attached at $s(b_i)$. 
% Because $h^1(\PP^1,f_i^*
% T_{\pi,\text{log}}(-e-1))$ is $0$, by the same long exact sequence as in
% the proof of Proposition ~\ref{prop-combs},
% $h^1(C_{\text{comb}},\mc{N}_j(-e(x_1 + \dots + x_N)))$ equals $0$.
% With some thought, 
% this vanishing implies
% that not only is the Hilbert scheme smooth at
% $[C_\text{comb}]$, but also the flag Hilbert scheme parameterizing
% curves together with 
% the $(e-1)$-jet of the curve at $N$ distinct smooth points of the
% curve is smooth 
% over the scheme parameterizing curvilinear $(e-1)$-jets at each of $N$
% points of $U$ at the point parameterizing $C_\text{comb}$ and the
%   $N$-points $x_1,\dots,x_N$.  
% Thus there exist deformations of $C_{\text{comb}}$
% smoothing all the nodes, containing the points $x_1,\dots,x_N$, and
% having prescribed \emph{general} curvilinear $(e-1)$-jet at each point
% $x_i$.   

% \mni
% Finally, given an integer $a$, choosing $e$ so that $eN-a \geq 2g-1$, 
% for every invertible sheaf $\mc{L}$ of degree $\leq a$, Riemann-Roch
% implies there exists an effective divisor $\Delta$ in $B$ such that 
% $$
% \mc{L}^\vee \cong \OO_B(-e(b_1 + \dots + b_N) + \Delta).
% $$
% Thus there is an injective sheaf map
% $$
% \sigma^* T_{\pi,\text{log}}(-e(b_1+\dots+b_N)) \hookrightarrow
% \sigma^* T_{\pi,\text{log}}(-e(b_1 + \dots + b_N) + \Delta) \cong
% \sigma^* T_{\pi,\text{log}}\otimes_{\OO_B} \mc{L}^\vee
% $$
% whose cokernel is a torsion sheaf $\mc{T}$. 
% The vanishing result from the last paragraph for $C_\text{comb}$
% implies that
% $$
% h^1(B,\sigma^* T_{\pi,\text{log}}(-e(b_1+ \dots + b_N))) \text{ equals
% } 0.
% $$
% Also, because $\mc{T}$ is a torsion sheaf on a curve, $h^1(B,\mc{T})$
% equals $0$.  Therefore, by the long exact sequence of cohomology,
% $$
% h^1(B,\sigma^* T_{\pi,\text{log}}\otimes_{\OO_B}\mc{L}^\vee) \text{
%   equals } 0.
% $$
% \end{proof}

\mni
Because the fibers of $\pi$ are rationally connected, every log
preflexible curve, resp. log flexible curve, extends to a linked log
preflexible curve, resp. linked log flexible curve.

\begin{lem} \label{lem-linkflex}
\marpar{lem-linkflex}
For every linked curve $C_{\text{link}}$ such that each point $b_i =
\pi_{C,\text{link}}(L_i)$ is disjoint from
$\pi_U(D_{\pi,\text{log}})$,
and for every $B$-morphism $j_0:C\rightarrow X$ mapping $C$
isomorphically to a log preflexible curve, resp. log flexible curve,
and mapping each fiber $C_{b_i}$ into the very free locus
$U_{\pi,\text{v.f.}}$ of $\pi_U$, 
there exists a $B$-morphism $j:C_\text{link} \rightarrow X$
which is linked log preflexible, resp. linked log flexible, and
restricting to $j_0$ on $C$.
\end{lem}

\begin{proof}
Let $L_i$ be a link of $C_{\text{link}}$.  Let $L_i$ intersect $C$ in
$m$ points $t_1,\dots,t_m$ contained in the fiber over a general point
$b_i$ of $B$.
Let $x_1,\dots,x_m$ be the images $j(t_1),\dots,j(t_m)$ in
$U_{b_i,\text{v.f.}}$.  
By Corollary \ref{cor-genWA}, there exists a morphism
$$
j_i:(L_i,t_1,\dots,t_m) \rightarrow ((U_{b_i,\text{v.f.}},x_1,\dots,x_m)
$$
such that 
$$
j_i^* T_{U_{b_i}} \cong \OO_{\PP^1}(a_1) \oplus \dots \oplus
\OO_{\PP^1}(a_n) \text{ for integers } a_1,\dots, a_n \geq m-1.
$$
Because of this,
$$
h^1(L_i,j_i^* T_{U_{b_i}}(-(t_1+\dots+t_m))) \text{ equals } 0.
$$

\mni
Define $j:C_\text{link} \rightarrow U$ to be the unique morphism 
restricting to $j_0$ on $C$ and restricting to $j_i$ on each link
$L_i$.  Because $j_i(t_k) = j_0(t_k)$ for every link $L_i$ and for
every node $t_k$ contained in $L_i$, this morphism is defined.  It is
clearly log preflexible.  

\mni
Next assume that $j_0$ is log flexible.  The claim is that $j$ is also
log flexible.  To see this, consider the short exact sequence
$$
\begin{CD}
0 @>>> \oplus_i j_i^* T_{U_{b_i}}(-C_{b_i}) @>>> j^*
T_{\pi,\text{log}} @>>> j_0^* T_{\pi,\text{log}} @>>> 0
\end{CD}.
$$  
By the hypothesis that $j_0$ is log flexible, 
the third term has vanishing $h^1$.  And by the construction of $j_i$,
 $j_i^* T_{U_{b_i}}(-C_{b_i})$, i.e., $j_i^*
T_{U_{b_i}}(-(t_1 + \dots + t_m))$, has vanishing $h^1$.  Thus, by the
long exact sequence of cohomology, also
$h^1(C_{\text{link}},j^* T_{\pi,\text{log}})$ equals $0$.  Therefore
$j:C_\text{link} \rightarrow U$ is a linked log flexible curve.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Existence of log deformations
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Existence of log deformations} \label{sec-exist}
\marpar{sec-exist}

\mni
There is a definition of one-parameter deformation that takes the
divisor $D_{\pi,\text{log}}$ into account.  Unfortunately, not every
curve over $B$ admits a log deformation specializing to a section
curve, e.g., \'etale covers of $B$ are rigid.  However, after
attaching a sufficient number of links, the linked curve does admit a
log deformation specializing to a section curve.

\begin{defn} \label{defn-Ddefo}
\marpar{defn-Ddefo}
Let $(C_{\text{link}},\pi_{C,\text{link}})$ be a linked curve with
handle $C$.  Let $D_C\subset C$ be an effective, reduced, 
Cartier divisor contained
in the smooth locus of $C_\text{link}$.  A \emph{one-parameter log
  deformation} of $(C_{\text{link}},\pi_{C,\text{link}},D_C)$ is a
one-parameter deformation of $(C_{\text{link}},\pi_{C,\text{link}})$,
$$
(\rho,\pi_{\mc{C}}):\mc{C} \rightarrow \Pi \times_k B
$$
together with an effective Cartier divisor $D_{\mc{C}} \subset \mc{C}$
such that 
\begin{enumerate}
\item[(i)]
the pullback of $D_{\mc{C}}$ to $\mc{C}_0 = C_{\text{link}}$ equals
$D_C$
\item[(ii)]
and $\pi_{\mc{C}}(D_{\mc{C}})$ equals $\pi_C(D_C)$, i.e., $D_{\mc{C}}$
is vertical over $B$.
\end{enumerate}
\end{defn}

\begin{lem} \label{lem-defo}
\marpar{lem-defo}
For every finite morphism of smooth, projective curves
$\pi_C:C\rightarrow B$ and for every effective, reduced, Cartier
divisor $D_C$ of $C$, after attaching sufficiently many links to $C$
over general points of $B$, there exists a one-parameter log
deformation specializing to a section curve.
\end{lem}

\begin{proof}
For all sufficiently positive integers $e$, 
for a general morphism $g:C\rightarrow \PP^1$ of degree $e$, the
induced morphism $(\pi_C,g):C\rightarrow B\times_k \PP^1$ is
unramified and is injective except for finitely many double points,
none of which intersects the image of $D_C$.  Denote by $\Sigma
\rightarrow B\times_k \PP^1$ the blowing up along the finitely many
double points of $(\pi_C,g)(C)$.  Then there is a $B$-morphism
$h:C\rightarrow \Sigma$ which is an embedding.  

\mni
For each point $p$ of $D_C$, denote by $m_p$ the multiplicity of $p$
in the Cartier divisor $\pi_C^*(\pi_C(p))$.
Denote by $\nu_p:\Sigma'_p \rightarrow
\Sigma$ the $m_p$-fold iterated blowup of $\Sigma$ first at $p$, then
at the image of $p$ in the strict transform of $h(C)$, etc.  Denote by
$E_p$ the final exceptional divisor of this sequence of blowups.  The
point of this construction is that
the strict transform of $h(C)$ intersects $E_p$ at $p$, and $E_p$
occurs with multiplicity $m_p$ in the Cartier divisor $\Sigma'_p
\times_B \{\pi_C(p)\}$.  
Denote by
$\nu:\Sigma' \rightarrow \Sigma$ the fiber product over all points $p$
in $D_C$ of $\nu_p:\Sigma'_p \rightarrow \Sigma$.  Denote by $E$ the
Cartier divisor in $\Sigma'$ being the
sum over all $p$ of the pullback of $E_p$ from $\Sigma'_p$.  
Denote by
$\pi_{\Sigma'}: \Sigma' \rightarrow B$ the composition of
$\Sigma'\rightarrow \Sigma \rightarrow B\times_k \PP^1$ with $\text{pr}_B$.
Denote by
$h':C\rightarrow \Sigma'$ the strict transform of $h(C)$.  The point
of this construction is that $E$ is a Cartier divisor in $\Sigma'$
which is vertical over $B$ and such that
$h^*E$ equals $D_C$.

\mni
Denote by $d$ the degree of $\pi_C$ and let $t_1,\dots,t_d$ be closed
points of $\PP^1$ such that the Cartier divisor
$B \times_k \{t_1,\dots,t_d \}$ of 
$B\times_k \PP^1$ is disjoint from all double points of $(\pi_C,g)(C)$
and disjoint from $(\pi_C,g)(D)$.  Denote by $T$ the strict transform
of $B\times_k \{t_1,\dots,t_d\}$ in $\Sigma'$.  Form the
invertible sheaf $\OO_{\Sigma'}(h'(C) - T)$ and the pushforward
$\mc{E} := \pi_{\Sigma',*}\OO_{\Sigma'}(h'(C)-T)$ on $B$.  
Because $\pi_{\Sigma'}$ is flat and because $\OO_{\Sigma'}(h'(C)-T)$
is locally free, $\mc{E}$ is torsion-free.  For every point
$b$ in $B - \pi_C(D_C)$, 
$\Sigma'_b$ is isomorphic to $\PP^1$ (via the projection $\Sigma'
\rightarrow B\times_k \PP^1 \rightarrow \PP^1$).  And $\Sigma'_b \cap
h'(C)$ and $\Sigma'_b \cap T$ are divisors of the same degree $d$.
Thus $\OO_{\Sigma'}(h'(C)-T)|_{\Sigma'_b}$ is isomorphic to
$\OO_{\Sigma'_b} \cong \OO_{\PP^1}$.  Therefore $\mc{E}|_b$ is
isomorphic to $H^0(\Sigma'_b,\OO_{\Sigma'_b})$, which is
one-dimensional.  Therefore $\mc{E}$ is an invertible sheaf.  

\mni
By Riemann-Roch and
Serre duality, for every sufficiently large degree, for a 
general effective divisor $\Delta$ on $B$ of that degree,
$\mc{E}\otimes_{\OO_B}\OO_B(\Delta)$ is
globally generated.  Choose $\Delta$ to be disjoint from $\pi_C(D_C)$
and from the image in $B$ of the finitely many intersection points of
$h'(C)$ and $T$.  
Since $\mc{E}\otimes_{\OO_B} \OO_B(\Delta)$ is globally generated, 
there exists a section which is
nonzero at every point of $\Delta$.
Of course a nonzero section of this sheaf (up to scaling)
is precisely the same thing as a divisor $V$ on $\Sigma'$ such that
$$
h'(C) + \pi_{\Sigma'}^* \Delta \sim T + V.
$$
For $b$ in $B-\pi_C(D_C)$, if the section is nonzero at $b$  
then $V$ does not intersect
$\Sigma'_b$.  The same does not necessarily hold for points $b$ of
$\pi_C(D_C)$ since $b$ may lie in the support of
$R^1\pi_{\Sigma',*}\OO_{\Sigma'}(h'(C)-T)$.
Therefore 
$V$ is a sum of finitely many irreducible  components of fibers
of $\pi_{\Sigma'}$ (possibly with multiplicity) 
lying over points not in $\Delta$.

\mni
The linked curve $(C_{\text{link}},\pi_{C,\text{link}})$ is $h'(C) +
\pi_{\Sigma'}^* \Delta$ together with the restriction of
$\pi_{\Sigma'}$.  Denote by $\Pi$ the pencil of 
divisors in $\Sigma'$ spanned by the divisors $h'(C) + \pi_{\Sigma'}^*
\Delta$ and $T + V$, with these two divisors marked as $0$ and
$\infty$ respectively.  Denote by $\mc{C} \subset \Pi\times_k \Sigma'$
the corresponding family of divisors.  By Bertini's theorem, the
general member $\mc{C}_t$ 
is smooth away from the base locus.  Now the only
singular points of $h'(C) + \pi_{\Sigma'}^* \Delta$ are the points
$h'(\pi_C^{-1}(\Delta))$.  Since $V$ does not intersect
$\pi_{\Sigma'}^*\Delta$, these singular points are not in the base
locus.  Since $\mc{C}_0$ is nonsingular at every basepoint, the same
is true for $\mc{C}_t$ for $t$ general.  Thus a general member
$\mc{C}_t$ is smooth everywhere. 

\mni
Define $D_{\mc{C}}$ to be the pullback to
$\mc{C}$ of the Cartier divisor $E$ in $\Sigma'$.  Because $E$ is
vertical over $B$ and because $h^* E$ equals $D_C$, the deformation
$\mc{C}$ together with the effective Cartier divisor $D_{\mc{C}}$ is a
one-parameter log deformation of
$(C_{\text{link}},\pi_{C,\text{link}},D_C)$.  And it specializes at
$t=\infty$ to a union of section curves and vertical curves.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Completion of the proof
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Completion of the proof} \label{sec-proof}
\marpar{sec-proof}

\mni
We are finally prepared for the proof of Theorem ~\ref{thm-GHS}.

\begin{proof}[Proof of Theorem ~\ref{thm-GHS}]
By Proposition \ref{prop-logflex}, there exists a log flexible curve
$j_0:C\rightarrow U$.  
Denote by $D_C$ the reduced scheme of the intersection $C\cap
D_{\pi,\text{log}}$.  By Lemma \ref{lem-defo}, after attaching
finitely many links to $C$ over the points of a general divisor
$\Delta$ of $B$, the linked
curve $C_\text{link}$ together with $D_C$ admits a one-parameter log
deformation 
$$
(\rho,\pi_\mc{C}):\mc{C} \rightarrow \Pi \times_k B, \ \
D_\mc{C}\subset \mc{C}
$$
of $(C_\text{link},D_C)$ specializing to a section curve (in fact
$\mc{C}_\infty$ is a union of section curves and vertical curves).

\mni
By Proposition ~\ref{prop-vfRC}, the relative very free locus 
$U_{\pi,\text{v.f.}}$ is open in $U$.  Thus $C\cap
U_{\pi,\text{v.f.}}$ is open in $C$.  So its complement is finitely
many points in $C$.  Thus a general divisor $\Delta$ is disjoint from the
finite set
$\pi_U(D_{\pi,\text{log}})$ and from the finite set $\pi_C(C - C\cap
U_{\pi,\text{v.f.}})$.  Then, by Lemma ~\ref{lem-linkflex}, there
exists an extension of $j_0$ to a linked log flexible curve
$$
j:C_\text{link} \rightarrow U.
$$

\mni
Form the fiber product
$$
U_\mc{C} := \mc{C} \times_{\pi_\mc{C},B,\pi_U} U.
$$
Since $\pi_U$ is flat, also the projection
$$
\text{pr}_{\mc{C}}:U_\mc{C} \rightarrow \mc{C}
$$
is flat.  Since $\pi_\mc{C}$ is surjective, the geometric generic
fiber of $\text{pr}_{\mc{C}}$ equals the geometric generic fiber of
$\pi_U$, which is integral.  Since $\text{pr}_\mc{C}$ is flat with
integral geometric generic fiber, $U_\mc{C}$ is integral.
Define
$$
\nu: \widetilde{U}_\mc{C} \rightarrow U_\mc{C}
$$
to be the blowing up of $U_\mc{C}$ along the closed subscheme
$D_\mc{C}\times_B D_{\pi,\text{log}}$.  Since $U_\mc{C}$ is integral,
also $\widetilde{U}_\mc{C}$ is integral.  And the composition
$$
\widetilde{U}_\mc{C} \rightarrow U_\mc{C} \rightarrow \mc{C}
\rightarrow \Pi
$$
is surjective.  Since $\Pi$ is a smooth curve, the morphism 
$$
\widetilde{\rho}:\widetilde{U}_\mc{C} \rightarrow \Pi
$$
is flat.  

\mni
Consider the graph,
$$
\Gamma_j: C_\text{link} = \mc{C}_0 \rightarrow \mc{C}_0 \times_B U =
U_{\mc{C},0}.  
$$
Because the links of $C_\text{link}$ do not intersect
$D_{\pi,\text{log}}$, the image of $\Gamma_j$ is smooth at every point
of intersection with $D_C \times_B D_{\pi,\text{log}}$.  Since $\nu$
is birational, $\Gamma_j$ gives a rational transformation from
$C_\text{link}$ to $\widetilde{U}_{\mc{C},0}$.  Since $\nu$ is proper,
and since $C_\text{link}$ is smooth at every point of intersection
with $D_C\times_B D_{\pi,\text{log}}$, the valuative criterion of
properness implies this rational transformation is actually a regular
morphism
$$
\widetilde{\Gamma}_j:\mc{C}_0 \rightarrow
\widetilde{U}_{\mc{C},0}.
$$
Clearly this is a section of the projection morphism
$$
\text{pr}_{\mc{C}_0}:\widetilde{U}_{\mc{C},0} \rightarrow \mc{C}_0.
$$

\mni
For every point $t$ in $C_\text{link}-D_C$, 
the morphism $\pi_U:U\rightarrow B$ is
smooth at $j(t)$.  Therefore also $U_{\mc{C},0} \rightarrow \mc{C}_0$
is smooth at $\Gamma_j(t)$.  And since $\nu$ is an isomorphism over
$\Gamma_j(t)$, also $\text{pr}_{\mc{C}_0}:\widetilde{U}_{\mc{C},0}
\rightarrow \mc{C}_0$ is smooth at $\widetilde{\Gamma}_j(t)$.  Also
the vertical tangent bundle equals the pullback of the vertical tangent
bundle of $\pi_U:U\rightarrow B$, which also equals
$T_{\pi,\text{log}}$ (since $j(t)$ is not in $D_{\pi,\text{log}}$).

\mni
Let $t$ be a point of $D_C$ and let $D_t$ be the unique irreducible
component of $D_{\pi,\text{log}}$ containing $j(t)$.  Give $D_t$ the
reduced structure.  Because $j_0(C)$ is transverse to $D_t$ at
$j_0(t)$, the ramification index $m_C-1$ of $\pi_C:C\rightarrow B$ at
$t$ equals the ramification index $m_D-1$ of $\pi_U$ along $D_t$.
Therefore, by Lemma ~\ref{lem-etale}, the projection
$$
\text{pr}_{\mc{C}_0}:\widetilde{U}_{\mc{C},0} \rightarrow \mc{C}_0
$$
is smooth over the preimage of $\{t\}\times D_t$ for every $t$ and the
vertical tangent bundle equals the pullback of $T_{\pi,\text{log}}$.  
Since
$\Gamma_j(t)$ is in $\{t\}\times D_t$, this implies that
$\text{pr}_{\mc{C}_0}$ is smooth at every point of the image of
$\widetilde{\Gamma}_j$ and the vertical tangent bundle of
$\text{pr}_{\mc{C}_0}$ equals the pullback of $T_{\pi,\text{log}}$.  

\mni
Since $\widetilde{\Gamma}_j$ is a section with image in the smooth
locus of $\text{pr}_{\mc{C}_0}$, the normal sheaf $\mc{N}$ equals the
restriction of the vertical tangent bundle.  Therefore
$\widetilde{\Gamma}_j^* \mc{N}$ equals $j^* T_{\pi,\text{log}}$.
Since $j:C_\text{link} \rightarrow U$ is log flexible,
$h^1(C_\text{link}, j^* T_{\pi,\text{log}})$ equals $0$.  Therefore,
by Proposition ~\ref{prop-Hilb}, the relative Hilbert scheme
$\text{Hilb}(\widetilde{U}_\mc{C}/\Pi)$ is smooth over $\Pi$ at the
point $0':= [\text{Image}(\widetilde{\Gamma}_j)]$.  Thus for a general
complete intersection curve $\Pi'$ containing $0'$, the morphism
$\Pi'\rightarrow \Pi$ is smooth at $0'$.  

\mni
Replace $\Pi'$ by the unique
irreducible component containing $0'$, and then replace this by its
normalization.  The result is that $\Pi'$ is a smooth, projective,
connected 
curve together with a morphism $\Pi' \rightarrow
\text{Hilb}(\widetilde{U}_\mc{C}/\Pi)$ so that the induced morphism
$\Pi'\rightarrow \Pi$ is smooth at $0'$.  In particular it is flat, so
surjective.  Let $\infty'$ denote a closed point of $\Pi'$ mapping to
$\infty$.  Then $(\Pi',0',\infty') \rightarrow (\Pi,0,\infty)$ is a
flat morphism of $2$-pointed smooth curves.  Thus, by Lemma
~\ref{lem-bc}, the base change $\Pi'\times_\Pi \mc{C}$ is a
one-parameter deformation of $C_\text{link}$ over $(\Pi',0',\infty')$
specializing to a section curve.

\mni
Denote by 
$$
Z\subset \Pi'\times_\Pi \widetilde{U}_\mc{C}
$$
the pullback of the universal closed subscheme
$\text{Univ}(\widetilde{U}_\mc{C}/\Pi)$ by the morphism
$\Pi'\rightarrow \text{Hilb}(\widetilde{U}_\mc{C}/\Pi)$.  
The
composition with $\text{pr}_\mc{C}$ is a projective morphism
$$
Z\subset \Pi'\times_\Pi \widetilde{U}_\mc{C} \rightarrow
\Pi'\times_\Pi \mc{C}
$$
of flat $\Pi'$-schemes.  Moreover, the fiber over $0'\in \Pi'$ is an
isomorphism since the projection $\widetilde{\Gamma}_j(C_\text{link})
\rightarrow C_\text{link}$ is an isomorphism.  Therefore the morphism
is an isomorphism over $N\times_\Pi \mc{C}$ for some open neighborhood
$N$ of $0'$ in $\Pi'$. (This is well-known; a complete proof is given
in \cite[Lemma 4.7]{dJS}.)
Invert
this isomorphism and compose it with the morphism
$$
\Pi'\times_\Pi \widetilde{U}_\mc{C} \rightarrow \widetilde{U}_\mc{C}
\rightarrow U_\mc{C} \rightarrow U.
$$
The result is precisely an extension
$$
j_N:N\times_\Pi \mc{C} \rightarrow X
$$
of $j$ for the one-parameter deformation $\Pi'\times_\Pi \mc{C}$.
Therefore, by Lemma ~\ref{lem-val}, there exists a section
$s:B\rightarrow X$ of $\pi$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Corollaries
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Corollaries} \label{sec-cor}
\marpar{sec-cor}

\mni
There are a number of consequences of Theorem ~\ref{thm-GHS} and its
generalization to positive characteristic in \cite{dJS}.  Many of
these consequences were recognized before Conjecture \ref{conj-KMM}
was proved.  

\begin{cor}\cite[Conjecture IV.5.6]{K} \label{cor-KMM} 
\marpar{cor-KMM}
Conjecture ~\ref{conj-KMM1} is true.  Moreover, for every smooth,
projective, irreducible variety $X$ over an algebraically closed field
of  characteristic $0$, there exists a dense open $X^0\subset X$ and a
projective, smooth morphism $q_0:X^0 \rightarrow Q^0$ such that every
fiber of $q_0$ is rationally connected, and every projective closure
of $Q^0$ is nonuniruled.
\end{cor}

\begin{cor} \cite[Corollary 1.7]{GHS} \label{cor-GHS}
\marpar{cor-GHS}
The uniruledness conjecture implies Mumford's conjecture.  To be
precise, assume that for every smooth, projective, irreducible variety
$X$ over an algebraically closed field $k$ of characteristic $0$, if $X$
is nonuniruled then $h^0(X,\omega_X^{\otimes n})$ is nonzero for some
$n>0$.  Then for every smooth, projective, irreducible variety $X$
over $k$, if $X$ is not rationally connected then
$h^0(X,\Omega_X^{\otimes n})$ is nonzero for some $n>0$.
\end{cor}

\mni
The next corollary is a fixed point theorem.  In characteristic $0$ it
can be proved using the Atiyah-Bott fixed point theorem.  But in
positive characteristic it is a new result.  There are examples
due to Shioda proving one cannot replace ``separably rationally
connected'' by ``rationally connected'', cf. \cite{Shioda74}.

\begin{cor} \cite{KollarRC} \label{cor-K}
\marpar{cor-K}
Let $Y$ be a smooth, projective, separably rationally connected
variety over a field $k$ and let $f:Y\rightarrow Y$ be a
$k$-automorphism.  If $\text{char}(k)$ is positive, say $p$, assume
in addition that
$f$ has finite order $n$ not divisible by $p^2$.
Then the fixed locus of $f$ is nonempty.  
\end{cor}

\begin{proof}
Of course it suffices to prove the case when $k$ is algebraically
closed, since the fixed locus of the base change equals the base
change of the fixed locus.  First assume 
$f$ has finite order $n$.  If $n$ is prime to $\text{char}(k)$, let
$B'$ denote $\PP^1$ and let $\ZZ/n\ZZ$ act on $\PP^1$ by
multiplication by a primitive $n^\text{th}$ root of unity.  Note this
action fixes $\infty$ and has trivial generic stabilizer.
If
$\text{char}(k) = p$ is positive and if $n=pm$ where $m$ is prime to
$p$, let $B'$ be the normal, projective completion of the affine curve 
$$
\mathbb{V}(y^m - (x^p-x)) \subset \AAA^2_k.
$$
Let $\zeta$ be a primitive $m^\text{th}$ root of unity, and let a
generator of $\ZZ/m\ZZ$ act by $(x,y) \mapsto (x,\zeta y)$.
Similarly, let a generator of $\ZZ/p\ZZ$ act by $(x,y) \mapsto
(x+1,y)$.  Clearly these actions commute, thus define an action of
$\ZZ/n\ZZ$ on $B'$.  Note this action fixes the unique point $\infty$
not in the affine chart above, and the action has trivial 
generic stabilizer.

\mni
Let $\ZZ/n\ZZ$ act
diagonally on $Y\times_k B'$, and let $X$ be the quotient.  Also
let $B$ be the quotient of the $\ZZ/n\ZZ$-action on $B'$.  The
projection $\pi:X\rightarrow B$ satisfies the hypotheses of Theorem
~\ref{thm-GHS} (or its generalization in \cite{dJS}).  Therefore there
exists a section.  This is the same as $\ZZ/n\ZZ$-equivariant
$k$-morphism $f:B' \rightarrow Y$.  In particular, since $\infty$ is a
fixed point in $B'$, $f(\infty)$ is a fixed point in $Y$.

\mni
Next assume $k$ has characteristic $0$.
By general limit arguments there
exists an integral, finitely generated
$\ZZ$-algebra $R$, a ring homomorphism $R\hookrightarrow k$, a smooth,
projective morphism $Y_R \rightarrow \SP R$ whose relative very free
locus is all of $Y_R$, and an $R$-automorphism $f_R:Y_R\rightarrow
Y_R$ such that the base change $Y_R\otimes_R k$ equals $Y$ and the
base change of $f_R$ equals $f$.  The intersection $(Y_R)^{f_R}$ of
the graph 
of $f_R$ and the diagonal of $Y_R\times_R Y_R$ is the fixed subscheme
of $f_R$ (actually its image under the diagonal morphism).  Since
$(Y_R)^{f_R}$ is
a proper scheme over
$\SP R$, the image in $\SP R$ is a closed subscheme of $\SP R$.  To
prove this closed subscheme equals all of $\SP R$, and thus contains
the image of $\SP k$, it suffices to prove it contains a Zariski dense
set of closed points.  

\mni
Choose an $f$-invariant very ample sheaf, choose a basis for the space
of global sections, and let $A$ be the $N\times N$ matrix with entries
in $R$ 
giving the action of $f$ on global sections with respect to this
basis.
The set of maximal ideal in $\SP R$ with
residue field of characteristic $p>N$ are Zariski dense in $\SP R$.
Every invertible matrix over a characteristic $p$ field with order
divisible by $p^2$ has a Jordan block with eigenvalue $1$ and size
divisible by $p$.  Thus, since $p>N$, the finite order of $f_R$ modulo
the prime is not divisible by $p^2$.  Therefore, by the previous case, 
the reduction of $f_R$
modulo the prime has nonempty fixed locus.  Therefore the original
automorphism $f$ has nonempty fixed locus.
\end{proof}

\mni
This fixed point theorem implies that separably rationally connected
varieties are simply connected.  When the field $k$ is $\CC$, this was
first proved by Campana using analytic methods, cf. the excellent
reference by Debarre,
\cite[Corollary 4.18]{De}.

\begin{cor}[Campana, Koll\'ar] \cite{C91}, \cite[3.6]{DBX}
\label{cor-K2}
\marpar{cor-K2}
Let $X$ be a smooth, projective, separably rationally connected
variety over an algebraically closed field $k$.  The algebraic
fundamental group of $X$ is trivial.  If $k=\CC$, then the topological
fundamental group of $X$ is also trivial.    
\end{cor}

\mni
Koll\'ar has generalized this considerably to prove a result for open
subschemes of rationally connected varieties, cf. \cite{KollarRC}.

\begin{proof}
The full proof is included in the beautiful survey by Debarre,
\cite[3.6]{DBX}.  Here is a brief sketch.  First of all, for every
quasi-projective, (not necessarily separably) 
rationally chain connected variety, Campana proved
that the algebraic fundamental group is finite and also the topological
fundamental group is finite when $k$ equals $\CC$ (so that the
topological fundamental group is defined).  
Thus
the universal cover
$\widetilde{X} \rightarrow X$ is finite.  Since $X$ is smooth,
projective and separably rationally connected, also
$\widetilde{X}$ is smooth, projective and
separably rationally connected.  If the fundamental group of $X$ is
nonzero, then it contains a cyclic subgroup $\ZZ/n\ZZ$ such that $p^2$
does not divide $n$.  Of course the action of this group on
$\widetilde{X}$ is fixed-point-free.  But Corollary ~\ref{cor-K}
implies there exists a fixed point.  Thus $X$ is simply connected.
\end{proof}

\mni
Theorem ~\ref{thm-GHS} also plays an important role in the proof of a
``converse'' to that same theorem.

\begin{thm} \cite{GHMS} \label{thm-GHMS}
\marpar{thm-GHMS}
Let $\pi:X\rightarrow B$ be a surjective morphism of normal,
projective, irreducible varieties over an algebraically closed field
$k$ of characteristic zero.  Assume that for some sufficiently large,
algebraically closed field extension $K/k$, for every $k$-morphism
$C\rightarrow B$ from a smooth, projective, $K$-curve to $B$, the
pullback $\pi_C:C\times_B X \rightarrow C$ has a section. Then there
exists a closed subvariety $Y\subset X$ such that the geometric
generic fiber of $\pi|_Y:Y\rightarrow B$ is nonempty, irreducible and
rationally connected.
\end{thm}

\mni
One corollary of this theorem, in fact the motivation for proving it,
was to answer a question first asked by Serre and 
left unresolved by Theorem ~\ref{thm-GHS}: could it be that a smooth,
projective variety $X$ over the function field of a curve has a
rational point if it is \emph{$\OO$-acyclic}, i.e., if $h^i(X,\OO_X)$
equals $0$ for all $i>0$?  One reason to ask this is that the
corresponding question has a positive answer if ``function field'' is
replaced by ``finite field'' thanks to N. Katz's positive
characteristic analogue of the Atiyah-Bott fixed point theorem,
\cite[Expos\'e XXII, Corollaire 3.2]{SGA7_2}, recently generalized by
Esnault, \cite{Esnault}.  Nonetheless, the answer is negative over
function fields.

\begin{cor} \cite{GHMS} \label{cor-GHMS}
\marpar{cor-GHMS}
There exists a surjective morphism $\pi:X\rightarrow B$ of smooth,
projective varieties over $\CC$ such that $B$ is a curve and the
geometric generic fiber of $\pi$ is an Enriques surface, but $\pi$ has
no section.  Thus, to guarantee a fibration over a curve has a
section, it is not sufficient to assume the geometric generic fiber is
$\OO$-acyclic.
\end{cor}

\mni
In fact G. Lafon found an \emph{explicit} morphism $\pi$ as in Corollary
\ref{cor-GHMS} where $B$ is
$\PP^1_\CC$, or in fact $\PP^1_k$ for any field $k$ with
$\text{char}(k)\neq 2$, and there does not even exist a power series
section near $0\in \PP^1_k$, cf. \cite{Lafon}.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Part: The Period-Index theorem of de Jong
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{The Period-Index theorem of de Jong} 
\label{pt-PI}
\marpar{pt-PI}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Statement of the theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Statement of the theorem} 
\label{sec-SoT}
\marpar{sec-SoT}

\mni
This chapter and the next are extracted from an earlier set of lecture notes,
\cite{SStrsbg}.  The results in this chapter
are joint work with A. J. de Jong.  In
particular, a full proof of Theorem ~\ref{thm-main} is presented in
\cite{SStrsbg}.  For this reason we will not reproduce the proof here,
but we will reproduce enough of the proof to prove de Jong's
Period-Index theorem.

\mni
Recall Problem ~\ref{prob-PI}: for a given field $K$, what is the
precise relationship between the index of division algebras and the
period, or exponent, of division algebras?  Of course if $K$ is
algebraically closed, or more generally if $K$ has cohomological
dimension $\leq 1$, Theorem ~\ref{thm-Galois}(i) implies that the only
division algebra over $K$ is $K$ itself.  Thus both the period and the
index equal $1$.  And in this sense the power $r(K)$ of $K$ equals
$0$.  In particular, by Chevalley's theorem and Tsen's theorem
discussed in Chapter \ref{pt-TL}, this holds if $K$ is a finite field
or if $K$ is the fraction field of a curve over an algebraically
closed field.  
The following question was explicitly asked by Colliot-Th\'el\`ene,
cf. \cite{CT}.

\begin{ques}\label{ques-CT}
\marpar{ques-CT}
Let $K$ be the function field of a variety of dimension $d\geq 2$ over
$\CC$.  For every division algebra $D$ over $K$, does
$\text{index}(D)$ divide $\text{period}(D)^{d-1}$?  In other words, is
$r(K)$ equal to $d-1$?
\end{ques}

\mni
There are examples showing $r(K)$ is $\geq d-1$, so this question is
sharp.  
One can also ask
this question with $\CC$ replaced by any algebraically closed field $k$,
perhaps with the restriction that $\text{char}(k)$ does not divide
$\text{period}(D)$.  Even more generally, one could ask this question
with $K$ replaced by any $C_d$ field, e.g., a function field of a
variety of dimension $d-1$ over a finite field.  

\mni
The first major advance in this direction (from which subsequent
advances have flowed) is the following theorem of A. J. de Jong.

\begin{thm}[de Jong's Period-Index theorem] \cite{dJ}
\label{thm-PI}
\marpar{thm-PI}
Let $k$ be an algebraically closed field.  Let $K/k$ be the function
field of a surface over $k$.  Then the power of $K$ is $1$, i.e., for
every central simple algebra $A$ over $K$, the period of $A$ equals
the index of $A$.
\end{thm}

\mni
Initially de Jong proved this under the additional hypothesis that
$\text{char}(k)$ does not divide the period of $A$.  Using the
technique of discriminant avoidance, described below, de Jong and I
removed this hypothesis, cf. \cite[Theorem 1.0.2]{dJS8}.
The original proof of de Jong used an analysis of the
moduli space of \emph{Azumaya algebras}, i.e., sheaves of central
simple algebras, on a smooth, projective surface.  The second proof of 
de Jong and myself will be presented below.
Lieblich used moduli spaces of $\alpha$-twisted sheaves, as introduced
by C\u{a}ld\u{a}raru and Yoshioka, 
to give a third proof.  Moreover, Lieblich's proof
proves more.  

\begin{thm}\cite{LieblichTwist} \label{thm-Lieblich}
For $K$ the function field of a surface over a finite field $k$, for
every division algebra $D$ over $K$, if $\text{char}(k)$ does not
divide $\text{period}(D)$, then $\text{index}(D)$ divides
$\text{period}(D)^3$.  If, moreover,
$D$ is the generic fiber of an Azumaya
algebra on a smooth, proper model of $K$, then $\text{index}(D)$
equals $\text{period}(D)$.  
\end{thm}

\mni
Before explaining the second proof due to de Jong and myself, there
are 3 more reformulations of Theorem ~\ref{thm-PI}.  The first
reformulation has to do with the map $\Delta_n$ from Theorem
~\ref{thm-BrBrprime}.

\begin{lem} \label{lem-PIDelta}
\marpar{lem-PIDelta}
For a field $K$, the power $r(K)$ equals $1$, i.e., the period equals
the index for every division algebra over $K$, if and only if for
every integer $n\geq 1$ the map
$$
\Delta_n:H^1(\G(K^s/K),\PGL{n,K}(K^s)) \rightarrow \text{Br}'(K)[n]
$$
is surjective, and thus bijective.
\end{lem}

\begin{proof}
Assume first that $r(K)$ equals $1$.  By Theorem ~\ref{thm-Br=Br'} and
Theorem ~\ref{thm-AW}, every element of $\text{Br}'(K)$ is of the form
$\Delta[D]$ for some division algebra $D$.  Since $r(K)$ equals $1$,
the order of $\Delta[D]$ equals the index of $D$.  Thus, if the order
$r$ of $\Delta[D]$ divides $n$, say $n=rl$,
then $A=D\otimes_K \Mat{l\times l,K}$ gives an element of
$H^1(\G(K^s/K),\PGL{n,K}(K^s))$ mapping to $\Delta[D]$ in
$\text{Br}'(K)[n]$.  Thus $\Delta_n$ is surjective.

\mni
Next assume that $\Delta_n$ is surjective for every $n$.  Let $D$ be a
division algebra over $K$, and denote by $n$ the order of
$\Delta(D)$.  Since $\Delta_n$ is surjective, there exists an Azumaya
algebra $A$ with $\text{dim}_K(A) = n^2$ such that $\Delta_n(A)$
equals $\Delta(D)$.  By Theorem ~\ref{thm-Br=Br'}, $A$ is Morita
equivalent to $D$.  By Theorem ~\ref{thm-AW}, $A = D\otimes_K
\Mat{l\times l,K}$ for some integer $l\geq 1$.  Comparing dimensions
of these $K$-vector spaces, the period $n$ is at least as large as the
index of 
$D$.  Since the period divides the index, also the index is at least
as large as the period, and thus both are equal.
\end{proof}

\mni
The second reformulation requires the auxiliary notion of
\emph{generalized Brauer-Severi varieties}.  
Let $A$ be a
central simple algebra over $K$ with $\text{dim}_K(A) = n^2$ 
and let $r\geq 1$ be an integer.  The
functor 
$$
K-\text{algebras} \rightarrow \text{Sets}, \ \ R \mapsto \{I\subset
A\otimes_K R | \text{ $I$ a free $R$-module of rank $rn$ and a right
  ideal }\}
$$
is representable.

\begin{defn}\cite{Blanchet}, \cite{SchoVdB} \label{defn-GBS}
\marpar{defn-GBS}
The \emph{generalized Brauer-Severi variety}, $\GR{r,A}$, is the
scheme representing the functor above.  Stated differently, if $E$
denotes the $\PGL{n,K}$-torsor associated to $A$, then $\GR{r,A}$
equals the twist
$\lsb{E}{\GR{K}(r,K^{\oplus n})}$ of the usual Grassmannian
$\GR{K}(R,K^{\oplus n})$ of locally free, rank $r$, 
locally direct summands of $K^{\oplus n}$.  
\end{defn}

\begin{lem} \label{lem-PIG}
\marpar{lem-PIG}
For a field $K$, the power $r(K)$ equals $1$ if and only if for every
central simple algebra $A$, for $r = \text{period}(A)$, $\GR{r,A}$ has
a $K$-point.
\end{lem}

\begin{proof}
Assume first that $r(K)$ equals $1$.  Let $A$ be a central simple
algebra with $\text{dim}_K(A) = n^2$.  
By Theorem ~\ref{thm-AW}, $A$ is isomorphic to $D\otimes_K
\Mat{l\times l,K}$ for some division algebra $D$ over $K$ and some
integer $l\geq 1$, i.e.,
$$
A = \text{End}_{D-\lmod}(D^{\oplus l}).
$$
Since $r(K)$ equals $1$, $\text{period}(A) =
\text{period}(D)$ equals $\text{index}(D)$.  Denote by $D^{\oplus 1}
\subset D^{\oplus l}$ the left $D$-submodule of $D^{\oplus l}$
generated by the first basis element.  Then 
$$
I = \text{Hom}_{D-\lmod}(D^{\oplus l},D^{\oplus 1})
$$
is a right ideal in $A$ with $\text{dim}_K(I) = rn$. Thus $I$ gives a
$K$-point of $\GR{r,A}$.

\mni
Next assume that $\GR{r,A}$ always has a $K$-point.  Let $D$ be a
division algebra of order $r$.  Since $\GR{r,D}$ has a $K$-point,
there is a right ideal $I$ in $D$ with 
$$
\text{dim}_K(I) = \text{period}(D)\cdot \text{index}(D).
$$
Since the only nonzero right ideal in $D$ is $D$ itself, $I$ equals
$D$.  Since $\text{dim}_K(D) = \text{index}(D)^2$, dividing gives
$$
\text{period}(D) = \text{index}(D).
$$    
\end{proof}

\mni
Among all the generalized Brauer-Severi varieties $\GR{r,A}$, the one
with $r=\text{period}(A)$ has a special property.  

\begin{lem} \label{lem-elemGBS}
\marpar{lem-elemGBS}
Let $K$ be a field and let $A$ be a central simple algebra over $K$
with $\text{dim}_K(A) = n^2$.  Let $r$ be an integer, $1\leq r \leq
n$.  There exists an invertible sheaf $\mc{L}$ on $\GR{r,A}$ such that
$\text{Pic}(\GR{r,A}\otimes_K K^s) = \ZZ\cdot(\mc{L}\otimes_K K^s)$,
i.e., the base change of $\mc{L}$ to $\GR{r,A}\otimes_K K^s =
\GR{K^s}(r,(K^s)^{\oplus n})$ generates the Picard group, if and only
if $\text{period}(A)$ divides $r$.  
\end{lem}

\begin{proof}
Consider the commutative diagram of central extensions of smooth group
schemes (similar to the one from the proof of Theorem
~\ref{thm-BrBrprime}). 
$$
\begin{CD}
1 @>>> \Gm{m,K} @>>> \GL{n,K} @>>> \PGL{n,K} @>>> 1 \\
& & @V (-)^r VV @VV q V @VV = V \\
1 @>>> \Gm{m,K} @>> s > \GL{n,K}/\mun{r,K}\cdot \text{Id}_{n\times n}
@>>> \PGL{n,K} @>>> 1
\end{CD}.
$$
Using Proposition ~\ref{prop-les6} and Remark ~\ref{rmk-comm},
$\Delta_n(A)$ has order dividing $r$ if and only if the associated
$\PGL{n,K}$-torsor of $A$ lifts to a torsor for
$\GL{n,K}/\mun{r,K}$.  On the other hand, the image of the natural
homomorphism
$$
\bigwedge^r:\GL{n,K} \rightarrow \GL{\binom{n}{r},K}
$$
is precisely the quotient $\GL{n,K}/\mun{r,K}$.  
Thus, from the commutative diagram
$$
\begin{CD}
1 @>>> \Gm{m,K} @>> s > \GL{n,K}/\mun{r,K} @>>> \PGL{n,K} @>>> 1 \\
& & @V = VV @VV \bigwedge^r V @VV \bigwedge^r V \\
1 @>>> \Gm{m,K} @>>> \GL{\binom{n}{r},K} @>>> \PGL{\binom{n}{r},K}
@>>> 1
\end{CD}
$$
a $\PGL{n,K}$-torsor lifts to a torsor for $\GL{n,K}/\mun{r,K}$ if and
only if the associated $\PGL{\binom{n}{r},K}$-torsor lifts to a torsor
for $\GL{\binom{n}{r},K}$, i.e., if it lifts from the group of
projective equivalences on $\PP(K^{\oplus \binom{n}{r}})$ the group of
equivalences of $(\PP(K^{\oplus \binom{n}{r}}),\OO(1))$.  Since the
restriction of $\OO(1)$ to $\GR{K}(r,K^{\oplus n})$ is the generator
of the Picard group, it follows that $\Delta_n(A)$ has order dividing
$r$ if and only if there exists an invertible sheaf $\mc{L}$ on
$\GR{r,A}$ whose base change to $K^s$ is a generator of the Picard
group.  
\end{proof}

\mni
This is relevant to Lemma ~\ref{lem-PIG} because of the
\emph{elementary obstruction} to the existence of $K$-points. 

\begin{defn}\cite{CTSansuc} \label{def-elem}
\marpar{def-elem}
Let $K$ be a field and let $X$ be a geometrically integral,
quasi-projective, smooth
$K$-scheme.  The \emph{elementary obstruction} to the existence of a
$K$-point of $X$ is the short exact sequence of $\G(K^s/K)$-modules,
$$
\begin{CD}
1 @>>> (K^s)^* @>>> K^s(X)^* @>>> K^s(X)^*/(K^s)^* @>>> 1
\end{CD}
$$ 
where $K^s(X)$ is the fraction field of $X\otimes_K K^s$.  Stated
differently, the elementary obstruction is zero if and only if there
exists a Galois-invariant splitting of $(K^s)^*\rightarrow K^s(X)^*$.  
\end{defn}

\begin{rmk} \label{rmk-elem}
\marpar{rmk-elem}
If $X$ has a $K$-point, or even a $0$-cycle of degree $1$ over $K$,
then the elementary obstruction is zero,
cf. \cite[Prop. 2.2.2]{CTSansuc}, \cite[Theorem 2.3.4]{Sk}.  
And if $X$ is
projective with $\text{Pic}(X\otimes_K K^s)$ isomorphic to $\ZZ$,
then the elementary obstruction vanishes if and only if there exists an
invertible sheaf $\mc{L}$ on $X$ such that $\mc{L}\otimes_K K^s$ is a
generator for the Picard group, cf. \cite[Lemma 2.2(v)]{BCTSk}.
\end{rmk}

\mni
This leads to the
final reinterpretation of Theorem ~\ref{thm-PI}

\begin{lem}
For a field $K$, the power $r(K)$ equals $1$ if and only if for every
central simple algebra $A$, denoting by $n$ the integer such that 
$\text{dim}_K(A)=n^2$, for every integer $r$ with $1\leq r \leq n$,
vanishing of the elementary obstruction for $\GR{r,A}$ is a sufficient
condition for existence of a $K$-point.
\end{lem}

\mni
This last reformulation suggests a problem, the refined version of
Problem \ref{prob-vague} for function fields of surfaces.

\begin{prob} \label{prob-vaguesurf}
\marpar{prob-vaguesurf}
Let $K$ be the function field of a surface over an algebraically
closed field $k$.  Let $X$ be a smooth, projective $K$-scheme.  Find
sufficient conditions on $X\otimes_K K^s$ so that $X$ has a $K$-point
when the elementary obstruction vanishes, e.g., when there exists an
invertible sheaf $\mc{L}$ on $X$ such that $\mc{L}\otimes_K K^s$ is a
generator for $\text{Pic}(X\otimes_K K^s)$.  
\end{prob}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Abel maps over curves and sections over surfaces
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Abel maps over curves and sections over surfaces} 
\label{sec-Abel}
\marpar{sec-Abel}

\mni
Let $\kappa$ be a characteristic $0$ field, not necessarily
algebraically closed.  Let 
$C$ be a smooth, projective, geometrically integral curve over
$\kappa$.  Let $\pi:X\rightarrow B$ be a projective, flat
morphism whose geometric generic fiber is irreducible and smooth.
And let $\mc{L}$ be a $\pi$-ample invertible sheaf on $X$.

\begin{defn} \label{defn-famsecs}
\marpar{defn-famsecs}
For a $\kappa$-scheme $S$, a 
\emph{family of sections of $\pi$ parameterized by $S$} is a morphism
of $C$-schemes
$$
\tau:S\times_\kappa C \rightarrow X.
$$
For an integer $e$,
the family of sections has \emph{degree $e$} if the invertible sheaf
$\tau^*\mc{L}$ on $S\times_\kappa C$ has relative degree $e$ over $S$,
i.e., for every geometric point $s$ of $S$, the basechange of the
invertible sheaf to $C_s$ has degree $e$.  A pair $(S,\tau)$ as above
is
\emph{universal} if for every $\kappa$-scheme $S'$ and for every
family of degree $e$ sections of $\pi$ parameterized by $S$, 
$$
\tau':S'\times_\kappa C \rightarrow X,
$$
there
exists a unique $\kappa$-morphism $f:S'\rightarrow S$ such that
$\tau'$ equals $\tau\circ (f,\text{Id}_C)$.  
\end{defn}

\begin{thm}[Grothendieck] \cite[Part IV.4.c, p. 221-19]{FGA}
  \label{thm-G}
\marpar{thm-G}
For every integer $e$ there
exists a pair $(\Sec^e(X/C/\kappa),\sigma)$ of a $\kappa$-scheme and a
family of degree $e$ sections of $\pi$ parameterized by $\Sec^e(X/C/\kappa)$,
$$
\sigma: \Sec^e(X/C/\kappa)\times_\kappa C \rightarrow X,
$$
such that $\sigma^*\mc{L}$ which is \emph{universal}. 
Moreover $\Sec^e(X/C/\kappa)$ is a quasi-projective $\kappa$-scheme.
\end{thm}

\mni
Invertible sheaves on $C$ of degree $e$ are parameterized by the Picard
scheme $\text{Pic}^e_{C/\kappa}$.  Thus, associated to the invertible
sheaf $\sigma^*\mc{L}$ there is a morphism of $\kappa$-schemes
$$
\alpha:\Sec^e(X/C/\kappa) \rightarrow \text{Pic}^e_{C/\kappa}.
$$
This morphism is the \emph{Abel map} associated to $\mc{L}$.  

\mni
The quasi-projective $\kappa$-scheme $\Sec^e(X/C/\kappa)$ is rarely
projective.  There is a projective scheme $\Sigma^e(X/C/\kappa)$
containing $\Sec^e(X/C/\kappa)$ as an open subscheme: the coarse
moduli space of the stack of \emph{stable sections}, i.e., stable maps
to $X$ whose stabilization under $\pi$ is an isomorphism to $C$.  
Stable maps are
an important tool in this area, but a discussion of the basics of
stable maps would carry us rather far from the main story.  Thus,
suffice it to say that $\Sigma^e(X/C/\kappa)$ is a projective scheme,
and there exists a $\kappa$-morphism
$$
\alpha:\Sigma^e(X/C/\kappa) \rightarrow \text{Pic}^e_{C/\kappa}.
$$
whose restriction to $\Sec^e(X/C/\kappa)$ is the Abel map from the
last paragraph.  Although the following problem 
may at first appear to have little to do
with Problem ~\ref{prob-vaguesurf}, in fact the two are intimately
connected.  

\begin{prob} \label{prob-vagueAbel}
\marpar{prob-vagueAbel}
Let $\kappa$, $\pi:X\rightarrow C$ and $\mc{L}$ be as above.  Assume
that the base change of $\mc{L}$ to the geometric generic fiber of
$\pi$ generates the Picard group.  Find conditions conditions under
which  
there exists a sequence of closed subschemes $(Z_e\subset
\Sigma^e(X/C/\kappa)_{e\gg 0}$ such that $Z_e$ intersects
$\Sec^e(X/C/\kappa)$ and such that the restriction of the Abel map
$$
\alpha|_{Z_e}:Z_e \rightarrow \text{Pic}^e_{C/\kappa}
$$
is surjective with rationally connected geometric generic fiber.
\end{prob}

\mni
In fact there do exist such conditions which capture, in some sense,
the notion of ``rational simple connectedness''.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Rational simple connectedness hypotheses
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rational simple connectedness hypotheses} \label{sec-RCAbel}
\marpar{sec-RCAbel}

\mni
\textbf{Hypothesis 1.}  The $\kappa$-scheme $X$ is smooth and
projective.

\mni
\textbf{Hypothesis 2.}  The invertible sheaf $\mc{L}$ is $\pi$-ample
and $\pi$-relatively globally generated. 

\mni
All of the hypotheses are insensitive to replacing $\mc{L}$ by
$\mc{L}\otimes_{\OO_X} \pi^* \mc{M}$ for an invertible sheaf $\mc{M}$
on $B$.  Thus, in Hypothesis 2, we may even assume $\mc{L}$ is ample
and globally generated.

\mni
The remaining hypotheses
are stated in terms of $Y$.

\mni
\textbf{Hypothesis 3.}  The basechange $\mc{L}_Y$ of $\mc{L}$ to $Y$
is very ample.  

\mni
From now on we consider $Y$ to be embedded in a
projective space $\PP^N_K$ by the complete linear system of
$\mc{L}_Y$.  In particular, $\Kgnb{0,1}(Y/K,1)$ is the
projective
$K$-scheme parameterizing pointed lines $(p,L)$ in $Y$, i.e., $L$ is a
line contained in $Y$ and $p$ is a point of $L$.  There is an
``evaluation'' $K$-morphism
$$
\text{ev}:\Kgnb{0,1}(Y/K,1) \rightarrow Y, \ \ (L,p) \mapsto p.
$$

\mni
\textbf{Hypothesis 4.}
The morphism $\text{ev}$ above is surjective, the geometric generic fiber of
$\text{ev}$  is irreducible and rationally connected, and the
fiber of $\text{ev}$ over every codimension $1$ point $\eta$ of $Y$ is
``Morse'', i.e., $\Kgnb{0,1}(Y/K,1)$ is smooth at every point of
$\text{ev}^{-1}(\eta)$ and the only singularities on the geometric
fiber over $\eta$ are at worst finitely many ordinary double point
singularities.  

\mni
Of course there is a dense open subset of $Y$, $Y_{\text{free}}$, such
that every line in $Y$ intersecting $Y_{\text{free}}$ is free.  Thus 
$$
\text{ev}:\text{ev}^{-1}(Y_{\text{free}}) \rightarrow Y_{\text{free}}
$$
is smooth and projective.  In particular, the fiber of $\text{ev}$
over the geometric generic point of $Y$ is automatically smooth.  

\mni
If one thinks of rational curves in
$Y$ as analogous to continuous paths in a CW complex, this
hypothesis is analogous to asking that the space of continuous
paths with one boundary point fixed is
itself path-connected.  Since continuous paths with one fixed
boundary point can be retracted
to a constant path to the basepoint, the condition always holds.
The Morse condition is technical, but often easy to verify. 
Notice also that this hypothesis implies that $\mc{L}_Y$ is not itself a
tensor power $\mc{K}^{\oplus d}$ for an integer $d> 1$. 

\mni
The next hypothesis uses the space of chains of free lines.  For an
integer $n\geq 1$, 
a
\emph{chain of $n$ free lines} is an $n$-tuple
$$
(C,p_1,q_n) = ((L_1,p_1,q_1),(L_2,p_2,q_2),\dots,(L_n,p_n,q_n))
$$ 
where each $L_i$ is a free line in $Y$, $p_1$ and $q_i$ are $K$-points
of $L_i$, and $q_i = p_{i+1}$ for $i=1,\dots,n-1$.  The points $p_1$
and $q_n$ are considered ``boundary points'' of the chain analogous to
the boundary points $0$ and $1$ of the closed unit interval $[0,1]$.

\mni
There is a quasi-projective $K$-scheme $\FCh_2(Y/K,n)$ parameterizing
chains of $n$ free lines.  And there is an ``evaluation'' $K$-morphism
$$
\text{ev}:\FCh_2(Y/K,n) \rightarrow Y\times_K Y, \ \ (C,p_1,q_n) \mapsto
(p_1,q_n).
$$

\mni
\textbf{Hypothesis 5.}
For some positive integer $n=n_0$,
the morphism $\text{ev}$ above is dominant and the geometric generic fiber
is ``birationally rationally connected'', i.e., it is isomorphic as a
scheme over $\overline{K(Y\times_K Y)}$ to an open subscheme of a
projective, rationally connected scheme.  

\mni
Under Hypotheses 3 and 4, 
$\text{ev}$ is dominant if and only if $Y$ is irreducible and
$\mc{L}_Y$ is a
generator of $\text{Pic}(Y)$. If the Picard number equals $1$, the
argument proving $\text{ev}$ is dominant appears, for instance,
in \cite[Corollary IV.4.14]{K}. 
And the opposite direction follows from
\cite[Proposition IV.3.13.3]{K}.

\mni
Together with Hypotheses 1 and 2, this implies a condition on
the singular fibers of $\pi$:
The
specializations of the geometric generic fiber occurring as
singular fibers of $\pi$ are sufficiently mild that the ample
generator of the Picard group of the geometric generic fiber extends
to an ample invertible sheaf on the special fiber.  In particular this
implies all singular geometric fibers are irreducible and more.

\mni
The hypothesis that the fibers of $\text{ev}$ are rationally
connected is very strong.  
The fiber of $\text{ev}$ is
analogous to the space of continuous paths in a path connected
CW complex connecting two points.  The CW complex is simply connected
if and only if this path space is simply connected.  In this way we
consider Hypothesis 5 to be a version of ``rational simple
connectedness''. 

\mni
The final hypothesis may seem technical, but it is essential.  There
are examples where Hypotheses 1 through 5 hold and yet the fibers of
the Abel maps are not rationally connected.  

\mni
\textbf{Hypothesis 6.}  There exists a \emph{very twisting scroll} in
$Y$, i.e., a pair $(R,D)$ consisting of a closed subscheme $R$ of
$\PP^1_K \times_K Y$ and a Cartier divisor class $D$ on $R$ such that
\begin{enumerate}
\item[(i)]
$R$ is a \emph{scroll}, i.e., $\text{pr}_{\PP^1}:R\rightarrow \PP^1$
is a smooth morphism whose geometric fibers are lines in $Y$,
\item[(ii)]
$\OO_R(D)$ is globally generated and $D$ has relative degree $1$ over
$\PP^1$, 
\item[(iii)]
the normal sheaf $N_{R/\PP^1\times Y}$ is globally generated, and
\item[(iv)]
$h^1(R,N_{R/\PP^1\times Y}(-D)\otimes_{\OO_{\PP^1}}\OO_{\PP^1}(-2))$
equals $0$.
\end{enumerate}

\mni
In fact this complicated condition has a
simple geometric meaning.  By Bertini's theorem, a general member of
the linear system $|D|$ is a smooth curve in $R$.  By (ii), this curve
is of the form $\tau(\PP^1)$ for a section
$\tau:\PP^1\rightarrow R$ of $\text{pr}_{\PP^1}$.  The scroll $R$
together with the section $\tau$ defines a family of pointed lines in
$Y$ parameterized by $\PP^1_K$.  Equivalently, it defines a
$K$-morphism
$$
\zeta:\PP^1_K \rightarrow \Kgnb{0,1}(Y,1).
$$
The conditions (ii)--(iv) say that $\zeta(\PP^1)$ is in the
smooth locus of both morphisms
$$
\text{ev}:\Kgnb{0,1}(Y,1) \rightarrow Y
$$
and
$$
\rho:\Kgnb{0,1}(Y,1) \rightarrow \Kgnb{0,0}(Y,1),
$$
that the pullback $\zeta^* T_\rho := \zeta^* (\Omega_\rho)^\vee$ is
globally generated, and that the pullback $\zeta^* T_{\text{ev}} := \zeta^*
(\Omega_{\text{ev}}^\vee$ is ample.  In fact if there exists a
morphism $\zeta$ with these properties, then there also exists a
morphism such that both $\zeta^* T_\rho$ and $\zeta^* T_{\text{ev}}$
are ample.  Then $\zeta$ is very free relative to both $\text{ev}$ and
$\rho$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Rational connectedness of the Abel map
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rational connectedness of the Abel map} \label{sec-bigthm}
\marpar{sec-bigthm}

\mni
These are all the hypotheses.  However, before stating the theorem
there is one more definition.

\begin{defn} \label{def-dfree}
\marpar{def-dfree}
Let $d\geq 0$ be an integer.
For an algebraically closed extension field $k$ of $\kappa$, a
\emph{$d$-free section} of $\pi$ defined over $k$ is a section
$\sigma:C\otimes_\kappa k \rightarrow X$ defined over $k$ such that
for one (and hence every sufficiently general) effective Cartier
divisor $D$ of $C\otimes_\kappa k$ of degree $d$, 
$$
h^1(C\otimes_\kappa k,\sigma^* N_{\sigma(C\otimes_\kappa
  k)/X\otimes_\kappa k}(-D)) \text{ equals } 0.
$$
\end{defn}

\mni
In particular, when $d \geq \max{2g,1}$ the first
cohomology vanishes if and only if 
the map on
global sections
$$
H^0(C\otimes_\kappa k,\sigma^* N_{\sigma(C\otimes_\kappa
  k)/X\otimes_\kappa k}) \rightarrow
H^0(C\otimes_\kappa k,\sigma^* N_{\sigma(C\otimes_\kappa
  k)/X\otimes_\kappa k}\otimes_{\OO_C} \OO_D)
$$
is surjective.  We call a section \emph{$(g)$-free} if it is $d$-free
for some $d\geq \max{2g,1}$, i.e., for $g\geq 1$ it is $2g$-free and
for $g=0$ it is $1$-free.  

\begin{thm}[de Jong, Starr] \label{thm-main}
\marpar{thm-main}
If $(X,\mc{L},\pi)$ satisfies Hypotheses 1 through 6, 
then there
exists a sequence $(Z_e)_{e\gg 0}$ of irreducible components $Z_e$ of
$\Sigma^e(X/C/\kappa)$ such that
\begin{enumerate}
\item[(i)]
for each $e\gg 0$, a general point of $Z_e$ parameterizes a free
section of $\pi$,
\item[(ii)]
for each $e\gg 0$, the Abel map
$$
\alpha|_{Z_e}:Z_e \rightarrow \text{Pic}^e_{C/\kappa}
$$
is surjective with rationally connected geometric generic fiber, and
\item[(iii)]
for every algebraically closed field extension $k$ of $\kappa$, for
every $(g)$-free section $\sigma_0:C\otimes_\kappa k \rightarrow X$ of
$\pi$ 
defined over $k$, and
for all $\delta \gg 0$, every stable section obtained by attaching
$\delta$ free lines in fibers of $\pi$ to $\sigma_0(C)$ gives a $k$-point
of $\Sigma^{e_0+\delta}(X/C/\kappa)$ lying in $Z_{e_0+\delta}$.  
\end{enumerate}
Moreover every geometric fiber $X_t$ of $\pi$ is integral.  And every
point 
of the smooth locus $(X_t)_{\text{smooth}}$ is contained in a very
free rational curve in $(X_t)_{\text{smooth}}$.
\end{thm}

\mni
Note that (iii) implies that the sequence
$(Z_e)_{e\gg 0}$ is \emph{canonical}.  In particular, it is Galois
invariant.  Thus, in order to verify the theorem over every
characteristic $0$ field $\kappa$, it suffices to verify the theorem
over algebraically closed, characteristic $0$ fields.  Moreover, as
the statements are geometric, it suffices to pass to ``sufficiently
large'' algebraically closed fields, e.g., uncountable fields.  This
will be technically convenient later (although, of course, the
interested reader can easily formulate all of our arguments over any
characteristic $0$ field).  From this point on, we will assume that
$\kappa$ is an uncountable, algebraically closed field of
characteristic $0$.  

\mni
The proof of this theorem goes beyond what we can present here.
However we will present a proof in the special case necessary for de
Jong's Period-Index theorem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section II.6 : Rationally simply connected fibrations over a surface
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rational simply connected fibrations over a surface}
\label{sec-R1Csurf}
\marpar{sec-R1Csurf}

\mni
Theorem ~\ref{thm-main} is important because of its application to the
existence of rational sections of fibrations over surfaces.

\begin{cor}[de Jong, Starr] \label{cor-exist}
\marpar{cor-exist}
Let $k$ be an algebraically closed field of characteristic $0$.  Let
$S$ be a smooth, integral, projective surface over $k$.  Let
$f:X\rightarrow S$ be a proper morphism.  Assume there exists a
Zariski open subset $U$ of $S$ and an invertible sheaf $\mc{L}$ on
$f^{-1}(U)$ such that
\begin{enumerate}
\item[(i)]
$S-U$ is a finite collection of $k$-points of $S$,
\item[(ii)]
$f^{-1}(U)$ is smooth,
\item[(iii)]
the restriction 
$$
f|_{f^{-1}(U)}:f^{-1}(U) \rightarrow U
$$
is flat and surjective
\item[(iv)]
$\mc{L}$ is $f$-ample and $f$-relatively globally generated, and
\item[(v)]
the geometric generic fiber $Y$ of $f$ together with the base change
$\mc{L}_Y$ of $\mc{L}$ satisfy Hypotheses 3--6 of Section
~\ref{sec-RCAbel}.
\end{enumerate}
Then there exists a rational section of $f$.
\end{cor}

\begin{proof}
There exists a Lefschetz pencil of ample divisors on $S$ whose
basepoints are all contained in $U$.  So after replacing $S$ by the
blowing up of the base locus, replacing $f$ and $\mc{L}$ by the
pullbacks over the blowing up, and replacing $U$ by its inverse image
in the blowing up, we may assume in addition that there exists a
surjective, projective morphism
$$
r:S \rightarrow \PP^1_k.
$$
Theorem ~\ref{thm-G} generalizes to give a pair
$(\Sec^e(X/S/\PP^1_k),\sigma)$ of a $\PP^1_k$-scheme
$\Sec^e(X/S/\PP^1_k)$ and an $S$-morphism
$$
\sigma:\Sec^e(X/S/\PP^1_k)\times_{\PP^1} S \rightarrow X
$$
which is universal among such pairs.  And there is a generalization of
the definition of $\Sigma^e(X/C/\kappa)$ 
giving a $\PP^1$-scheme
$\Sigma^e(X/S/\PP^1_k)$ and a family of stable sections
$$
(\rho:\mc{C}\rightarrow \Sigma_e(X/S/\PP^1_k), h:\mc{C} \rightarrow X)
$$
of $\pi$ having $\mc{L}$-degree $e$.  And again using the methods of
\cite{detdiv}, there is an Abel map of $\PP^1_k$-schemes
$$
\alpha:\Sigma_e(X/S/\PP^1_k) \rightarrow \text{Pic}^e_{S/\PP^1_k},
$$
at least when we restrict over the dense open subset
$(\PP^1_k)_{\omega,\text{good}}$ -- the maximal open subset over which
$\omega$ is smooth (over the finitely many $k$-points of $\PP^1_k -
(\PP^1_k)_{\omega,\text{good}}$ there are some technical difficulties
in representing the relative Picard functor by a projective scheme).

\mni
Denote the function field $k(\PP^1_k)$ by $\kappa$ (of course it is
not algebraically closed).  So $\SP \kappa
\rightarrow \PP^1_k$ is the generic point of $\PP^1_k$.
The fibers of these $\PP^1_k$ schemes and morphisms of
$\PP^1_k$-schemes over $\SP \kappa$ are
$$
\Sigma^e(X/S/\PP^1_k)\times_{\PP^1_k} \SP \kappa =
\Sigma^e(X_\kappa/C/\kappa),
$$
$$
\text{Pic}^e(X/S/\PP^1_k)\times_{\PP^1_k} \SP \kappa =
\text{Pic}^e_{C/\kappa}
$$
where $X_\kappa$ denotes $X\times_{\PP^1_k} \SP \kappa$ and $C$
denotes $S\times_{\PP^1_k} \SP \kappa$.  Moreover the basechange of
the $\PP^1_k$-morphism $\alpha$ above is the Abel map
$$
\alpha:\Sigma^e(X_\kappa/C/\kappa) \rightarrow \text{Pic}^e_{C/\kappa}
$$
defined in Section ~\ref{sec-Abel}.

\mni
The hypotheses on $X,\mc{L},U$ and $Y$ imply Hypotheses 1--6 for
$X_\kappa, \mc{L}_\kappa$ and $Y$ (which hasn't actually changed).
Thus, by Theorem ~\ref{thm-main}, there exists a sequence
$(Z_e)_{e\geq e_0}$ of irreducible components $Z_e$ of
$\Sigma^e(X_\kappa/C/\kappa)$ satisfying Conditions (i)--(iii) of
Theorem ~\ref{thm-main}.  For every $e\geq e_0$, the closure of $Z_e$
in $\Sigma^e(X/S/\PP^1_k)$ is an irreducible component $Z_{\PP^1_k,e}$
of $\Sigma^e(X/S/\PP^1_k)$.  Because of Conditions (i) for
$(Z_e)_{e\geq e_0}$, a general point of $Z_{\PP^1_k,e}$ intersects
$\Sec^e(X/S/\PP^1_k)$ is a dense open subset of $Z_{\PP^1_k,e}$.  And
because of Condition (ii), for $e\gg 0$ the Abel map
$$
\alpha:Z_{\PP^1_k,e} \rightarrow \text{Pic}^e_{S/\PP^1_k}
$$
is surjective (at least over $(\PP^1_k)_{\omega,\text{good}}$), and
the geometric generic fiber is rationally connected.

\mni
For every integer $e$ there exist sections 
$$
\sigma:(\PP^1_k)_{\omega,\text{good}} \rightarrow
\text{Pic}^e_{S/\PP^1_k}.
$$
Indeed, the exceptional divisor over the blowing up of every basepoint
of the Lefschetz pencil of divisors on $S$ is a Cartier divisor on $S$
which defines a section to $\text{Pic}^1_{S/\PP^1_k}$.  So $e$ times
this section gives a section $\sigma$.

\mni
For $e\gg 0$, the geometric generic fiber of
$$
\alpha:Z_{\PP^1_k,e} \rightarrow \text{Pic}^e_{S/\PP^1_k}
$$
is rationally connected.
It may happen, nonetheless, that the fiber of 
$$
\alpha:Z_{\PP^1_k,e} \rightarrow \text{Pic}^e_{S/\PP^1_k}
$$
over the geometric generic point of $\sigma(\PP^1_k)$ is not
rationally connected. There is a standard way to deal with this issue.
We
can always realize $\sigma(\PP^1_k)$ as an irreducible component of
multiplicity $1$ in a complete intersection curve $C_0$ in
$\text{Pic}^e_{S/\PP^1_k}$, i.e., $C_0$ is a curve which is a complete
intersection of $g=g(C)$ very ample divisors in
$\text{Pic}^e_{S/\PP^1_k}$.  And we can deform these divisors to
obtain a family of complete intersection curves $(C_t)$ in
$\text{Pic}^e_{S/\PP^1_k}$ parameterized by a DVR $R=k\Sem{t}$.  By
deforming the divisors appropriately, we can arrange that the fiber of
$C_t$ over the geometric generic point $\ol{k\Semr{t}}$ is smooth, and the
fiber of
$\alpha$ over the geometric generic point of $C_t$ is rationally
connected.  

\mni
Thus after performing a base change from $k$ to
$L=\ol{k\Semr{t}}$, the restriction of $\alpha$ over the geometric
generic fiber 
$$
C_L:= C\otimes_{k\Sem{t}} \ol{k\Semr{t}}
$$
is a rationally connected fibration over the smooth, projective curve
$C_L$ over the algebraically closed field $L$.  Thus, by Theorem
~\ref{thm-GHS}, there exists a section.  But by the usual limit
theorems, every section over $L$ can, in fact, be defined over the
fraction field $k(R') = k\Semr{t^{1/N}}$ of some
finite extension $R'=k\Sem{t^{1/N}}$ of $R$.  

\mni
Since $\sigma(\PP^1_k)$
is an irreducible component of multiplicity $1$ in $C_0$,
$\sigma(\PP^1_k)$ is still an irreducible component of multiplicity
one in the base change $C_t\otimes_R R'$.  Thus $C_t\otimes_R R'$ is
normal at the generic point of $\sigma(\PP^1_k)$, which is a
codimension $1$ point of this $2$-dimensional scheme.  Finally, by the
valuative criterion of properness applied to the proper $\PP^1$-scheme
$Z_{\PP^1_t,e}$, the section defined over $C_t\otimes_R k(R')$ extends
to the generic point of $\sigma(\PP^1_k)$.  Thus there exists a
rational section 
$$
\tau:\sigma(\PP^1_k) \dashrightarrow Z_{\PP^1_k,e}
$$
of $\alpha$.  Again by the valuative criterion of properness, this
extends to all of $\sigma(\PP^1_k)$.

\mni
It may happen that $\tau(\SP \kappa)$ parameterizes a stable section
rather than a section.  But every stable section has a unique handle,
i.e., the unique component which is the image of a section.  Because
the handle is canonical, it satisfies the cocycle condition for Galois
descent, and thus is defined over the same field of definition as the
stable section -- in this case $\kappa$.  Thus finally, quite possibly
for some integer $e' < e$, there exists a $\kappa$-point of
$\Sec^{e'}(X/S/\PP^1_k)$.  By the universal property of
$\Sec^{e'}(X/S/\PP^1_k)$, this is equivalent to a morphism
$$
\SP \kappa \times_{\PP^1_k} S \rightarrow X.
$$
The closure of the image of this morphism is an integral 
closed subscheme $Z$ of $X$ such that the projection $Z\rightarrow S$
is a proper morphism which is an isomorphism over $C$.  Thus it is
birational, i.e., $Z$ is the closure of the image of a rational
section of $X\rightarrow S$.  Therefore $f:X\rightarrow S$ has a
rational section. 
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Discriminant avoidance
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Discriminant avoidance} \label{sec-DA}
\marpar{sec-DA}

\mni
Even though the theorems above require that $\text{char}(\kappa)$
equals $0$, there is a technique for ``lifting'' from positive
characteristic.  It is recalled very briefly below, but the proof will
not be discussed.

\mni
Let $\mc{G}$ be a reductive group scheme over some base scheme $T$.  
Let $\mc{X}$ be a smooth, projective 
$T$-scheme on which $\mc{G}$ acts.  For every
$T$-scheme $S$ and every $\mc{G}$-torsor $\mc{T}$ over $S$, there is
an associated $S$-scheme 
$$
\mc{X}_{\mc{T}} := \mc{X} \times_T \mc{T}/\mc{G},
$$ 
the quotient by the free action of $\mc{G}$.  Let $U$ be a dense open
subscheme of $T$.  Let $c$ be a nonnegative integer.  
Consider the following two
properties of the datum $(T,\mc{G},\mc{X})$ and the integer $c$.

\begin{prpty} \label{prpty-1}
\marpar{prpty-1}
For every algebraically closed field over $U$, $\SP k \rightarrow U$,
for every projective, integral $k$-scheme $S$ of dimension $c$, 
and for
every $\mc{G}$-torsor $\mc{T}$ over $S$, the projection 
$$
\mc{X}_{\mc{T}} \rightarrow S
$$
admits a rational section.
\end{prpty}

\begin{prpty} \label{prpty-2}
\marpar{prpty-2}
For every algebraically closed field over $T$, $\SP k \rightarrow U$,
for every quasi-projective, 
integral $k$-scheme $S$ of dimension $c$, and for
every $\mc{G}$-torsor $\mc{T}$ over $S$, the projection 
$$
\mc{X}_{\mc{T}} \rightarrow S
$$
admits a rational section.
\end{prpty}

\mni
The basic technique of ``discriminant avoidance'' proves the
following.

\begin{prop}\cite{dJS8} \label{prop-DA}
\marpar{prop-DA}
If Property 1 holds, then Property 2 holds.
\end{prop}

\begin{cor} \label{cor-DAPI}
\marpar{cor-DAPI}
To prove Theorem ~\ref{thm-PI}, it suffices to prove Theorem
~\ref{thm-main} in the special case that 
the morphism
$$
X\otimes_\kappa \ol{\kappa} \rightarrow C\otimes_\kappa \ol{\kappa}
$$
is of the form $\GR{C\otimes_\kappa \ol{\kappa}}(r,\mc{E})$ for a
locally free sheaf $\mc{E}$ of rank $n$ divisible by $r$ 
on $C\otimes_\kappa
\ol{\kappa}$.  
\end{cor}

\begin{proof}
By Lemma ~\ref{lem-PIG}, to prove Theorem ~\ref{thm-PI} it suffices to
prove that for every pair of positive integers $r,n$ with $r$ dividing
$n$, for every field $K$ as in Theorem ~\ref{thm-PI}, for every
central simple algebra $A$ with $\text{dim}_K(A)=n^2$ and
$\text{period}(A)$ dividing $r$, the generalized Brauer-Severi variety
$\GR{r,A}$ has a $K$-point.  

\mni
Define $T=\SP \ZZ_p$ and define $U=\SP \QQ_p$.  
Let $\mc{G}$
be the group scheme $\GL{n,T}/\mun{r,T}$ appearing in the proof of
Lemma ~\ref{lem-elemGBS}.  And let $(\mc{X},L)$ be $(\GR{T}(r,\OO_T^{\oplus
  n}),\OO(1))$.  By the proof of Lemma ~\ref{lem-elemGBS}, for every
algebraically closed field $k$, for every fraction field $K$ of a
surface over $k$, and for every central simple algebra $A$ over $K$
with $\text{dim}_K(A)=n^2$ and $\text{period}(A)$ dividing $r$, there
exists a quasi-projective surface $S$ over $k$ with fraction field
$K$, and there exists a $\mc{G}$-torsor $\mc{T}$ over $S$ such that
$\GR{r,A}$ is the generic fiber of the $S$-scheme $\mc{X}_{\mc{T}}$.
Thus by Proposition ~\ref{prop-DA}, to prove $\GR{r,A}$ always has a
$K$-point, it suffices to prove that for every \emph{characteristic
  $0$} algebraically closed field $k$, for every \emph{projective}
surface $S$ over $k$, and for every $\mc{G}$-torsor $\mc{T}$ over $S$,
$\mc{X}_{\mc{T}}$ has a rational section over $S$.  Define $\mc{L}$ on
$\mc{X}_{\mc{T}}$ to be the twist of $L$ on $\mc{X}$.  

\mni  
By resolution of singularities on surfaces in characteristic $0$
(which is reasonably easy), we may as well assume $S$ is smooth.  Then
the proof of Corollary ~\ref{cor-exist} reduces existence of a
rational section to Theorem ~\ref{thm-main} for the restriction of
$\mc{X}_{\mc{T}}$ and $\mc{L}$ to the generic fiber $C$ of a Lefschetz
pencil of divisors on $S$.  By Corollary ~\ref{cor-Tsen} and by
Proposition ~\ref{prop-division}(i), the base change of $\mc{T}$ to
$C\otimes_\kappa \ol{\kappa}$ lifts to a torsor for
$\GL{n,\ol{\kappa}}$.  This is equivalent to a locally free sheaf
$\mc{E}$ of rank $n$ on $C\otimes_\kappa \ol{\kappa}$.  Thus the base
change of $\mc{X}_{\mc{T}}$ to $C\otimes_\kappa \ol{\kappa}$ equals
$\GR{C\otimes_\kappa \ol{\kappa}}(r,\mc{E})$.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Proof of the main theorem for Grassmann bundles
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of the main theorem for Grassmann bundles} 
\label{sec-Grass}
\marpar{sec-Grass}

\mni
We will not prove Theorem ~\ref{thm-main}, but we will prove it for
Grassmann bundles coming from Azumaya algebras over $C$.  The
following proof is an \emph{ad hoc} argument using the well-known fact
that the moduli space of stable sheaves of fixed rank and determinant
on a curve is unirational.

\begin{prop} \label{prop-cheat}
\marpar{prop-cheat}
Let $\kappa$ be a characteristic $0$ field, let $C$ be a smooth,
projective, geometrically integral $\kappa$-scheme, let 
$$
\pi:X\rightarrow C
$$
be a flat, projective morphism, and let $\mc{L}$ be a $\pi$-ample
invertible sheaf on $X$ whose base change to the geometric generic
fiber of $\pi$ generates the Picard group.
Assume further that 
the morphism
$$
\pi\otimes \text{Id}:
X\otimes_\kappa \ol{\kappa} \rightarrow C\otimes_\kappa \ol{\kappa}
$$
is a Grassmann bundle $\GR{C\otimes_\kappa \ol{\kappa}}(r,\mc{E})$ for a
locally free sheaf $\mc{E}$ of rank $n$ divisible by $r$ 
on $C\otimes_\kappa
\ol{\kappa}$.  Then the conclusion of Theorem ~\ref{thm-main} holds.
Therefore, by Corollary ~\ref{cor-DAPI}, de Jong's Period-Index
theorem, Theorem ~\ref{thm-PI}, also holds.
\end{prop}

\begin{proof}
Thus $\Sec^e(X/C/\kappa)\otimes_\kappa \ol{\kappa}$ is the
scheme parametrizing locally free, rank $r$ quotients $\mc{Q}$ of
$\mc{E}^\vee$ of degree $e$.  And the Abel map associates to $\mc{Q}$
the determinant $\text{det}(\mc{Q}) = \bigwedge^r \mc{Q}$ (up to a
constant translation which makes no difference in the following). 

\mni 
Denote by $V_e$ the subscheme $V_e$ of $\Sec^e(X/C/\kappa)\otimes_\kappa
\ol{\kappa}$ parametrizing quotients with are stable sheaves.  
Since stability
is an open condition in families, $V_e$ is an open subscheme.  Moreover,
since $\mc{Q}$ is stable if and only if every twist of $\mc{Q}$ by an
invertible sheaf is stable, $V_e$ is independent of the choice of lift
of the torsor $(\mc{T}\times_S C)\otimes_\kappa \ol{\kappa}$ to a
$\GL{n,L}$-torsor over $C\otimes_\kappa L$, where $\mc{T}$ is as in
the proof of Corollary ~\ref{cor-DAPI}.  Therefore $V_e$ is invariant
under the action of $\G(\ol{\kappa}/\kappa)$, i.e., 
$V_e$ equals $U_e\otimes_\kappa \ol{\kappa}$
for some open subscheme $U_e$ of $\Sec^e(X/C/\kappa)$.  Finally, for
$e\gg 0$, there exist quotients $\mc{Q}$ which are stable so that $U_e$
is nonempty.  Define $Z_e$ to be the closure of $U_e$.

\mni
By standard arguments it is not hard to show that for $e\gg 0$
\emph{every} stable sheaf of rank $r$ and degree $e$ on
$C\otimes_\kappa L$ appears as a quotient $\mc{Q}$, and also that
$$
h^1(C,\textit{Hom}_{\OO_C\otimes_\kappa L}(\mc{E},\mc{Q}) \text{
  equals } 0.
$$
Therefore the forgetful morphism 
$$
\Phi:U_e \rightarrow \text{Bun}_{r,e,\text{stable}}(C\otimes_\kappa L)
$$
to the moduli space of stable rank $r$, degree $e$, locally free
sheaves on $C\otimes_\kappa L$ 
is surjective and the fiber over every point $\mc{Q}$ is an open
subset of the constant rank affine space
$\text{Hom}_{\OO_C\otimes_\kappa L}(\mc{E},\mc{Q})$, i.e., $\Phi$ is
an open subscheme of an affine bundle over the moduli space of stable
sheaves.  And the Abel map factors through the ``usual'' Abel map
$$
\text{det}: \text{Bun}_{r,e,\text{stable}}(C\otimes_\kappa L)
\rightarrow \text{Pic}^e_{C\otimes_\kappa L/L}.
$$

\mni
As is well-known, the morphism $\text{det}$ above 
has rationally connected (and even unirational)
fibers of dimension $(r^2-1)(g(C)-1)$, i.e., the moduli space of
stable sheaves of rank $r$ and fixed determinant is unirational.  
Thus the geometric generic fiber of 
$$
\alpha|_{U_e}:U_e \rightarrow \text{Pic}^e_{C/\kappa}
$$
is isomorphic to an open subscheme of an affine bundle over a
rationally connected variety, which is itself an open subscheme of a
rationally connected projective variety. 
But it is also
isomorphic to a dense open subset of the projective variety which is the
geometric generic fiber of 
$$
\alpha|_{Z_e}:Z_e \rightarrow \text{Pic}^e_{C/\kappa}.
$$
Since rational connectedness is a birational invariant among
projective varieties, the geometric
generic fiber of $\alpha|_{Z_e}$ is rationally connected.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Part: Rational simple connectedness and Serre's
%% ``Conjecture II''
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Rational simple connectedness and Serre's
  ``Conjecture II''} 
\label{pt-S2}
\marpar{pt-S2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Generalized Grassmannians are rationally simply connected
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Generalized Grassmannians are rationally simply connected}
\label{sec-GP}
\marpar{sec-GP}

\mni
This chapter is extracted from an earlier set of lecture notes,
\cite{SStrsbg}.  The omitted proofs can be found there.
The results of this chapter are joint work with
A. J. de Jong and Xuhua He, heavily relying on a strategy proposed by
Philippe Gille.

\mni
The proof of Proposition ~\ref{prop-cheat} used an \emph{ad hoc}
argument.  But in fact one can directly verify the hypotheses of
Theorem ~\ref{thm-main} for Grassmannians.  The most difficult
hypothesis to verify is the existence of a very twisting scroll.  This
is not too bad for Grassmannians.
But for \emph{generalized Grassmannians}, i.e., 
minimal (positive dimensional) homogeneous spaces,
for other groups, e.g., groups of type
$E_8$, it becomes unreasonable to verify this ``by hand''.  Thus it is
extremely fortunate that Xuhua He has given a very simple, elegant
proof that every generalized Grassmannian satisfies the hypotheses.
Moreover, the proof uses only general results about root systems -- it
does not use the classification of simple algebraic groups.  The
conclusion is the following.

\begin{cor}[de Jong, He, Starr] \label{cor-GPtwistR1C}
\marpar{cor-GPtwistR1C}
Let $k$ be an algebraically closed field of characteristic $0$.  Let
$X$ be a smooth, connected, projective $k$-scheme of positive
dimension which is a homogeneous space for a linear algebraic group
scheme $G$ over $k$. Assume there exists an invertible sheaf
$\mc{L}$ on $X$ which is an ample generator for
$\text{Pic}(X)$.  Then $X$ and $\mc{L}$ satisfy Hypotheses 3--6 of
Section ~\ref{sec-RCAbel}.
\end{cor}

\begin{cor}[de Jong, He, Starr] \label{cor-GPexist}
\marpar{cor-GPexist}
Let $k$ be an algebraically closed field of characteristic $0$.  Let
$S$ be a smooth, integral, projective surface over $k$.  Let
$f:X\rightarrow S$ be a proper morphism.  Assume there exists a
Zariski open subset $U$ of $S$ and an invertible sheaf $\mc{L}$ on
$f^{-1}(U)$ such that
\begin{enumerate}
\item[(i)]
$S-U$ is a finite collection of $k$-points of $S$,
\item[(ii)]
$f^{-1}(U)$ is smooth,
\item[(iii)]
the restriction 
$$
f|_{f^{-1}(U)}:f^{-1}(U) \rightarrow U
$$
is flat and surjective
\item[(iv)]
$\mc{L}$ is $f$-relatively globally generated, and
\item[(v)]
the geometric generic fiber $Y$ of $f$ is a homogeneous space for a
linear algebraic group over $K=\ol{k(S)}$ and $\mc{L}_Y$ is an ample
generator for $\text{Pic}(Y)$.
\end{enumerate}
Then there exists a rational section of $f$.
\end{cor}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Statement of the theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Statement of the theorem}
\label{sec-S2state}
\marpar{sec-S2state}

\mni
Serre's Conjecture II, Conjecture ~\ref{conj-S2}, has been proved for
some important classes of fields, but in general it remains open.  One
class of fields satisfying the hypotheses of the conjecture is the
class of function fields $K$ of surfaces over an algebraically closed
field $k$.  Notice that if $\text{char}(k)$ is positive, then $K$ is
not perfect but does satisfy the conditions of Remark ~\ref{rmk-S}.  

\mni
For these fields, there has been tremendous progress.  In particular, 
work of Merkurjev--Suslin,
Ojanguren--Parimala, Colliot-Th\'el\`ene--Gille--Parimala and Gille
has reduced Serre's Conjecture II for all function fields $K$ of
surfaces over an algebraically closed field $k$ to the special case
for such fields when $G$ is an simple algebraic group of type $E_8$,
cf. \cite[Theorem 1.2(v)]{CTGP}.  The main application of Theorem
~\ref{thm-main} and Corollary ~\ref{cor-exist} settles the ``split
case'' of Serre's Conjecture II for function fields $K$.

\begin{thm}[de Jong, He, Starr] \label{thm-SIIsplit}
\marpar{thm-SIIsplit}
Let $k$ be an algebraically closed field and let $K/k$ be the function
field of a surface.  Let $G$ be a connected, simply connected,
semisimple algebraic group over $k$.  Every $G$-torsor over $K$ is
trivial.

\mni
In particular, if $G_K$ is a simple algebraic group over $K$ of type
$E_8$, then $G_K$ is itself split and thus every $G_K$-torsor over $K$
is trivial.  So Serre's Conjecture II holds over function fields for
groups of type $E_8$.
\end{thm}

\begin{rmk} \label{rmk-Gille}
\marpar{rmk-Gille}
As we explain further below, in addition to Corollary
~\ref{cor-GPexist}, the main input to Theorem ~\ref{thm-SIIsplit} is a
strategy due to Philippe Gille.
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section : Reductions of structure group
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Reductions of structure group}
\label{sec-red}
\marpar{sec-red}

\mni
There is a statement that implies both Theorem ~\ref{thm-SIIsplit} as
well as again proving Theorem \ref{thm-PI} (it is essentially the same
proof as above).  The statement below is a generalization of a
consequence of Theorem ~\ref{thm-main} proved by Philippe Gille.  
Gille's strategy is absolutely fundamental in the proof of Theorem
~\ref{thm-SIIsplit}.  
Let $k$ be an algebraically closed
field.  Let $G$ be a (smooth)
connected, simply connected,
semisimple algebraic group over $k$, and let $P$ be a
(reduced) parabolic subgroup of $G$.  The center $Z_G$ of $G$ is a
finite group scheme which
is contained in
$P$.  There is a maximal quotient $P\twoheadrightarrow T_P$ which is an
algebraic torus.  Denote by $Z_{G,P}$ the kernel of the induced
homomorphism $Z_G \rightarrow T_P$.  The natural action of $G$ on
$G/P$ lifts canonically to a linear action on every invertible sheaf over
$G/P$.  The finite subgroup scheme $Z_{G,P}$ is the maximal subgroup
acting trivially on $G/P$ and on every invertible sheaf over $G/P$.
Thus $G/Z_{G,P}$ is the maximal quotient of $G$ acting on $G/P$ whose
action lifts to a linear action on every invertible sheaf over $G/P$.  

\begin{thm} \label{thm-main2}
\marpar{thm-main2}
Let $K/k$ be the function field of a surface over $k$.  For every
torsor $\mc{T}$ for $G/Z_{G,P}$ over $K$, the associated $K$-variety
$X= \mc{T}/P$ has a $K$-point.  Equivalently, the torsor $\mc{T}$ has
a reduction of structure group to $P/Z_{G,P}$.  
\end{thm}

\begin{proof}
Let $T$ denote Spec of the Witt ring of $k$, e.g., $T=\SP k$ if $k$
has characteristic $0$ and $T=\SP \ZZ_p$ if $k$ equals $\ZZ/p\ZZ$.
And denote by $U$ the open subset of $T$ which consists of the generic
point only.

\mni
Associated to the root datum for $G$, we can construct a smooth,
linear algebraic group scheme $G_T$ over $T$.  And associated to
the parabolic $P$, we can construct a closed subgroup scheme $P_T$
over $T$.  The definition of $Z_{G,P}$ extends to give a finite, flat
group scheme $Z_{G_T,P_T}$ over $T$.  The quotient group scheme
$\mc{G} = G_T/Z_{G_T,P_T}$ is a reductive group scheme since it is
$T$-flat and the closed fiber is reductive.  And the $T$-scheme
$\mc{X} = G_T/P_T$ is smooth and quasi-projective.  Since the closed
fiber is proper over $k$, $\mc{X}$ is projective over $T$.  Thus
$\mc{G}$ and $\mc{X}$ satisfy the hypotheses in Section
~\ref{sec-DA}.  The goal is to prove Property ~\ref{prpty-2} for
$c=2$.  Because of Proposition ~\ref{prop-DA}, it suffices to prove
Property ~\ref{prpty-1} for $c=2$.  In particular, to prove the
theorem it suffices to assume that $k$ has characteristic $0$.  

\mni
Now we use an induction argument proposed by Philippe Gille.  The
induction is on the corank,
$\text{rank}(G)-\text{rank}(P)$.  The base case is
when $P$ is a maximal parabolic.  Again applying Proposition
~\ref{prop-DA}, to prove the result for all $G/Z_{G,P}$-torsors over
fraction fields $K$ of surfaces over $k$, it suffices to prove the
result for each torsor which is the generic fiber of a
$G/Z_{G,P}$-torsor over a smooth, projective, connected surface $S$
over $k$.  
In this case the $S$-scheme
$\mc{X}_S = [\mc{T}_S \times (G/P)]/(G/Z_{G,P})$ satisfies the
hypotheses of Corollary ~\ref{cor-GPexist}.  Thus Corollary
~\ref{cor-GPexist} implies the result in this case.

\mni
By way of induction, assume the corank is $>1$ and the result is known
for all smaller values of the corank.  Since the corank is $>1$, $P$
is not a maximal parabolic.  Let $Q$ be a maximal parabolic containing
$P$.  Then $Z_{G,P}$ is contained in $Z_{G,Q}$.  For every
$G/Z_{G,P}$-torsor over $K$, by the base case, the associated
$G/Z_{G,Q}$-torsor has a reduction of structure group to $Q/Z_{G,Q}$.
Thus the original $G/Z_{G,P}$-torsor has a reduction of structure
group to $Q/Z_{G,P}$ (observe $(G/Z_{G,P})/(Q/Z_{G,P})$ is the same as
$(G/Z_{G,Q})/(Q/Z_{G,Q})$ since both are just $G/Q$).

\mni
Now $Q$ has
a filtration by normal subgroup schemes,
$$
Q = Q_0 \supset Q_1 \supset Q_2,
$$
where $Q_2$ is the unipotent radical of $Q$ and where
$Q_0/Q_1$ is the maximal quotient of $Q$ which is of
multiplicative type, i.e., isomorphic to $\mathbf{G}_{m,k}$.  By
Hilbert's Theorem 90, every $Q_0/Q_1$-torsor over $K$ is trivial, thus
there is a reduction of structure group to $Q_1/Z_{G,P}$ (by
construction $Z_{G,P}$ is contained in $P\cap Q_1$).
And over a
characteristic $0$ field, every torsor for a unipotent group is
trivial.  Thus this $Q_1/Z_{G,P}$-torsor has a reduction of structure
group to 
$(P\cap Q_1)/Z_{G,P}$ 
if and only if the associated $Q_1/Z_{G,P}Q_2$ torsor has a
reduction of structure group to $(P\cap Q_1)/Z_{G,P}(P\cap Q_2)$.  But
$Q_1/Q_2$ is again a semisimple, simply connected algebraic group, $G'$
$(P\cap Q_1)/(P\cap Q_2)$ is a parabolic subgroup $P'$, and
$Z_{G',P'}$ equals the image of $Z_{G,P}$.  Since the corank of $P'$
in $G'$ is $1$ less than the corank of $P$ in $G$, by the induction
hypothesis every $G'/Z_{G',P'}$-torsor over the fraction field of a
surface
$K$
has a reduction of structure
group to $P'/Z_{G',P'}$.  Thus every $G/Z_{G,P}$-torsor over $K$ has a
reduction of structure group to $P/Z_{G,P}$.  Therefore the result is
proved by induction on the corank.
\end{proof}

\begin{proof}[Proof of Theorem ~\ref{thm-PI}]
This is roughly the same as the proof already presented.  We include
this proof to illustrate how Theorem ~\ref{thm-main2} connects the
Period-Index problem, Problem ~\ref{prob-PI}, and Serre's Conjecture
II, Conjecture ~\ref{conj-S2}.
Let $G$ be $\mathbf{SL}_{n,k}$.  Let $m$ be an integer $1<m<n$ and
which divides $n$.
Let $P$ be the maximal parabolic
subgroup of $\mathbf{SL}_{n,k}$ consisting of upper block matrices
with the upper right block of size $m$ and another diagonal block of
size $n-m$.  The center $Z_G$ of $\mathbf{SL}_{n,k}$ is the group scheme
$\mathbf{\mu}_n$ of $n^\text{th}$ roots of unity.  And $Z_{G,P}$ is
the subgroup scheme $\mathbf{\mu}_m$.  

\mni
One can prove that a torsor for $G/Z_G = \mathbf{PGL}_{n,k}$ has a
reduction of structure group to $G/Z_{G,P}$ if and only if the order
of the corresponding element in the Brauer group
$H^2(\text{Gal}(K),\mathbf{\mu}_n)$ divides $m$.  And then, by
Theorem ~\ref{thm-main2}, there is a reduction of structure group to
$P/Z_{G,P}$.  

\mni
Here is a reformultation in terms of central simple algebras.
Let $C$ be a central simple algebra with center
$K$ and with $\text{dim}_K(C) = n^2$.  Let $\mc{T}_K$ be the
$K$-scheme whose set of $A$-points for each commutative
$K$-algebra $A$ equals the
set of $A$-algebra isomorphisms of $C\otimes_K A$ with
$\text{Mat}_{n\times n,A}$.  Since the automorphism group of
$\text{Mat}_{n\times n}$ is $\mathbf{PGL}_n$, $\mc{T}_K$ is a
$\mathbf{PGL}_{n,k}$-torsor over $K$.  By the previous paragraph, if
the order of $[C]$ in the Brauer group of $K$ divides $m$,
then there is a reduction of structure group to $P/Z_{G,P}$.  But this
is the same thing as an isomorphism of $C$ with $\text{Mat}_{m\times
  m,K} \otimes_K B$ for some central simple algebra $B$.  In
particular, if $D$ is a division algebra over $K$ with
$\text{dim}_K(D) = n^2$, then $D$ is not isomorphic to
$\text{Mat}_{m\times m,K} \otimes_K B$.  Thus the order of $[D]$ does
not divide $m$.  Since this holds for every proper divisor $m$ of $n$,
the conclusion is that the order of $[D]$ equals $n$.  
\end{proof}

\begin{proof}[Proof of Theorem ~\ref{thm-SIIsplit}]
By the same argument as in the proof of Theorem ~\ref{thm-main2}, it
suffices to prove the case when $k$ has characteristic $0$.  
Denote by $B$ a Borel subgroup of $G$.  Then $Z_{G,B}$ is the trivial
group scheme.  So by Theorem ~\ref{thm-main2}, every $G$-torsor over
$K$ has a reduction of structure group to a $B$-torsor over $K$.
Denote by $R_u(B)$ the unipotent radical of $B$.  Since $B$ is
connected and solvable, $B/R_u(B)$ is of multiplicative type, i.e.,
isomorphic to $\mathbf{G}_{m,k}^r$ where $r$ is the rank of $G$.  By
Hilbert's Theorem 90, every torsor for $B/R_u(B)$ is trivial.  Thus
there is a reduction of structure group to $R_u(B)$.  But every torsor
for a unipotent group over a characteristic $0$ field is trivial.
Thus the $B$-torsor is trivial, and hence the original $G$-torsor was
also trivial.
\end{proof}

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