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\title
{Arithmetic over function fields}

\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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\begin{abstract}
These notes accompany lectures presented at the 
Clay Mathematics Institute 2006 Summer School on
Arithmetic Geometry.  The lectures summarize some recent
progress on existence of rational points of projective
varieties defined over a function field over an
algebraically closed field.  
\end{abstract}

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

\mni
These notes accompany lectures presented at the Clay Mathematics
Institute 2006 Summer School on Arithmetic Geometry.  They are
more complete than the lectures themselves.  Exercises assigned
during the lectures are proved as lemmas or propositions in these notes.
Hopefully this makes the notes useful to a wider audience than the
original participants of the summer school.

\mni
This report describes some recent progress on questions
in the interface between arithmetic geometry and
algebraic geometry.  In fact the questions come from arithmetic
geometry: what is known about existence and ``abundance'' of points on
algebraic varieties defined over a non-algebraically closed field
$K$.  But the answers are in algebraic geometry, i.e., they apply only
when the field $K$ is the function field of an algebraic variety over
an algebraically closed field.  For workers in number theory, such
answer are of limited interest.  But hopefully the techniques will be
of interest, perhaps as simple analogues for more advanced techniques
in arithmetic.  With regards to this hope, the reader is encouraged
to look at two articles on the arithmetic side, \cite{GHMS2} and \cite{GHMS3}. 
Also, of course, the answers have interesting consequences
within algebraic geometry itself. 

\mni
There are three sections corresponding to the three lectures I
delivered in the summer school.  The first lecture proves the
classical theorems of
Chevalley-Warning and Tsen-Lang: complete intersections in projective
space of sufficiently low degree defined over
finite fields or over function fields always have rational points.
These theorems imply corollaries about the Brauer group and Galois
cohomology of these fields, which are also described.

\mni
The second section introduces rationally connected varieties and
presents the proof of Tom Graber, Joe Harris and myself of a
conjecture of Koll\'ar, Miyaoka and Mori: every rationally connected
fibration over a curve over an algebraically closed field of
characteristic $0$ has a section.  The proof presented here
incorporates simplifications due to A. J. de Jong.  Some effort is
made to indicate the changes necessary to prove A. J. de Jong's
generalization to separably rationally connected fibrations over
curves over fields of arbitrary characteristic.  In the course of the
proof, we give a thorough introduction to the ``smoothing combs''
technique of Koll\'ar, Miyaoka and Mori and its application to weak
approximation for ``generic jets'' in smooth fibers of rationally
connected fibrations.  
This has been significantly generalized to weak approximation for
\emph{all} jets in smooth fibers 
by Hassett and Tschinkel, cf. \cite{HT06}.
Some corollaries of the Koll\'ar-Miyaoka-Mori conjecture to
Mumford's conjecture, fixed point theorems, and fundamental groups are
also described (these were known to follow before the conjecture was proved).

\mni
Finally, the last section hints at the beginnings of a generalization
of the Koll\'ar-Miyaoka-Mori conjecture
to
higher-dimensional function fields (not just function fields of curves).
A rigorous result in this area is a
second proof of A. J. de Jong's \emph{Period-Index Theorem}: for a
division algebra $D$ whose center is the function field $K$ of a surface,
the index of $D$ equals the order of $[D]$ in the Brauer group of
$K$.  This also ties together the first and second sections.
Historically the primary motivation for the theorems of Chevalley,
Tsen and Lang had to do with Brauer groups and Galois cohomology.  The
subject has grown beyond these first steps.  But the newer results do
have consequences for Brauer groups and Galois cohomology in much the
same vein as the original results in this subject.


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%% Section: The Tsen-Lang theorem
%% 
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\section{The Tsen-Lang theorem} \label{sec-TL}
\marpar{sec-TL}

\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
$$
p(X_0,\dots,X_n) = p_{q-1}\cdot X_n^q + \dots  + p_0 X_n^0, \ \
p_0,\dots,p_{q-1} \in K[X_0,\dots,X_{n-1}].
$$
(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 
$$
p_0(a_0,\dots,a_{n-1})=\dots=p_{q-1}(a_0,\dots,a_{n-1}).
$$
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 algorithms 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}

\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 the 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}

\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}.
Let $K$ be a field.  
A \emph{division algebra with center $K$} is a $K$-algebra $D$ with
center $K$ such that every nonzero element of $D$ has a (left-right)
inverse.  Although this is not always the case, we will also demand
that $\text{dim}_K(D)$ is finite.  

\mni
Denote by $\overline{K}$ the separable closure of $K$.  Every division
algebra with center $K$ is an example of a \emph{central simple
  algebra over $K$}, i.e., a $K$-algebra $A$ with center $K$ and 
$\text{dim}_K(A)$ finite
such that $A\otimes_K \overline{K}$ is isomorphic as a
$\overline{K}$-algebra to the algebra $\text{Mat}_{n\times
  n}(\overline{K})$ of $n\times n$ matrices with entries in
$\overline{K}$ for some integer $n$.  In particular, $\text{dim}_K(A)
= n^2$ for a unique integer $n$.  For a division algebra $D$ with
center $K$, the unique integer $n$ is called the \emph{index} of $D$.

\mni
Let $\phi:A\otimes_K \overline{K} \rightarrow \text{Mat}_{n\times
  n}(\overline{K})$ be an isomorphism of $\overline{K}$-algebras.
There is an induced homogeneous, degree $n$, polynomial map of
$\overline{K}$-vector spaces
$$
\text{det}\circ \phi:A \rightarrow \text{Mat}_{n\times
  n}(\overline{K}) \rightarrow \overline{K}.
$$
By the Skolem-Noether theorem, every other isomorphism
$\phi':A\otimes_K \overline{K} \rightarrow \text{Mat}_{n\times
  n}(\overline{K})$ is of the form $\text{conj}_a\circ \phi$ where
$a\in \text{Mat}_{n\times n}(\overline{K})$ is an invertible element
and
$$
\text{conj}_a:\text{Mat}_{n\times n}(\overline{K}) \rightarrow
\text{Mat}_{n\times n}(\overline{K}), \ \ \ \text{conj}_a(b) = aba^{-1}
$$
is conjugation by $a$.  But $\text{det}\circ \text{conj}_a$ equals
$\text{det}$.  Thus the map $\text{det}\circ\phi$ is independent of
the particular choice of $\phi$.  Since the Galois group of
$\overline{K}/K$ acts on the polynomial map through its action on
$\phi$, the polynomial map is also Galois invariant.  Therefore there
exists a unique homogeneous, degree $n$, polynomial map of $K$-vector
spaces
$$
\text{Nrm}_{A/K}:A \rightarrow K
$$
such that for every isomorphism of $\overline{K}$-algebras $\phi$,
$\text{det}\circ \phi$ equals $\text{Nrm}_{A/K}\otimes 1$.  

\mni
The homogeneous, polynomial map of $K$-vectors spaces
$\text{Nrm}_{A/K}$ is the \emph{reduced norm} of
$A$.  It is multiplicative, 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.  
\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}
Let $D$ be a division algebra with center $K$.  Denote by $n$ the
index of $D$.
Because $\text{Mat}_{n\times n}(\overline{K})$ 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
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.  The
\emph{cohomological dimension} of a field $K$ is the smallest integer
$\text{cd}(K)$ such that for every Abelian, discrete, torsion Galois
module $A$ and for every integer $m > \text{cd}(K)$,
$$
H^m(\overline{K}/K,A) = \{0\}.
$$

\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% 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 a certain class 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, whose proof was sketched in the lectures of Hassett in this
same Clay Summer School,
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
$n\geq 1$, the analogous property holds after replacing $\PP^1_k$ by the
proper, connected,
nodal, reducible curve $C_n$ which is a chain of $n$ smooth rational curves.
\end{defn}

\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 \emph{very free
  rational curve}, i.e., 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.,
$$
f^* T_X \cong \OO_{\PP^1_k}(a_1) \oplus \dots \oplus
\OO_{\PP^1_k}(a_n), \ a_1,\dots,a_n > 0.
$$

\begin{defn} \label{defn-vflocus}
\marpar{defn-vflocus}
The \emph{very free locus} $X_\text{v.f.}$ 
of $X$ is the union of the images in $X$ 
of all very free rational curves to $X\otimes_k K$ as $K/k$ varies
over all algebraically closed extensions.
\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 quasi-projective 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 $\pi:X\rightarrow B$ (not necessarily of quasi-projective
varieties over a field)
having irreducible geometric
fibers, there exists an open subset $X_{\pi,\text{v.f.}}$ of $X$
such that for every geometric fiber $X_b$ of $\pi$,
$(X_b)_{\text{v.f.}}$ 
equals $X_{\pi,\text{v.f.}} \times_B X_b$.  In
particular, the  
subset of $B$ parameterizing points whose geometric fiber
contains a very free rational curve is an open subset,
$\pi(X_{\pi,\text{v.f.}})$.
\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
$$
f^* T_X \cong \OO_{\PP^1_k}(a_1) \oplus \dots \oplus
\OO_{\PP^1_k}(a_n), \ a_1,\dots,a_n \geq a.
$$
\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, irreducible 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 $m$ be a
closed point
of $M$ such that $u_m:\PP^1 \rightarrow X$ has image in
$X_\text{smooth}$ and such that $u^{(2)}$ is smooth at $(m,t_1,t_2)$
for some $t_1,t_2\in \PP^1_k$.  Then the morphism $u_m$ 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 $\pi_X:X\rightarrow B$ 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 $\pi_X\circ s$ 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.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Outline of the proof
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Outline of the proof} \label{ssec-out}
\marpar{ssec-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 $\pi_X|_C:C\rightarrow B$ 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 $\pi_X$.  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
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 $\pi_X$.
\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 $\pi_X:X\rightarrow B$ 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 $\pi_X$.
\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 $\pi_X|_C:C\rightarrow B$
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 $\Pi' \rightarrow \Pi$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Hilbert schemes and smoothing combs
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Hilbert schemes and smoothing combs} \label{ssec-Hilb}
\marpar{ssec-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.

\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
$$
h^1(Z_2,\mc{N}_{Z_2/Y_s}) = 0, h^1(Z_1,\mc{N}_{Z_1/Z_2}) = 0, \text{
  and if } h^i(Z_2,\mc{I}_{Z_1/Z_2}\cdot \mc{N}_{Z_2/Y_2}) = 0 \text{
  for } i=1,2,
$$
$\text{fHilb}(Y/S)$ 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 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 is 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 $jC\rightarrow Y$ 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 $\mc{N}_{C/Y} \rightarrow \mc{T}$.
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 smooth, irreducible, 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 smooth,
quasi-projective morphism $\pi:Y\rightarrow B$, the relative very free
locus $Y_{\pi,\text{v.f.}}$ is an open subset of $Y$.

\mni
Again in the absolute case,
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 $t_1$ 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}
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
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 union of two copies $C_1$ and $C_2$
of $\PP^1$
attached at $\infty$ and let $f:C\rightarrow Y$ be the morphism whose
restriction to each $C_i$ 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 $U$ be a smooth, quasi-projective $k$-scheme
and let $\pi:U \rightarrow B$ be a good morphism (in the sense of
Definition ~\ref{defn-U}).   
Assume there exists a section $s:B\rightarrow U$.  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 an 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}

\begin{proof}
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 Prop ~\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 
$$
f_i^* T_{\pi,\text{log}} \cong \OO_{\PP^1}(a_1)\oplus \dots \oplus
\OO_{\PP^1}(a_n), \ \ a_1,\dots,a_n \geq 1
$$
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 normal
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 normal 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 $U = \PP^1\times_k Y$ 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 closed points
$b_1,\dots,b_M$ of $\PP^1$, 
for every collection of closed points
$y_1,\dots,y_M$ of $Y_\text{v.f.}$, 
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 $h^1(B,\sigma^*\mc{N}_{\sigma(B)/U}(-2))$ equals $0$.  In
other words, $\phi:\PP^1_k \rightarrow Y$ is a morphism with $\phi(b')
= f(b')$ and with $h^1(\PP^1,\phi^*T_Y(-2))$ equals $0$.  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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Ramification issues
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Ramification issues} \label{ssec-ram}
\marpar{ssec-ram}

\mni
The argument sketched in Subsection ~\ref{ssec-out} and the powerful
smoothing combs technique from Subsection ~\ref{ssec-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 $\pi_X:X\rightarrow B$ is not smooth.  In other words,
finitely many scheme-theoretic fibers of $\pi_X$ 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.

\begin{defn} \label{defn-U}
\marpar{defn-U}
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.  
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
$\pi_X^{-1}(b)\cap U$ is smooth.  Denote the restriction of $\pi_X$ to
$U$ by $\pi_U$.  The morphism $\pi$ is \emph{good} if the good locus
equals all of $X$.

\mni
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.
Notice this is actually a finite sum over those 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 $\pi_X$ 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).

\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
$\pi_X(x)$ 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 $\widehat{\OO}_{U,x}$, 
$$
\pi_X^* t = a_m r^m + a_{m+1} r^{m+1} + \dots
$$
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 $\widehat{\OO}_{U,x}$ such that $\pi_X^*$ is
given by
$$
\pi_X^* t = r^m, \ \ \pi_X^*(dt) = mr^{m-1} dr.
$$ 
Thus $\pi^*$ locally factors through $\pi_X^*\Omega_B((e-1)D) =
\pi_X^*\Omega_B(D_{\pi,\text{log}})$.  Moreover it is clear that locally
the
cokernel is the locally free sheaf 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
of relative differentials of $\widetilde{U\times_B V}/V$ equals the
pullback of $\Omega_{\pi,\text{log}}$.  Thus the relative
deformation theory of
the $\widetilde{U\times_B V}$ 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 blowdown 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 $0$.
\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}

\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)]
$C$ intersects the very free locus
$U_{\pi,\text{v.f.}}$ of $\pi_U$, 
\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}

\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 $\mc{E}$, the restriction 
$h^1(\mc{C}_t,T_{\pi,\text{log}}|_{\mc{C}_t})$ 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 $j_C$ 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 such
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
$j:C_\text{comb} \rightarrow U$ is a linked log flexible curve.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Existence of log deformations
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Existence of log deformations} \label{ssec-exist}
\marpar{ssec-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 intersect 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
$b$ $B - \pi_C(D_C)$, 
$\Sigma'_b$ 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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Completion of the proof
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Completion of the proof} \label{ssec-proof}
\marpar{ssec-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 $C\cap
D_{\pi,\text{log}}$.  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 $\widetilde{\Gamma}_j(C_\text{flex})
\rightarrow C_\text{flex}$ 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 $\pi_X$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Subsection: Corollaries
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Corollaries} \label{ssec-cor}
\marpar{ssec-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}

\mni
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 $B$ 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 $f_R:Y_R\rightarrow
XYR$ such that the base change $Y_R\otimes_R k$ equals $Y$ and the
base change of $f_R$ equals $f$.  Since the intersection of the graph
of $f_R$ and the diagonal of $Y_R\times_R Y_R$ 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.

\mni
Corollary ~\ref{cor-K} is particularly interesting in combination with
a result which is attributed to Campana.

\begin{prop}[Campana] \label{prop-C}
\marpar{prop-C}
Let $Y$ be a quasi-projective, rationally chain connected variety over
an algebraically closed field $k$.  The algebraic
fundamental group of $Y$ is finite.  Moreover, if $k=\CC$, then the
topological fundamental group of $Y$ is finite.
\end{prop}

\mni
The basic point is this: there exists a choice of $u:M\times_k C_n
\rightarrow Y$ such that the induced morphism $u^{(2)}:M\times_k C_n
\times_k C_n \rightarrow Y\times_k Y$ is generically finite.  For a
general point $y$ of $Y$, the fiber over $\{y\}\times Y$ has an
irreducible component $N$ which is generically finite over $Y$.  Thus
there exists a morphism 
$$
v:N\times_k C_n \rightarrow Y
$$ 
and two points
$0,\infty \in C_n$ such that 
$$
v_0:N=N\times \{0\} \rightarrow Y
$$
the constant map with image $\{Y\}$ and 
$$
v_\infty: N = N\times \{\infty\} \rightarrow Y
$$
is dominant and generically finite.  Let $w:\widetilde{Y} \rightarrow
Y$ be a connected, unramified, cover in either 
the algebraic category or in the
complex analytic category.  Let $\widetilde{y}$ be a point over $y$.
The morphism $v_0$ lifts to a constant morphism $\widetilde{v}_0$.
Since $C_n$ is simply connected, this implies that there is a unique
morphism 
$$
\widetilde{v}:N\times_k C_n \rightarrow \widetilde{Y}
$$
whose restriction to $N\times \{0\}$ equals $\widetilde{v}_0$ and
which is compatible with $v$.  In particular, $v$ factors through
$w$.  Therefore the degree of $w$ is bounded by the degree of $v$ so
that $Y$ has fundamental group bounded by the degree of $v$.

\begin{cor}\cite{C91} \cite{KollarRC} \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
This was proved over $\CC$ by Campana.  The general case was proved by
Koll\'ar.  By Proposition ~\ref{prop-C}, the universal cover
$\widetilde{X} \rightarrow X$ is finite.  Moreover, by the proof of
the proposition, $\widetilde{X}$ is also 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.
Koll\'ar has generalized this considerably to prove a result for open
subschemes of rationally connected varieties.

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

\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 $X$, 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).  

\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}.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: The Period-Index theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Period-Index theorem} \label{sec-PI}
\marpar{sec-PI}

\mni
Theorem ~\ref{thm-GHS} is a generalization of Tsen's theorem,
Corollary ~\ref{cor-Tsen}, because a sufficiently 
general complete intersection
$\mathbb{V}(F_1,\dots,F_r) \subset \PP^n$ with $d_1 + \dots + d_r \leq
n$ is smooth, projective and separably rationally connected (the proof
of this is nontrivial, as is the specialization argument reducing
Tsen's theorem to the case of complete intersections which are
sufficiently general).
Is there a similar generalization of the Tsen-Lang theorem? 

\mni
Joint
work with Harris, \cite{HS2}, proves the spaces of rational curves on
general low degree hypersurfaces are rationally connected.  This was
later generalized in joint work with A. J. de Jong: 
complete
intersections $X=\mathbb{V}(F_1,\dots,F_r) \subset \PP^n$ with $d_1^2 +
\dots + d_r^2 \leq n+1$ are rationally simply connected in
the sense that the space of ``good'' rational curves
in $X$ containing
two fixed, general points is itself a rationally connected variety.
This is analogous to simple connectedness in topology: a path connected
topological space is simply connected if the space of paths connecting
two fixed points is itself path connected. 

\mni
Moreover, de Jong gave a heuristic argument suggesting that for a
rationally simply connected fibration over a surface, the only
obstruction to existence of a rational section is the elementary
obstruction.  Given a geometrically integral scheme $X$ 
defined over a
field $K$, the \emph{elementary obstruction} to existence of a
$K$-point is the existence of a $\text{Gal}(\ol{K}/K)$-invariant
splitting of the homomorphism of Abelian Galois modules,
$$
\ol{K}^* \hookrightarrow \text{Frac}(X\otimes_K \ol{K})^*,
$$
where $\ol{K}$ is the separable closure of $K$ and $\text{Frac}$ is
the function field.
If there exists a
$K$-point of $X$, evaluation at this point gives a Galois-invariant
splitting.  
The elementary obstruction 
was introduced by Colliot-Th\'el\`ene and Sansuc, \cite{CTSansuc}.  Its
vanishing implies the vanishing of other known obstructions.  In
particular, it implies the vanishing of a Brauer obstruction
$$
\delta:\text{Pic}(X\otimes_K \ol{K})^{\text{Gal}(\ol{K}/K)}
\rightarrow \text{Br}(K)
$$
measuring whether or not a Galois-invariant invertible sheaf
$\ol{\mc{L}}$ on $X\otimes_K \ol{K}$ is the pullback of an invertible
sheaf $\mc{L}$ on $X$.  

\mni
At the moment, in order to give a rigorous proof, de Jong's heuristic
argument requires several additional hypotheses on the rationally
simply connected fibration.  One case where the hypotheses hold is
when all fibers of the fibration are Grassmannian varieties.  Although
this is very special, it is also quite interesting since it gives a
second proof of de Jong's \emph{period-index theorem}.

\begin{thm} \cite{dJS8}
  \label{thm-Grass}
\marpar{thm-Grass}
Let $K$ be the function field of a surface over an algebraically
closed field $k$.  Let $(X,\mc{L})$ be a pair of a $K$-scheme and an
invertible $\OO_X$-module $\mc{L}$.  If $(X\otimes_K \ol{K},
\mc{L}\otimes_K \ol{K})$ is isomorphic to
$(\text{Grass}(r,\ol{K}^n),\OO(1))$, then $X$ has a $K$-point.
\end{thm}

\begin{cor}[de Jong's Period-Index theorem] \cite{dJ}
\label{cor-PI}
\marpar{cor-PI}
Let $A$ be a central simple $K$-algebra with $A\otimes_K \ol{K} \cong
\text{Mat}_{n\times n}(\ol{K})$.  Let $r< n$ be an integer such
that $r[A]$ equals $0$ in $\text{Br}(K)$.  Then there exists a left
ideal $I\subset A$ such that $\text{dim}_K(I) = rn$.  In particular,
if $A = D$ is a division algebra then $[D]$ has order $n$ in
$\text{Br}(K)$, i.e., the period of $D$ equals the index of $D$.  
\end{cor}

\mni
Corollary ~\ref{cor-PI} follows from Theorem ~\ref{thm-Grass} by
setting $X$ to be the \emph{generalized Brauer-Severi variety}
parameterizing left ideals in $A$ of rank $rn$.  Since $A\otimes_K
\ol{K}$ equals $\text{Mat}_{n\times n}(\ol{K})$, $X\otimes_K \ol{K}$
equals $\text{Grass}(r,\ol{K}^n)$.  
The Brauer obstruction
to the existence of an invertible sheaf $\mc{L}$ with $\mc{L}\otimes_K
\ol{K} \cong \OO(1)$ is precisely the element $r[A]$ in
$\text{Br}(K)$.  

% \mni
% In fact Corollary ~\ref{cor-PI} is almost equivalent to Theorem
% ~\ref{thm-Grass}.  The distinction comes from outer automorphisms.
% Every pair $(X,\mc{L})$ determines a torsor for the automorphism group
% scheme of $(\text{Grass}(r,K^n),\OO(1))$.  This is a smooth, reductive
% group scheme whose identity component is a central extension of
% $\text{PGL}_n$ by $\mu_{n/r}$.  If $r\neq n/2$, then the group scheme
% is connected.  However, if $r=n/2$, the group scheme is disconnected
% and the group of components is $\ZZ/2\ZZ$.  Given a torsor, the
% associated pair $(X,\mc{L})$ arises from a central simple algebra
% $A$ if and only if the associated torsor for the group of
% components has a rational point.  From this point on, assume that
% $(X,\mc{L})$ arises from a central simple algebra $A$.  This
% hypothesis allows 
% the most difficult step in de Jong's heuristic argument to be
% replaced by a simple result about moduli of vector bundles on a smooth
% curve.  

\mni
The first reduction is ``discriminant avoidance'', i.e., reduction to
the case that the variety $X$ is the generic fiber of a smooth,
projective morphism over a smooth, projective surface.  Let $T$ be a
quasi-compact, 
integral scheme and let $\mc{G}$ be a smooth, affine group scheme over
$T$ whose geometric fibers are reductive.

\begin{lem} \label{lem-da}
\marpar{lem-da}
For every
integer $c$ there exists a datum $(U,\ol{U},\mc{T}_U)$ of a projective,
flat $T$-scheme $\ol{U}$ with integral geometric fibers, an open
subset $U$ of $\ol{U}$ and a $\mc{G}$-torsor $\mc{T}_U$ over $U$ such
that
\begin{enumerate}
\item[(i)]
$U$ is smooth over $T$,
\item[(ii)]
the complement $\ol{U}-U$ has codimension $\geq c$ in $\ol{U}$, 
\item[(iii)]
and
for every $\mc{G}$-torsor $\mc{T}_k$ over
an infinite field $K$ over $T$,
there exists a $T$-morphism
$i:\SP K \rightarrow U$ and an isomorphism of $\mc{G}$-torsors over $K$,
$i^* \mc{T}_U \cong \mc{T}_K$.
\end{enumerate}
\end{lem}

\mni
The idea is to form the GIT quotient $\ol{U}$ of a linear
action of $\mc{G}$ on $(\PP^N_T,\OO(1))$.  If the linear
representation is ``sufficiently large'', then $\mc{G}$ acts properly
and freely on an open subset $V$ of $\PP^N_T$ of
codimension $\geq c$.  Take $U$ to
be the quotient of $V$.  Then $U$ is smooth over $T$ and
$\ol{U}-U$ has codimension $\geq c$.  Finally, for every field $K$
over $\OO_T$ and for every $\mc{G}$-torsor $\mc{T}_K$ over $K$, the
twist $\PP^N_T \times_T \mc{T}_K/\mc{G}$ is isomorphic to $\PP^N_K$.
Thus there exists a $K$-point.  If $K$ is infinite, then the set of
$K$-points is Zariski dense so that there exists a point in the image
of $V$.  This point is only well-defined up to the action of $\mc{G}$,
but the associated morphism $i:\SP K \rightarrow U$ 
is well-defined.  Chasing definitions, $i^* \mc{T}_U$ is isomorphic to
$\mc{T}_K$.  

\begin{prop} \label{prop-reduct}
\marpar{prop-reduct}
Let $k$ be an algebraically closed field of characteristic $0$, let
$S$ be a smooth, projective surface over $k$, let
$\pi:\mc{X}\rightarrow S$ be a smooth, projective morphism, and let
$\mc{L}$ be an invertible $\OO_{\mc{X}}$-module.  Let $K/k$ be the
fraction field of $S$ and let $X$ be the generic fiber of $\pi$.
\begin{enumerate}
\item[(i)]
If $(X\otimes_K \ol{K},\mc{L}\otimes_K \ol{K})$ is isomorphic to
$(\text{Grass}(r,\ol{K}^n),\OO(1))$, then $X$ has a $K$-point.
\item[(ii)]
Item (i) implies Theorem ~\ref{thm-Grass}
\end{enumerate}
\end{prop}

\mni
The point of this proposition is Item (ii), i.e., to prove Theorem
~\ref{thm-Grass} it suffices to assume that $X$ is the generic fiber
of a proper and \emph{everywhere smooth} morphism.  The point is that
for every algebraically closed field $k$, there exists the spectrum of
a DVR, $T$, whose residue field is $k$ and whose fraction field has
characteristic $0$.  Now let $\mc{G}$ be the automorphism group scheme
of $(\text{Grass}(r,\OO_T^n),\OO(1))$.  This satisfies the hypotheses
of Lemma ~\ref{lem-da}.  Taking $c=3$, there exists a datum
$(U,\ol{U},\mc{T}_U)$ as in Lemma ~\ref{lem-da} such that $\ol{U}-U$
has codimension $\geq 3$.  For the original field $K/k$, there exists
a morphism $i:\SP K\rightarrow U$ inducing the pair $(X,\mc{L})$.
Because $\text{tr.deg.}(K/k) = 2$, the 
closure of $\text{Image}(i)$ in $\ol{U}$ has dimension $\leq 2$.  
Because $\ol{U}-U$ has
codimension $\geq 3$, there exists a locally closed subscheme $\mc{S}$
of $\ol{U}$ such that
\begin{enumerate}
\item[(i)]
$\mc{S} \rightarrow T$ is flat,
\item[(ii)]
the closed fiber $S_0$ of $\mc{S}$ is irreducible with generic point $i(\SP
K)$,
\item[(iii)]
and the generic fiber $\mc{S}_\eta$ of $\mc{S}$ is a closed subscheme
of $\ol{U}_\eta$ completely contained in $U_\eta$.
\end{enumerate}
Using specialization arguments, to prove Theorem ~\ref{thm-Grass} for
the restriction of $\mc{T}_U$ to the
generic point of the closed fiber $S_0$, it suffices to prove
Theorem ~\ref{thm-Grass} for the restriction of $\mc{T}_U$ to the
generic point of the geometric generic fiber $S_{\ol{\eta}}$.  By
construction, this satisfies the additional hypotheses in Proposition
~\ref{prop-reduct}.  

\mni
Thus, assume the additional hypotheses of Proposition
~\ref{prop-reduct} are satisfies.  After replacing $S$ by the blowing
up at the base locus of a Lefschetz pencil of divisors, and replacing
$\mc{X}$ by its base change, assume there exists a flat, proper
morphism $\rho:S\rightarrow B$ with smooth, connected generic fiber.
Denote by $B^0$ the maximal open subscheme of $B$ over which $\rho$ is
smooth, and denote $S^0 = B^0\times_B S$ and $\mc{X}^0 = B^0\times_B
\mc{X}$.  There is a pair
$$
(\rho_\text{Sect}:\text{Section}(\mc{X}^0/S^0/B^0) \rightarrow B^0,
\sigma: \text{Section}(\mc{X}^0/S^0/B^0) \times_{B^0} S^0 \rightarrow
\mc{X}^0)
$$
which is universal among all pairs $(T,\sigma_T)$ of a $B^0$-scheme
$T$ and an $S_0$-morphism $\sigma_T:T\times_{B^0} S^0\rightarrow
\mc{X}^0$.  The universal pair can be constructed in terms of the
relative Hilbert scheme.  In Grothendieck's terminology, it is
$\Pi_{S^0/B^0}\mc{X}_0$, cf. \cite[p. 195-13]{FGA}.

\mni
There is a relative Picard scheme $\text{Pic}(S^0/B^0)$ of $S^0$ over
$B^0$.  Associated to the invertible sheaf $\mc{L}$ on $\mc{X}^0$,
there is an invertible sheaf $\sigma^* \mc{L}$ on
$\text{Section}(\mc{X}^0/S^0/B^0)\times_{B^0} S^0$.  This induces an
\emph{Abel morphism}
$$
\alpha: \text{Section}(\mc{X}^0/S^0/B^0) \rightarrow
\text{Pic}(S^0/B^0).
$$
Of course $\text{Pic}(S^0/B^0)$ breaks up according the relative
degree of the line bundle,
$$
\text{Pic}(S^0/B^0) = \sqcup_{d\in \ZZ} \text{Pic}^d(S^0/B^0).
$$
Pulling this back via the Abel morphism gives a decomposition
$$
\text{Section}(\mc{X}^0/S^0/B^0) = \sqcup_{d\in \ZZ}
\text{Section}^d(\mc{X}^0/S^0/B^0)
$$
together with Abel morphisms
$$
\alpha_d: \text{Section}^d(\mc{X}^0/S^0/B^0) \rightarrow
\text{Pic}^d(S^0/B^0).
$$

\mni
For all $d\geq 0$, there are sections of the projection
$\text{Pic}^d(S^0/B^0) \rightarrow B^0$.  The image of this section is
a curve $B'_0$
isomorphic to the smooth curve $B^0$.  If the generic fiber of
$\alpha_d$ is a dense open subset of a rationally connected variety,
then Theorem ~\ref{thm-GHS} together with the generic version of weak
approximation, Proposition ~\ref{prop-genWA}, implies there exists a
rational section of the restriction of $\alpha_d$ over $B'_0$ (this
uses a slight specialization argument, because the restriction of
$\alpha_d$ may not be a rationally connected fibration).  Thus it
suffices to prove that for $d\gg 0$,
\begin{enumerate}
\item[(i)]
the fiber of $\alpha_d$ over the geometric generic point of
$\text{Pic}^d(S^0/B^0)$ is nonempty,
\item[(ii)]
the fiber is also irreducible,
\item[(iii)]
the fiber is also isomorphic to an open subset of a rationally
connected variety.
\end{enumerate}
Moreover, and this will be important, it suffices to prove there
exists a canonically defined open subscheme $W\subset
\text{Section}^d(\mc{X}^0/S^0/B^0)$ such that (i)--(iii) hold for
$\alpha_d|_W$.  

\mni
Note that (i)--(iii) are really statements about the morphism
$\mc{X}_{\ol{\eta_B}} \rightarrow S_{\ol{\eta_B}}$ of fibers over the
geometric generic point of $B$.  Thus, it is again a question about a
fibration over a projective curve, namely the curve $C=S_{\ol{\eta_B}}$
over the algebraically closed field $\kappa = \ol{k(B)}$.
So Proposition ~\ref{prop-reduct} (i) follows from.

\begin{prop} \label{prop-reduct2}
\marpar{prop-reduct2}
Let $\kappa$ be an algebraically closed field of characteristic $0$.  Let
$C$ be a smooth, projective, connected curve over $\kappa$.  Let
$\pi:\mc{X}_C \rightarrow C$ be a smooth, projective morphism and let
$\mc{L}$ be an invertible sheaf on $\mc{X}_C$.  Assume the fiber of
$(\mc{X}_C,\mc{L})$  over the
geometric generic point of $C$ is isomorphic to
$(\text{Grass}(r,\ol{\kappa(C)}),\OO(1))$.  Then for $d\gg 0$ there
exists a canonically defined open subset $W_d \subset
\text{Section}^d(\mc{X}_C/C/\SP \kappa)$ such that 
the
geometric generic fiber of
$$
\alpha_d:W_d \hookrightarrow \text{Section}^d(\mc{X}_C/C/\SP \kappa)
\rightarrow \text{Pic}^d(C/\SP \kappa)
$$
satisfies (i), (ii) and (iii) above.
\end{prop}

\mni
In order to prove Corollary ~\ref{cor-PI}, it suffices to prove this
in the special case that $\mc{X}_C \rightarrow C$ 
is the parameter scheme for rank $rn$ left
ideals in an Azumaya algebra $\mc{A}$ over $C$ with
$\mc{A}\otimes_{\OO_C} \ol{\kappa(C)} \cong \text{Mat}_{n\times
  n}(\ol{\kappa(C)})$.  An \emph{Azumaya algebra} over a scheme $T$ is
a coherent $\OO_T$-algebra which is \'etale locally isomorphic to
$\text{Mat}_{n\times n}(\OO_T)$ for some integer $n$.  

\mni
Because of Tsen's theorem, Corollary ~\ref{cor-Tsen}, and Proposition
~\ref{prop-division}(i), there exists a locally free $\OO_C$-module
$\mc{E}$ of rank $n$ such that $\mc{A} \cong \textit{End}(\mc{E})$.
The locally free sheaf $\mc{E}$ is only well-defined up to the
operation $\mc{E} \mapsto \mc{E}\otimes_{\OO_C} \mc{N}$, for any
invertible sheaf $\mc{N}$.  
The choice of a locally free sheaf $\mc{E}$ and an isomorphism of
algebras gives an isomorphism of 
$\text{Sect}(\mc{X}_C/C/\SP \kappa)$ with the parameter scheme of
locally free quotients $\mc{E} \twoheadrightarrow \mc{Q}$ of rank
$r$.  
To see this, associate to each quotient the left ideal of
endomorphisms that factor as
$$
\mc{E}\twoheadrightarrow \mc{Q}\xrightarrow{\phi} \mc{E}
$$
as $\phi$ varies over all $\OO_C$-module homomorphisms.  
Replacing $\mc{E}$ by $\mc{E}\otimes_{\OO_C} \mc{N}$ gives a new
isomorphism sending the original quotient to the twist
$\mc{E}\otimes_{\OO_C} \mc{N} \twoheadrightarrow
\mc{Q}\otimes_{\OO_C}\mc{N}$.  The real effect of this change has to
do with the Abel map.  
Up to a
constant translation by a point of $\text{Pic}(C/\SP \kappa)$, which
does not change (i)--(iii), the Abel map is identified with the map
sending a quotient to $\text{det}(\mc{Q})$ in $\text{Pic}(C/\SP
\kappa)$.  After replacing $\mc{E}$ by $\mc{E}\otimes_{\OO_C} \mc{N}$,
the constant changes by adding $[\mc{N}^{\otimes r}]$.  Thus, it is
clear that the Abel map really should only be considered as
well-defined up to an additive constant.

\mni
If $d$ is sufficiently large, there exist quotients such that $\mc{Q}$
is stable.  Note that $\mc{Q}$ is stable if and only if
$\mc{Q}\otimes_{\OO_C} \mc{N}$ is stable.  Therefore the open subset
$W_d$ of $\text{Section}(\mc{X}_C/C/\SP \kappa)$ parameterizing stable
quotients is well-defined and canonical.  Fix an integer $d_0$.  For
every integer $e$, the moduli space of stable, rank $r$ locally free
sheaves on $C$ of degree $d_0$ is isomorphic to the moduli space for
degree $d_0+re$ via the map sending $\mc{Q}$ to $\mc{Q}(D)$, where $D$
is any fixed Cartier divisor of degree $e$.  For each fixed locally
free sheaf $\mc{Q}$ of rank $r < n$, 
if $e$ is sufficiently large, there exists a surjection
$\mc{E}\twoheadrightarrow \mc{Q}(D)$ for all Cartier divisors of degree $e$.
Because the moduli space of stable bundles is quasi-compact, it
follows that there exists a single integer $e_0$ such that for every
$e\geq e_0$ and every stable locally free sheaf $\mc{Q}$ of rank $r$
and degree $d_0$, for every Cartier divisor $D$ of degree $e$ there
exists a surjection $\mc{E} \twoheadrightarrow \mc{Q}(D)$.  
By the same sort of argument, if $e$ is sufficiently large then
$h^1(C,\textit{Hom}_{\OO_C}(\mc{E},\mc{Q}(D)))$ equals $0$.

\mni
Repeating this argument with $d_0$ replaced by each of
$d_0,d_0+1,d_0+2,\dots, d_0+r-1$, there exists an integer $d_1$ such
that for every $d\geq d_1$ (i.e., $d=d_0 + re$, etc.), for every
stable, locally free sheaf $\mc{Q}$ of rank $r$ and degree $d$, there
exists a surjection
$$
\mc{E} \twoheadrightarrow \mc{Q}
$$
and also $h^1(C,\textit{Hom}_{\OO_C}(\mc{E},\mc{Q}))$ equals $0$.  In other
words, the forgetful morphism from the space of quotients $\mc{E}
\twoheadrightarrow \mc{Q}$ to the space of stable sheaves $\mc{Q}$ is
smooth and surjective, and the geometric fibers are each isomorphic to
an open subset of an affine space
$\text{Hom}_{\OO_C}(\mc{E},\mc{Q})$.  Moreover, the fiber of the Abel
map $\alpha_d$ is the inverse image of the space of stable sheaves
with fixed determinant.  As is well-known, the moduli space of stable
sheaves over $C$ of fixed rank $r$ and fixed determinant is a
unirational variety of dimension $(r^2-1)(g(C)-1)$.  Thus the fiber of
the Abel map fibers over a rationally connected variety and the fibers
are rationally connected.  By Corollary ~\ref{cor-KMM}, it follows
that a general fiber of
$$
\alpha_d|_W:W_d \hookrightarrow \text{Section}^d(\mc{X}_C/C/\SP
\kappa) \rightarrow \text{Pic}^d(C/\SP \kappa)
$$
is isomorphic to an open subset of a rationally connected variety,
i.e., (i), (ii) and (iii) hold.  Therefore Corollary ~\ref{cor-PI} is
true.

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