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\title
{Weak approximation and R-equivalence over function fields of curves}



\author[Roth]{Mike Roth}
\address{
Department of Mathematics and Statistics \\
Queen's University \\ 
Kingston, Ontario, K7L 3N6}
\email{mikeroth@mast.queensu.ca}

\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}
  For a rationally connected fibration over a complex curve, and for a
  closed point of the curve, we prove that every power series section
  of the fibration near the point is approximated to arbitrary order
  by polynomial sections provided the ``Laurent fiber'', i.e., the
  deleted power series neighborhood of the fiber, as a variety over
  the Laurent series field is \emph{$R$-connected} -- the analogue of
  rational connectedness when working over a non-algebraically closed
  field such as Laurent series.  In other words, we prove the
  Hassett-Tschinkel conjecture when the Laurent fiber is
  $R$-connected.  For varieties over the fraction field of a complete
  DVR, we introduce a ``continuous variant'' of $R$-connectedness
  called \emph{$\pse$ $R$-connectedness} and we prove $\pse$
  $R$-connectedness of the Laurent fiber also implies the
  Hassett-Tschinkel conjecture.  Our theorem implies all of the known
  cases of the Hassett-Tschinkel conjecture, and we also prove some
  new cases.  The key is a new object, a ``pseudo ideal sheaf'', which
  generalizes Fulton's notion of effective pseudo divisor.
\end{abstract}

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

\mni
\textbf{Disclaimer.}  This is the current, working draft of this
article.  This draft is incomplete, but has a complete list of the
results.  An earlier, complete draft has been posted to the arXiv
preprint server.

\mni 
Let $\pi:\mcX \rightarrow B$ be a surjective morphism from a smooth,
projective, complex variety to a smooth, projective, complex curve.
Considered differently, $\pi$ is a $1$-parameter family $\{ \mcX_b
\}_{b \in B}$ of projective varieties.  The morphism $\pi$ satisfies
\emph{weak approximation} if for every finite sequence
$(b_1,\dots,b_m)$ of distinct closed points of $B$, for every sequence
$(\whts_1,\dots,\whts_m)$ of elements $\whts_i$ in
$\mcX(\whtOO_{B,b_i})$, i.e., formal power series section of $\pi$
near $b_i$, and for every positive integer $N$, there exists a regular
(i.e., polynomial) section $s$ of $\pi$ which is congruent to
$\whts_i$ modulo $\mfm_{B,b_i}^{N}$ for every $i=1,\dots,m$.

\mni
Hassett observed that if $\pi$ satisfies weak approximation then every
sufficiently general fiber of $\pi$ is \emph{rationally connected},
i.e., every pair of points in the fiber is in the image of a morphism
from $\PP^1$ to the fiber.  Hassett and Tschinkel conjectured the
converse.

\begin{conj}[Hassett-Tschinkel conjecture]
\cite[Conjecture 2]{HT06} \label{conj-HT}
\marpar{conj-HT}
If a general fiber of $\pi$ is
rationally connected, then $\pi$ satisfies weak approximation.  
\end{conj}

\mni
We prove this conjecture assuming an extra condition on the ``Laurent
fibers'' of $\pi$.  For every closed point $b$ of $B$, the fraction
field $K = \Fr(\whtOO_{B,b})$ of the completed local ring is
isomorphic to the Laurent series field $\CC\Semr{t}$.  The
\emph{Laurent fiber} is the projective scheme $\mcX_K :=
\mcX\otimes_{\OO_B}K$.  This is an algebraic version of the
``deleted tubular neigborhood'' of the closed fiber $\mcX_b
:=\pi^{-1}(b)$.  Even when $\mcX_b$ is singular, the $K$-scheme
$\mcX_K$ is smooth.  Two $K$-points $s$ and $t$ of the Laurent fiber
are \emph{directly $R$-equivalent} if there exists a $K$-morphism from
$\PP^1_K$ sending $0$ to $s$ and sending $\infty$ to $t$,
$$
f:(\PP_K,0,\infty) \rightarrow (\mcX \otimes_{\OO_B} K, s,t).
$$
The Laurent fiber is \emph{$R$-connected} if all
$K$-points are directly $R$-equivalent.  
When $\mcX_b = \pi^{-1}(b)$ is smooth, the
Laurent fiber $\mcX_K$ is $R$-connected if and only if the closed
fiber $\mcX_b$ is rationally connected.  (The usual definition of
$R$-connected is a bit different.  But Koll\'ar proved the equivalence
of the two definitions when the geometric generic fiber is 
separably rationally connected, as it is here.)

\mni
We prove Conjecture \ref{conj-HT} when the Laurent fibers of
$\pi$ are $R$-connected.  In fact the proof requires a weaker, 
``continuous'' variant of $R$-connnectedness which is
technical but useful.  As above, the underlying hypothesis is that the
geometric generic fiber of $\pi$ is rationally connected.  
Denote by $\whtOO$ the complete local ring
$\whtOO_{B,b}$, and let $u$ be a uniformizer of $\whtOO$.  This is a
Henselian (even a complete) DVR.
Denote by $\mcX_{\whtOO}$ the base change $\SP (\whtOO) \times_B
\mcX$.  
The ``bidisk'', $\Boo$, is the $\whtOO$-scheme
$$
\Boo := \SP (\whtOO\Sem{v}).
$$
The ``punctured bidisk'' is the complement of the closed point,
$$
\Boos := \Boo \setminus \{\langle u,v \rangle \}.
$$
Denote by $\Dv$ the divisor in $\Boo$
$$
\Dv := \SP (\whtOO\Sem{v}/\langle v \rangle) \cong \SP (\whtOO).
$$
And for every positive integer $N$, denote by $\Du{N}$ the divisor in $\Boo$
$$
\Du{N} := \SP (\whtOO\Sem{v}/\langle u^N \rangle) \cong
\SP((\whtOO/\langle u^N \rangle)\Sem{v}).
$$
Finally denote the intersections with $\Boos$ by 
$\Dvs = \Dv \cap \Boos$, $\Dus{N} = \Du{N}\cap \Boos$.  

\mni
Let $\whts$ be an $\whtOO$-point of $\mcX_{\whtOO}$.  
The restriction of $\whts$ 
to $\SP(\Fr(\whtOO))$ 
determines an
$\whtOO$-morphism from $\Dvs \rightarrow \mcX_{\whtOO}$ 
via the identification $\Dvs = \SP
(\Fr(\whtOO))$.  Similarly, the restriction of $\whts$ to
$\SP(\whtOO/\langle u^N \rangle)$ composed with the ``projection'' 
$$
\SP((\whtOO/\langle u^N \rangle)\Semr{v}) \rightarrow \SP (\whtOO/\langle
u^N \rangle)
$$
determines an $\whtOO$-morphism $\Dus{N} \rightarrow \mcX_{\whtOO}$.
The morphisms $\Dvs \rightarrow \mcX_{\whtOO}$ and $\Dus{N}
\rightarrow \mcX_{\whtOO}$ defined in this way are the
\emph{associated morphisms} of $\whts$.  

\begin{defn} \label{defn-psiRprelim}
\marpar{defn-psiRprelim}
Two $\whtOO$-points $\whts$ and $\whtt$ of $\mcX_{\whtOO}$ are \emph{directly
  $\pse$ $R$-equivalent} if for every positive integer $N$ there exist 
$\whtOO$-morphisms
$$
q_N, r_N:\Boos \rightarrow \mcX_{\whtOO}
$$
such that the restriction of $q_N$, resp. $r_N$, 
to $\Dvs$ is the morphism associated to $\whts$, resp. the morphism
associated to $\whtt$, and such that the
restriction of $q_N$, resp. $r_N$, to $\Dus{N}$ is the morphism associated
to $\whtt$, resp. the morphism associated to $\whts$.  
And $\mcX_{\whtOO}$ is \emph{$\pse$ $R$-connected} if every pair
of $\whtOO$-points is directly $\pse$ $R$-equivalent.
\end{defn}

\mni
In fact $\pse$ $R$-connectedness depends only on the
Laurent fiber $\mcX_K = \SP K \times_{\SP \whtOO} \mcX_{\whtOO}$.
Also $R$-connectedness implies $\pse$ $R$-connectedness.
Here is our main theorem.

\begin{thm} \label{thm-first}
\marpar{thm-first}
The morphism $\pi$ satisfies weak approximation
if the Laurent fiber of $\pi$ over every closed point of $B$ is
$\pse$ $R$-connected, e.g., if it is $R$-connected.
\end{thm}

\mni
\textbf{$\Pse$ ideal sheaves.}
The key is a new sheaf-theoretic object which we call a ``$\pse$ ideal
sheaf''.  Let $S$ be a scheme or an algebraic space and let $Y$ be a flat,
projective $S$-scheme or a flat, proper, algebraic space over $S$.

\begin{defn} \label{defn-pisprelim}
\marpar{defn-pisprelim}
For every $S$-scheme $T$, denoting $Y_T = T\times_S Y$, 
a \emph{$\pse$ ideal sheaf on
  $Y/S$} parameterized by $T$ is
a pair $(\mc{F},\phi)$ of a $T$-flat, locally finitely presented,
quasi-coherent $\OO_{Y_T}$-module
$\mc{F}$ together
with an $\OO_{Y_T}$-module homomorphism, $\phi:\mc{F} \rightarrow \OO_{Y_T}$,
such that the induced $\OO_{Y_T}$-module homomorphism,
$$
\phi':\bigwedge^2 \mc{F} \rightarrow \mc{F}, \ \ f_1 \wedge f_2 \mapsto
\phi(f_1)f_2 - \phi(f_2)f_1,
$$
is the zero homomorphism.  A \emph{morphism} from a $\pse$ ideal sheaf
$(\mc{F}_1,\phi_1)$ to a $\pse$ ideal sheaf $(\mc{F}_2,\phi_2)$ is an
isomorphism of $\OO_{Y_T}$-modules, $\psi:\mc{F}_1\rightarrow
\mc{F}_2$, such that $\phi_2\circ \psi$ equals $\phi_1$.  The identity
morphisms and composition of morphisms are defined in the obvious way.
For an
$S$-morphism $g:T'\rightarrow T$ and a $\pse$ ideal sheaf
$(\mc{F},\phi)$ parameterized by $T$, the \emph{pullback by $g$} is
$(g^*\mc{F},g^*\phi)$.   
\end{defn}

\mni
As proved later, this category is an Artin stack $\pis{Y/S}$.  And 
the Hilbert scheme $\Hilb_{Y/S}$ (or ``Hilbert algebraic space'', more
generally) 
is an open substack of $\pis{Y/S}$.  
Most importantly, when $Y$
is a Cartier divisor in a flat, projective $S$-scheme $X$ (or a flat,
proper algebraic space over $S$), there is a \emph{restriction morphism}
$$
\iota_Y:\Hilb_X \rightarrow \pis{Y}
$$
associating to a flat family of closed subschemes of $X$ with ideal
sheaf $\mc{I}$, the restriction $\mc{I}\otimes_{\OO_X} \OO_Y$ with its
natural map to $\OO_Y$.  
For a geometric point $s$ of $S$ and for
an ideal sheaf $\mc{I}$ on the fiber $X_s$ which is
generated by a regular sequence, the
morphism $\iota_Y$ is smooth at $[I]$ 
if 
$H^1(X,\OO_X(-Y)\cdot \textit{Hom}_{\OO_X}(I/I^2,\OO_X/I))$ 
equals
$0$.  And in this case, the Zariski tangent space of the fiber of
$\iota_Y$ at $[I]$ is canonically isomorphic to 
$H^0(X,\OO_X(-Y)\cdot \textit{Hom}_{\OO_X}(I/I^2,\OO_X/I))$. 
Therefore, given a closed subscheme of $X$ and given an infinitesimal
deformation of the intersection of the scheme with a ``tubular 
neighborhood'' of a Cartier divisor in $X$, this result allows us to
find an infinitesimal deformation of the closed subscheme of $X$ whose
intersection with the tubular neighborhood is as prescribed.  We will
usually apply this when $S$ equals $\SP(\CC)$, when $X$ equals
$\mcX$, and when $Y$ is an infinitesimal neighborhood of a fiber
$\mcX_b$ of $\pi$.

\mni
\textbf{$R$-connectedness and strong rational connectedness.}
As mentioned, $\mcX_K$ is $R$-connected 
if $\mcX_b$ is smooth and rationally connected.  
More generally, $\mcX_K$ is $R$-connected
if the smooth locus $(\mcX_b)_{\text{smooth}}$ is \emph{strongly
  rationally connected}, i.e., if every pair of closed points of
$(\mcX_b)_{\text{smooth}}$ is connected by a (complete) rational curve
in $(\mcX_b)_{\text{smooth}}$.  Thus Theorem ~\ref{thm-first} gives a
new proof of a theorem of Hassett and Tschinkel, \cite{HT06}: $\pi$
satisfies weak approximation if the smooth locus of every fiber
$\mcX_b$ is strongly rationally connected.

\mni
Additionally, we prove a structural result about strong rational
connectedness using $\pse$ ideal sheaves.  
Let $X$ be a normal, projective, complex variety and
let $D$ be a closed subset of $X$ containing the singular locus.  
Assume there exists a very
free curve in $X$ which is disjoint from $D$.  Denote by $U$ the dense
open subset of $X$ which is the union of all very free curves which
are disjoint from $D$, i.e., the largest open subset of $X\setminus D$
which is strongly rationally connected.
Then every connected component of $X\setminus
U$ intersects $D$.  In particular, 
$(X\setminus D)\setminus U$ has no isolated
points.  This generalizes one of the steps in 
Chenyang Xu's proof that log Del
Pezzo surfaces are strongly rationally connected, \cite{XuSRC}.  Since
Xu's theorem is an important advance in this area, we present our own
interpretation of his proof in an appendix to this article.

\mni
\textbf{Special cases of $R$-connectedness.}
There are many other cases where the Laurent fibers $X_K$
are known to be $R$-connected,
cf. \cite[Section 10]{C-TCIME}:  
\begin{enumerate}
\item[(i)] $X_K$ is a compactification of a connected linear group
  \cite[Corollaire 6]{CTSansucRequiv},
\item[(ii)] $X_K$ is a conic bundle over $\PP^1$ with discriminant
  divisor of degree $4$ \cite{C-TSkor} (such surfaces include Del
  Pezzo surfaces of degree $4$ and Chatelet surfaces),  
\item[(iii)] $X_K$ is a smooth complete intersection of type $(2,2)$
  in $\PP^n$ with $n\geq 5$ \cite[Theorem 3.27(ii)]{C-TSansucSw-D}, and
\item[(iv)] $X_K$ is a smooth cubic hypersurface in $\PP^n$ with
  $n\geq 5$ \cite[Footnote, p. 927]{Madore}.
\end{enumerate}
So if the generic fiber of $\pi$ satisfies one of (i) -- (iv), then
Theorem ~\ref{thm-first} implies $\pi$ satisfies weak approximation.
We generalize (i) below.  In fact (ii) is one example of a
\emph{fibration}, which we consider below.  We will also say more
about Del Pezzo surfaces.
Case (iv) improves a result of Hassett and Tschinkel,
\cite{HTSeattle}, who proved weak approximation for cubic
hypersurfaces in $\PP^6$.

\mni
\textbf{$R$-connectedness and rational simple connectedness.}
Hassett proved $\pi$ satisfies weak approximation if the geometric
generic fiber satisfies a suitably strong version of ``rational simple
connectedness''.  Using \cite{GHS} the Laurent fibers are
$R$-connected if they satisfy a weak version of rational
simple connectedness discussed below.  This refines Hassett's theorem.
In
particular, 
de Jong and the second author proved that every smooth
complete intersection of type $(d_1,\dots,d_c)$ in $\PP^n$ satisfies
the weak version of rational simple connectedness provided that
$$
n+1 \geq d_1^2 + \dots + d_c^2.
$$
We will review the proof of this quickly below.
Thus $\pi$ satisfies weak approximation if the geometric generic fiber
of $\pi$ is a complete intersection as above.

\mni
\textbf{$\Pse$ $R$-connectedness and fibrations.}
A \emph{fibration} over $\CC(B)$ is a surjective morphism
$\rho:\mcX \rightarrow \mcY$ of smooth, projective $\CC(B)$-schemes
whose geometric generic fiber is connected.  
If $\mcY$ satisfies weak approximation, and if the fiber of $\rho$
over each sufficiently general $\CC(B)$-point of $\mcY$ 
satisfies weak
approximation, then also $\mcX$ satisfies weak approximation.
The analogous
statement for $R$-connectedness is probably false: 
the analogue would contradict the Fano conjecture that some conic bundles over
$\PP^2_{\CC}$ are non-unirational.
However, $\pse$ $R$-connectedness does satisfy the analogous property
for a fibration of $K$-schemes.  

\mni
\textbf{$\Pse$ $R$-connectedness of homogeneous spaces.}
Because of the fibration property, 
conic bundles as in (ii) are $\pse$ $R$-connected.  
This also applies to a compactification
of a homogeneous space $G/H$ under a connected, linear algebraic group
$G$.  When $H$ is connected, i.e., when the connected component of the
identity $H_0$ equals all of $H$,
then $H$ is (geometrically) rationally connected.  Hence
$R$-connectedness of $G/H$ follows from $R$-connectedness of $G$.
For general $H$, we do not know if $G/H$ is $R$-connected.  However,
it is $\pse$ $R$-connected.
The
Deligne-Mumford stack $B(\pi_0H)$ is $R$-connected in a suitable
sense.  Thus considering $G/H$ as fibered over $B(\pi_0 H)$ with fibers of
the form $G/H_0$ (more correctly, $G/\widetilde{H}_0$ where
$\widetilde{H}_0$ is a twist of $H_0$), it follows that every
compactification of $G/H$ is pseudo $R$-connected.  
By Theorem ~\ref{thm-first}, if the generic fiber of $\pi$ is a
compactification of $G/H$ then $\pi$ satisfies weak approximation.
This gives a new proof of a theorem of Colliot-Th\'el\`ene and Gille,
\cite{C-TG}.  

\mni
\textbf{The Hassett-Tschinkel Conjecture.}
In fact the main importance of Theorem ~\ref{thm-first} is a new
perspective on the Hassett-Tschinkel conjecture.
We believe this conjecture fails as stated.
As we understand it from conversations with experts, 
there is an \emph{a priori} obstruction to weak approximation defined
using notions of $\AAA^1$-homotopy theory.  The two authors do not
know how to compute whether or not this obstruction is zero.  But we
are quite hopeful that this will be possible in the near future, based
on the work of Asok, Morel and Bhatt relating rational
connectedness to notions of $\AAA^1$-homotopy theory. 
Our understanding is that the obstruction is
analogous to the Brauer-Manin
obstruction to weak
approximation for varieties defined over number fields.  Just as the
``Brauer pairing'' is actually an obstruction to
$R$-connectedness of $p$-adic fibers -- and even to $\pse$
$R$-connectedness of $p$-adic fibers --
so too any obstruction defined
using a cohomology theory with a ``homotopy axiom'' should factor
through $R$-equivalence.  And if the cohomology theory 
has a ``continuous nature'', then the obstruction should even factor
through $\pse$ $R$-equivalence.
Thus $R$-connectedness and $\pse$ $R$-connectedness
of Laurent fibers is
really a stand-in for vanishing of these cohomological obstructions,
at the moment beyond our reach.  Each positive result listed above
supporting the 
Hassett-Tschinkel conjecture
has a 
hypothesis implying $R$-connectedness or $\pse$ $R$-connectedness
of Laurent fibers. 
Therefore the conjecture which these results more properly support is
that the only 
obstruction to weak approximation for a rationally connected fibration
is a cohomological
obstruction.  This is an analogue for $\CC(B)$ of a well-known
conjecture of Colliot-Th\'el\`ene for number fields,
\cite{CTBudapest}. 

\mni
\textbf{New questions and other fields.}
This perspective leads to an array of enticing questions.  
Is weak approximation preserved by finite base
change?  Is it an open condition in families?  Is it a closed
condition?  Is it preserved by ``twists''?  
Is it preserved under quotients by finite groups?  
Moreover, what can one say about the set of formal sections of $\pi$
which are 
\emph{approximable}, i.e., 
approximated to arbitrary order by rational sections?
We will discuss these questions further at the end of the article.  

\mni
Of course all of these results remain valid when $\CC$ is replaced by
any algebraically closed field of characteristic $0$.  We will discuss
which results remain valid in positive characteristic, and the issues in
extending the other results.

% Theorem
% ~\ref{thm-first} has something to say for each of these questions.  
% Here are two results related to the 
% question of describing approximable formal sections.
% First, for every formal section
% $\whts$ there exists an integer $N$ such that for every formal section
% $\wht{t}$ agreeing with $\whts$ to order $N$, $\whts$ is approximable
% if and only if $\wht{t}$ is approximable.  Second, there exists a
% positive integer $R$ such that for every branched cover 
% $g:B'\rightarrow B$ whose ramification index is divisible by $R$ at every
% point over the discriminant locus of $\pi$, every formal
% section $\whts$ of $\mc{X}$ over $B$ pulls back to an approximable
% formal section of $\mc{X}'$ over $B'$.
% We emphasize that $R$ is
% independent of the choice of $\whts$, thus weak approximation of
% $\mcX$ over $B$ is ``achieved'' by a finite base change $B'/B$.  

\mni
\textbf{Cubic surfaces with integral special fibers.}
Finally we prove a new weak approximation result for Del Pezzo
surfaces which is elementary and which has nothing to do with Theorem
~\ref{thm-first}.  If every fiber $\mcX_b$ of $\pi$ is an integral
scheme whose dualizing sheaf $\omega_{\mcX_b}$ is antiample with degree
$c_1(\omega_{\mcX_b})^2 \geq 3$, then weak approximation holds.  In
particular 
this implies all previously known weak approximation
results for cubic surfaces.  For Del Pezzo
surfaces of degrees 2 and 1, the results of Amanda Knecht and the
results of Chenyang Xu give the best current results.  In an appendix
we present our interpretation of the proof of Xu's theorem.  

\mni
\textbf{Overview of this paper.}
[This still needs to be written]

\mni
\textbf{Acknowledgments.}  We are very grateful to Universita' Roma
Tre, Queen's University, Stony Brook University and Columbia
University where this work was carried out.  We are also very grateful
to A. J. de Jong who made major contributions to this work,
particularly regarding the notion of pseudo $R$-equivalence, but
who preferred not to be a coauthor.  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Extension of two results of Grothendieck
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Extension of two results of Grothendieck}
\label{sec-G}
\marpar{sec-G}

\mni
This section proves technical results which are useful in the
following section.  But this section is
not strictly necessary for understanding the
main ideas.  The reader can safely skip this section by assuming all
proper morphisms of algebraic spaces
in later sections are actually projective morphisms
of schemes.

\mni
One of Grothendieck's results which is frequently useful in studying
coherent sheaves on proper schemes is \cite[Corollaire 7.7.8]{EGA3},
stated below.
The original proof includes a hypothesis which Grothendieck refers to
as ``\emph{surabondante}'' (Grothendieck's emphasis).  This hypothesis
has subsequently been removed by Lieblich, cf. ~\cite[Proposition
2.1.3]{Lieblich}.  We would like to briefly explain another way to
remove these hypotheses based on the representability of the Quot
functor.  Martin Olsson proved a very general result about
representability of the Quot functor.
This leads to the following version of Grothendieck's result.

\begin{thm}\cite[Corollaire 7.7.8]{EGA3}, \cite[Proposition
  2.1.3]{Lieblich} \label{thm-Hom}
\marpar{thm-Hom}
Let $Y$ be an algebraic space.
Let $f:\mcX\rightarrow Y$ be a separated, locally finitely
presented, algebraic 
stack over the category $(\text{Aff}/Y)$ of affine $Y$-schemes.  Let
$\mc{F}$ and $\mc{G}$ be locally finitely presented, quasi-coherent 
$\OO_{\mcX}$-modules.  Assume that $\mc{G}$ is $Y$-flat and has
proper support over $Y$.  Then the covariant functor of quasi-coherent
$\OO_Y$-modules,
$$
\mc{T}(\mc{M}) :=
f_*(\textit{Hom}_{\OO_{\mcX}}(\mc{F},\mc{G}\otimes_{\OO_Y} \mc{M})),
$$
is representable by a locally finitely presented, quasi-coherent
$\OO_Y$-module $\mc{N}$, i.e., there is a natural equivalence of
functors
$$
\mc{T}(\mc{M}) \xrightarrow{\cong}
\textit{Hom}_{\OO_Y}(\mc{N},\mc{M}).
$$
\end{thm}  

\mni
Grothendieck deduced his version of this result as a corollary of
another theorem.  In fact the theorem also follows from the corollary.

\begin{cor}\cite[Th\'eor\`eme 7.7.6]{EGA3} \label{cor-Hom}
\marpar{cor-Hom}
Let $f:\mcX \rightarrow Y$ be as in Theorem ~\ref{thm-Hom}.  Let
$\mc{G}$ be a locally finitely presented, quasi-coherent
$\OO_{\mcX}$-module which is $Y$-flat and has proper support over
$Y$.  Then there exists a locally finitely presented, quasi-coherent
$\OO_Y$-module $\mc{Q}$ and a natural equivalence of covariant
functors of quasi-coherent $\OO_Y$-modules $\mc{M}$,
$$
f_*(\mc{G}\otimes_{\OO_Y}\mc{M}) \xrightarrow{\cong}
\textit{Hom}_{\OO_Y}(\mc{Q},\mc{M}).
$$
\end{cor}

\mni
Grothendieck in turn deduces \cite[Th\'eor\`eme 7.7.6]{EGA3} 
as a special case of a more general 
representability
result regarding the hyperderived pushforwards of a bounded below 
complex of locally
finitely presented, quasi-coherent
$\OO_{\mcX}$-modules which are $Y$-flat and have proper support over
$Y$.  Undoubtedly this general representability result extends to the
setting of stacks.  However, to deduce \cite[Corollaire 7.7.8]{EGA3}
from \cite[Th\'eor\`eme 7.7.6]{EGA3}, Grothendieck requires a
hypothesis that $\mc{F}$ has a presentation by locally free
$\OO_{\mcX}$-modules.  This is quite a restrictive hypothesis.

\mni
Instead one can try to prove Theorem ~\ref{thm-Hom} directly using
Artin's representability theorems.  This is precisely what Lieblich
does in ~\cite[Proposition 2.1.3]{Lieblich}.  Here we point
out that Theorem ~\ref{thm-Hom} also follows from
from representability of the Quot functor as proved by Olsson,
~\cite[Theorem 1.5]{OProper}.  

\begin{lem} \label{lem-Hom1}
\marpar{lem-Hom1}
Let $f:\mcX \rightarrow Y$ be as in Theorem ~\ref{thm-Hom}.  Let
$\mc{F}$ and $\mc{G}$ be locally finitely presented, quasi-coherent
$\OO_{\mcX}$-modules.  And let 
$$
\phi:\mc{F} \rightarrow \mc{G}
$$ 
be a
homomorphism of $\OO_{\mcX}$-modules.  
Assume $\mc{G}$ has proper support over $Y$,
resp. $\mc{F}$ and $\mc{G}$ have proper support over $Y$ and $\mc{G}$
is $Y$-flat.  Then there exists an open subspace $U$, resp. $V$, of
$S$ with the following property.  For every morphism $g:Y'\rightarrow
Y$ of algebraic spaces, the pullback morphism of sheaves on
$Y'\times_Y \mcX$, 
$$
(g,\text{Id}_{\mcX})^* \phi:
(g,\text{Id}_{\mcX})^* \mc{F} \rightarrow (g,\text{Id}_{\mcX})^*
\mc{G}
$$
is surjective, resp. an isomorphism, if and only if $g$ factors
through $U$, resp. $V$.
\end{lem}

\begin{proof}
Since $\text{Coker}(\phi)$ is a locally finitely presented,
quasi-coherent sheaf supported on $\text{Supp}(\mc{G})$,
$\text{Supp}(\text{Coker}(\phi))$ is closed in 
$\text{Supp}(\mc{G})$.  
Since
$\text{Supp}(\mc{G})$ is proper over $Y$, 
$f(\text{Supp}(\text{Coker}(\phi)))$ is a
closed subset of 
$Y$.  Define $U$ to be the open complement.  For a morphism
$g:Y'\rightarrow Y$, $(g,\text{Id}_{\mcX})^* \phi$ is surjective if
and only if $\text{Coker}((g,\text{Id}_{\mcX})^* \phi)$ is zero,
i.e., if and only if the support of
$\text{Coker}((g,\text{Id}_{\mcX})^* \phi)$ is empty.
Formation of the cokernel
is compatible with pullback, i.e.,
$$
\text{Coker}((g,\text{Id}_{\mcX})^* \phi) \cong
(g,\text{Id}_{\mcX})^*\text{Coker}(\phi). 
$$
Thus the support of $\text{Coker}((g,\text{Id}_{\mcX})^* \phi)$ is
empty if and only if $g(Y')$ is disjoint from
$f(\text{Supp}(\text{Coker}(\phi)))$, i.e., if and only if $g(Y')$ is 
contained in $U$.  

\mni
Next assume that $\mc{F}$, $\mc{G}$ each have proper support over $Y$
and $\mc{G}$ is $Y$-flat.  
Since $\mc{G}$ is $Y$-flat, the kernel of $\phi$ is locally
finitely presented by \cite[Lemme 11.3.9.1]{EGA4} (the property of
being locally finitely presented can be checked locally in the fppf
topology on $\mcX$, thus reduces to the case of a morphism of
schemes).  
The support of the
kernel is contained in the support of the kernel of $\mc{F}$.  Thus
$\text{Ker}(\phi)$ is a locally finitely presented, quasi-coherent
sheaf on the support of $\mc{F}$, which is proper over $Y$.  So the
support of $\text{Ker}(\phi)$ is also proper over $Y$.  Thus its image
under $f$ is a closed subset of $Y$.  Define $V$ to be the open
complement of this closed subset in $U$.

\mni
For every morphism $g:Y'\rightarrow U$, since $\mc{G}$ is $Y$-flat,
the following sequence is exact
$$
0 \rightarrow (g,\text{Id}_{\mcX})^*\text{Ker}(\phi) \rightarrow
(g,\text{Id}_{\mcX})^* \mc{F} \rightarrow (g,\text{Id}_{\mcX})^*
\mc{G} \rightarrow 0.
$$
Thus $(g,\text{Id}_{\mcX})^*\phi$ is an isomorphism if and only if
$(g,\text{Id}_{\mcX})^* \text{Ker}(\phi)$ is zero, i.e., if and only
if $g(Y')$ is contained in $V$.
\end{proof}

\begin{lem} \label{lem-Hom2}
\marpar{lem-Hom2}
Let $f:\mcX \rightarrow Y$, $\mc{F}$ and $\mc{G}$ be as in Theorem
~\ref{thm-Hom}.  There exists a locally finitely presented, separated
morphism of algebraic spaces $h:Z\rightarrow Y$ and a morphism of
quasi-coherent sheaves on $Z\times_Y \mcX$ 
$$
\phi:(h,\text{Id}_{\mcX})^*\mc{F} \rightarrow
(h,\text{Id}_{\mcX})^* \mc{G}
$$
which represents the contravariant functor associating to every
morphism $g:Y'\rightarrow Y$ the set of morphisms of quasi-coherent
sheaves on $Y'\times_Y \mcX$
$$
\psi:(g,\text{Id}_{\mcX})^*\mc{F} \rightarrow
(g,\text{Id}_{\mcX})^* \mc{G}.
$$
\end{lem}

\begin{proof}
By \cite[Theorem 1.5]{OProper}, there exists a locally finitely
presented, separated morphism $i:W\rightarrow Y$ of algebraic spaces
and a quotient
$$
\theta:(i,\text{Id}_{\mcX})^*(\mc{F}\oplus \mc{G}) \rightarrow \mc{H}
$$
representing the Quot functor of flat families of locally finitely
presented, quasi-coherent quotients of the pullback of $\mc{F}\oplus
\mc{G}$ having proper support over the base.  Denote by
$$
\theta_{\mc{G}}:(i,\text{Id}_{\mcX})^* \mc{G} \rightarrow \mc{H}
$$
the composition of the summand
$$
e_{\mc{G}}:(i,\text{Id}_{\mcX})^* \mc{G} \hookrightarrow
(i,\text{Id}_{\mcX})^*(\mc{F}\oplus \mc{G})
$$ 
with $\theta$.  

\mni
By Lemma ~\ref{lem-Hom1}, there is an open subspace $Z$ of $W$ such
that for every morphism $j:Y'\rightarrow W$, the pullback
$$
(j,\text{Id}_{\mcX})^* \theta_{\mc{G}}: (i\circ
j,\text{Id}_{\mcX})^* \mc{G} \rightarrow (j,\text{Id}_{\mcX})^*
\mc{H}
$$
is an isomorphism if and only if $j(Y')$ factors through $Z$.  
Denote by $h:Z\rightarrow Y$ the restriction of $i$ to $Z$.  
If $j(Y')$ factors through $Z$, then
$(j,\text{Id}_{\mcX})^* \theta$ equals the composition
$$
(i\circ j,\text{Id}_{\mcX})^*(\mc{F}\oplus \mc{G})
\xrightarrow{(\psi,\text{Id})} (i\circ j,\text{Id}_{\mcX})^*\mc{G}
\xrightarrow{j^*\theta_{\mc{G}}} (j,\text{Id}_{\mcX})^*\mc{H}
$$
for a unique morphism of quasi-coherent sheaves
$$
\psi:(i\circ j,\text{Id}_{\mcX})^* \mc{F} \rightarrow (i\circ
j,\text{Id}_{\mcX})^* \mc{G}.
$$
In particular, applied to $\text{Id}_Z:Z\rightarrow Z$, this produces
the homomorphism $\phi$.  And it is straightforward to see that the
natural transformation associating to every morphism $j:Y'\rightarrow
Z$ the homomorphism $\psi$ is an equivalence of functors, i.e.,
$(h:Z\rightarrow Y,\phi)$ is universal.
\end{proof}

\begin{proof}[Proof of Theorem ~\ref{thm-Hom}]
Let $h:Z\rightarrow Y$ and $\phi$ be as in Lemma ~\ref{lem-Hom2}.
The zero homomorphism $0:\mc{F} \rightarrow \mc{G}$
defines a $Y$-morphism $z:Y \rightarrow Z$.  Since $Z$ is separated
over $Y$, the $Y$-morphism $z$ is a closed immersion.  Since $Z$ is
locally finitely presented over $Y$, the ideal sheaf $\mc{I}$ of this
closed immersion is a locally finitely presented, quasi-coherent 
$\OO_Z$-module.
Thus $z^*\mc{I}$ is a locally finitely presented, quasi-coherent 
$\OO_Y$-module.  Denote this $\OO_Y$-module by $\mc{N}$.  

\mni
Consider the closed subspace $Z_1$ of $Z$ with ideal sheaf
$\mc{I}^2$.  The restriction of $h$ to $Z_1$ is a finite morphism,
thus equivalent to the locally finitely presented $\OO_Y$-algebra $h_*
\OO_{Z_1}$.  Of course this fits into a short exact sequence
$$
0 \rightarrow z^* \mc{I} \rightarrow h_*\OO_{Z_1} \rightarrow \OO_Y
\rightarrow 0
$$
where the injection is an ideal sheaf and the surjection is a
homomorphism of $\OO_Y$-algebras.  And the morphism $z$ defines a
splitting of this surjection of $\OO_Y$-algebras.  Thus, as an
$\OO_Y$-algebra, there is a canonical isomorphism
$$
h_*\OO_{Z_1} \cong \OO_Y \oplus z^* \mc{I}.
$$
The restriction of $\phi$ to $Z_1$ together with
adjunction of $h^*$ and $h_*$ defines a homomorphism of $\OO_Y$-modules
$$
\mc{F}\rightarrow \mc{G}\otimes_{\OO_Y} h_*\OO_{Z_1}.
$$
Using the canonical isomorphism, this homomorphism is of the form
$$
\mc{F} \xrightarrow{(0,\chi)} \mc{G} \oplus (\mc{G}\otimes_{\OO_Y} \mc{N}).
$$
The homomorphism $\chi$ defines a natural transformation of covariant
functors of quasi-coherent $\OO_Y$-modules $\mc{M}$
$$
\textit{Hom}_{\OO_Y}(\mc{N},\mc{M}) \rightarrow
f_*(\textit{Hom}_{\OO_{\mcX}}(\mc{F}, \mc{G}\otimes_{\OO_Y}\mc{M}).
$$
By the same argument used to prove the equivalence of parts (a) and
(d) of \cite[Th\'eor\`eme 7.7.5]{EGA3}, this is an equivalence of
functors and the induced $Y$-morphism
$$
\underline{\text{Spec}}_Y(\text{Sym}^\bullet(\mc{N})) \rightarrow Z
$$
is an isomorphism.  
\end{proof}

\begin{proof}[Proof of Corollary ~\ref{cor-Hom}]
This follows from Theorem ~\ref{thm-Hom} by taking $\mc{F}$ to be
$\OO_{\mcX}$.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: $\Pse$ ideal sheaves
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$\Pse$ ideal sheaves} \label{sec-pis}
\marpar{sec-pis}

\mni
Given a fibration over a curve, given a fiber, and given a section,
the $N$-jet of the section at the fiber is simply the intersection of
the section and the $N^{\text{th}}$ infinitesimal neighborhood of the
fiber, considered as a closed subscheme of the infinitesimal
neighborhood.  The rule associating to a section its $N$-jet gives a
morphism from the parameter scheme of sections to the Hilbert scheme
of the $N^{\text{th}}$ infinitesimal neighborhood of the fiber.  In
order to prove Theorem ~\ref{thm-first}, it is necessary to extend
this morphism to the locus parameterizing ``combs'' whose intersection
with the fiber may not be transverse.  This is done using ``$\pse$
ideal sheaves'', a generalization of Fulton's notion of effective 
pseudo divisor.

\mni
Let $f:X \rightarrow Y$ be a flat, locally finitely presented,
proper morphism of algebraic spaces.  For every morphism of algebraic
spaces, $g:Y'\rightarrow Y$, denote by $f_{Y'}:X_{Y'}\rightarrow Y'$ the
basechange of $f$.  Using results of Martin Olsson,
the following definitions and results are still valid whenever $X$ is
a flat, locally finitely presented, proper algebraic stack over $Y$.
But since our application is to algebraic spaces, we leave the case of
stacks to the interested reader.

\begin{defn} \label{defn-pis}
\marpar{defn-pis}
For every morphism $g:Y'\rightarrow Y$ of algebraic spaces, a flat
family of \emph{$\pse$ ideal sheaves of $X/Y$ over $Y'$} is a pair
$(\mc{F},u)$ consisting of
\begin{enumerate}
\item[(i)]
a $Y'$-flat, locally finitely presented, quasi-coherent
$\OO_{X_{Y'}}$-module $\mc{F}$, and
\item[(ii)]
an $\OO_{X_{Y'}}$-homomorphism $u:\mc{F} \rightarrow \OO_{X_{Y'}}$
\end{enumerate}
such that the following induced morphism is zero,
$$
u':\bigwedge^2 \mc{F} \rightarrow \mc{F}, \ \ f_1\wedge f_2 \mapsto
u(f_1)f_2 - u(f_2)f_1.
$$

\mni
For every pair $g_1:Y'_1 \rightarrow Y$, $g_2:Y'_2 \rightarrow Y$ of
morphisms of algebraic spaces, for every pair $(\mc{F}_1,u_1)$, resp.
$(\mc{F}_2,u_2)$, of flat families of $\pse$ ideal sheaves of $X/Y$
over $Y'_1$, resp. over $Y'_2$, and 
for every $Y$-morphism $h:Y'_1 \rightarrow Y'_2$, a \emph{pullback map} 
from $(\mc{F}_1,u_1)$ to $(\mc{F}_2,u_2)$ over $h$ is an
isomorphism of $\OO_{X_{Y'_1}}$-modules
$$
\eta:\mc{F}_1 \rightarrow h^* \mc{F}_2
$$
such that $h^* u_2 \circ \eta$ equals $u_1$.  

\mni
The \emph{category of $\pse$ ideal sheaves of $X/Y$},
$\pis{X/S}$, 
is the category
whose objects are data $(g:Y'\rightarrow Y,(\mc{F},u))$ of an affine $Y$-scheme
$Y'$ together with a flat family of $\pse$ ideal sheaves of $X/Y$ over
$Y'$, and whose Hom sets
$$
\text{Hom}((g_1:Y'_1\rightarrow Y,(\mc{F}_1,u_1)),(g_2:Y'_2\rightarrow
Y,(\mc{F}_2,u_2)))
$$
are the sets of pairs $(h,\eta)$ of a $Y$-morphism $h:Y'_1\rightarrow
Y'_2$ together with a pullback map $\eta$ from $(\mc{F}_1,u_1)$ to
$(\mc{F}_2,u_2)$ over $h$.  Identity morphisms and composition of
morphisms are defined in the obvious manner.  There is an obvious
functor from $\pis{X/Y}$ to the category $(\text{Aff}/Y)$ of
affine $Y$-schemes sending every object $(g:Y'\rightarrow Y,(\mc{F},u))$
to $(g:Y'\rightarrow Y)$ and sending every morphism $(h,\eta)$ to $h$.  
\end{defn}

\begin{prop} \label{prop-algebraic}
\marpar{prop-algebraic}
The category $\pis{X/Y}$ is a limit-preserving 
algebraic stack over the category $(\text{Aff}/Y)$ of affine $Y$-schemes.
Moreover, the diagonal is quasi-compact and separated.  
\end{prop}

\begin{proof}
Denote by $\text{Coh}_{X/Y}$ the category of coherent sheaves on
$X/Y$, cf. \cite[(2.4.4)]{LM-B}.  
There is a functor $G:\pis{X/Y} \rightarrow \text{Coh}_{X/Y}$ sending every
object
$(g:Y'\rightarrow Y,(\mc{F},u))$ to $(g:Y'\rightarrow Y,\mc{F})$ and
sending every morphism $(h,\eta)$ to $(h,\eta)$.  This is a $1$-morphism of
categories over $(\text{Aff}/Y)$.  
By \cite[Th\'eor\`eme
4.6.2.1]{LM-B} and \cite[Proposition 4.1]{Smoduli}, $\text{Coh}_{X/Y}$
is a limit-preserving algebraic stack over $(\text{Aff}/Y)$ with
quasi-compact, separated diagonal.  Thus to prove the proposition, it
suffices to prove that $G$ is representable by locally finitely
presented, separated algebraic spaces.

\mni
Let $Y'$ be a $Y$-algebraic space and let $\mc{F}$ be a locally
finitely presented, quasi-coherent
$\OO_{X_{Y'}}$-module.  Since $f_{Y'}:X_{Y'}\rightarrow Y'$ is flat,
locally finitely presented and proper, by Lemma ~\ref{lem-Hom2},
there exists a locally finitely presented, separated morphism
$h:Z\rightarrow Y'$ of algebraic spaces and a universal homomorphism
$$
u:(h,\text{Id}_{X_{Y'}})^* \mc{F} \rightarrow \OO_{X_Z}.
$$
In fact, as follows from the proof of Theorem ~\ref{thm-Hom}, there is
a 
locally finitely presented, quasi-coherent
$\OO_{Y'}$-module $\mc{N}$ and a homomorphism of
$\OO_{X_{Y'}}$-modules
$$
\chi:\mc{F} \rightarrow \OO_{X_{Y'}}\otimes_{\OO_{Y'}} \mc{N}
$$
such that $Z =
\underline{\text{Spec}}_{Y'}(\text{Sym}^\bullet(\mc{N}))$ and such
that $u$ is the homomorphism induced by $\chi$.  

\mni
Since $\mc{F}$ is a locally finitely presented, quasi-coherent
$\OO_{X_{Y'}}$-module which is $Y'$-flat and has proper support over
$Y'$, the same is true of $(h,\text{Id}_{X_{Y'}})^* \mc{F}$ relative
to $Z$.  Thus, by Theorem ~\ref{thm-Hom}, there exists a locally
finitely presented, quasi-coherent $\OO_Z$-module $\mc{N}$ and a
universal homomorphism
$$
\chi:\bigwedge^2 (h,\text{Id}_{X_{Y'}})^* \mc{F} \rightarrow
(h,\text{Id}_{X_{Y'}})^* \mc{F}\otimes_{\OO_Z} \mc{N}. 
$$
Thus there exists a unique homomorphism of $\OO_Z$-modules,
$$
\zeta:\mc{N} \rightarrow \OO_Z
$$
such that the induced homomorphism of $\OO_{X_Z}$-modules
$$
\bigwedge^2 (h,\text{Id}_{X_{Y'}})^* \mc{F} \xrightarrow{\chi}
(h,\text{Id}_{X_{Y'}})^* \mc{F}\otimes_{\OO_Z} \mc{N}
\xrightarrow{\text{Id}\otimes \zeta} (h,\text{Id}_{X_{Y'}})^* \mc{F}
$$
equals $u'$.  Let $P$ denote the closed subspace of $Z$ whose ideal
sheaf equals $\zeta(\mc{N})$.  Chasing diagrams, the restriction
morphism $h|_P:P\rightarrow Y'$ together with the pullback of $u$ to
$P$ is a pair representing $Y'\times_{\text{Coh}_{X/Y}} \pis{X/Y}
\rightarrow Y'$.  Since $Z\rightarrow Y'$ is a locally finitely
presented, separated morphism of algebraic spaces (also schematic),
and since $P$ is a closed subspace of $Y'$ whose
ideal sheaf is locally finitely generated (being the image of the
locally finitely presented sheaf $\mc{N}$), also $P \rightarrow Y$ is
a locally finitely presented, separated morphism of algebraic spaces
(also schematic).
\end{proof}

\mni
Denote by $(g:\text{Hilb}_{X/Y}\rightarrow Y,C)$ a universal pair of a
morphism of algebraic spaces $g$ and a closed subspace $C$ of
$\text{Hilb}_{X/Y}\times_Y X$ which is flat, locally finitely
presented, and proper over $\text{Hilb}_{X/Y}$, i.e., an object
representing the Hilbert functor, cf. \cite[Corollary 6.2]{Artin}.
Denote by 
$$
0 \rightarrow \mc{I} \xrightarrow{u} \OO_{\text{Hilb}_{X/Y}\times_Y X}
\rightarrow \OO_C \rightarrow 0.
$$
the natural exact sequence, where $\mc{I}$ is the ideal sheaf of $C$
in $\text{Hilb}_{X/Y}\times_Y X$.  

\begin{prop} \label{prop-Hilbopen}
\marpar{prop-Hilbopen}
The pair $(\mc{I},u)$ is a family of 
$\pse$ ideal sheaves of $X/Y$ over $\text{Hilb}_{X/Y}$.  The induced
$1$-morphism
$$
\iota:\text{Hilb}_{X/Y} \rightarrow \pis{X/Y}
$$
is representable by open immersions.  
\end{prop}

\begin{proof}
Since the kernel of a surjection of flat modules is flat, $\mc{I}$ is
flat over $\text{Hilb}_{X/Y}$.  By \cite[Lemme 11.3.9.1]{EGA4},
$\mc{I}$ is a locally finitely presented, quasi-coherent sheaf.  Since
the homomorphism $u$ is injective, to prove that $u'$ is zero it
suffices to prove the composition $u\circ u'$ is zero.  This follows
immediately from the definition of $u'$.  
Thus
the pair $(\mc{I},u)$ is a family of $\pse$ ideal sheaves of $X/Y$
over $\text{Hilb}_{X/Y}$.

\mni  
To prove that $\iota$ is representable by open immersions, it suffices
to prove that it is representable by quasi-compact, \'etale
monomorphisms of schemes.  To this end, let $Y'$ be an affine
$Y$-scheme and let $(\mc{F},v)$ be a flat family of $\pse$ ideal
sheaves of $X/Y$ over $Y'$.  Denote the cokernel of $v$ by 
$$
w:\OO_{X_{Y'}} \rightarrow \mc{G}.
$$ 
By \cite[Theorem 3.2]{OS}, there is a morphism of algebraic spaces
$\sigma:\Sigma\rightarrow Y'$ such that
$(\sigma,\text{Id}_{X_{Y'}})^*\mc{G}$ 
is flat over $\Sigma$ and such that $\Sigma$ is universal among $Y'$-spaces
with this property.  Moreover, $\Sigma$ is a surjective,
finitely-presented, quasi-affine monomorphism (in particular
schematic).  As an aside, please note that the remark preceding
\cite[Theorem 3.2]{OS} is incomplete -- \cite[Proposition 3.1]{OS}
should be properly attributed to Laumon and Moret-Bailly,
\cite[Th\'eor\`eme A.2]{LM-B}.  

\mni
By \cite[Lemma 11.3.9.1]{EGA4},
$\text{Ker}((\sigma,\text{Id}_{X_{Y'}})^*w)$ is a locally finitely
presented, quasi-coherent sheaf.  Moreover, because it is the kernel
of a surjection of sheaves which are flat over $\Sigma$, it is also
flat over $\Sigma$.  
By Lemma ~\ref{lem-Hom1}, there is an open subscheme $W$ of $\Sigma$
such that a morphism $S\rightarrow \Sigma$ factors through $W$ if and
only if the pullback of
$$
(\sigma,\text{Id}_{X_{Y'}})^* \mc{F} \rightarrow 
\text{Ker}((\sigma,\text{Id}_{X_{Y'}})^*w)
$$
is an isomorphism.  Chasing universal properties, it is clear that $W
\rightarrow Y'$ represents
$$
Y' \times_{\pis{X/Y}} \text{Hilb}_{X/Y}.
$$
Thus $\iota$ is representable by finitely-presented, quasi-affine
monomorphisms of schemes.  

\mni
It only remains to prove that $\iota$ is \'etale.  Because $\iota$ is
a finitely-presented it remains to prove
that $\iota$ is formally \'etale.  Thus, let $Y'=\SP A'$ where $A'$ is
a local Artin $\OO_Y$-algebra with maximal ideal $\mathfrak{m}$ and residue
field $\kappa$.  
And let 
$$
0 \rightarrow J \rightarrow A' \rightarrow A \rightarrow 0
$$
be an infinitesimal extension, i.e., $\mathfrak{m}J$ is zero.
Let $(\mc{F},u)$ be a $\pse$ ideal sheaf of
$X/Y$ over $Y'$, and assume the basechange to $\SP A$ is an ideal
sheaf with $A$-flat cokernel.  Since $\iota$ is a monomorphism,
formal \'etaleness for $\iota$
precisely says that $Y'\rightarrow \pis{X/Y}$ factors through $\iota$,
i.e., $u$ is injective and $\text{Coker}(u)$ is $A'$-flat.

\mni
To prove this use the local flatness criterion,
e.g., as formulated in ~\cite[Proposition 11.3.7]{EGA4}.  This
criterion is an equivalence between the conditions of
\begin{enumerate}
\item[(i)]
injectivity of $u$ is injective and 
$A'$-flatness of $\text{Coker}(u)$
\item[(ii)]
and injectivity of
$$
u\otimes_{A'}\kappa: \mc{F}\otimes_{A'}\kappa \rightarrow
\OO_{X_{Y'}}\otimes_{A'} \kappa
$$
\end{enumerate}
By hypothesis, (i) holds after
basechange to $A$.  Thus (ii) holds after
basechange to $A$.  But since $A'/\mfm$ equals $A/\mfm$, (ii) for
the original family over $A'$ is precisely the same as (ii) for the
basechange family over $A$.
Thus also (i) holds over $A'$.
\end{proof}

\mni
The significance of $\pse$ ideal sheaves has to do with restriction to
Cartier divisors.  Let $D$ be an effective Cartier divisor in $X$,
considered as a closed subscheme of $X$, and assume $D$ is flat over
$Y$.  Denote by $\mc{I}_D$ the pullback 
$$
\mc{I}_D :=
\mc{I}\otimes_{\OO_X} \OO_D
$$ on $\text{Hilb}_{X/Y}\times_X D$.  And denote by
$$
u_D:\mc{I}_D \rightarrow \OO_{\text{Hilb}_{X/Y}\times_X D}
$$
the restriction of $u$.

\begin{prop} \label{prop-flat}
\marpar{prop-flat}
The locally finitely presented, quasi-coherent sheaf $\mc{I}_D$ is
flat over $\text{Hilb}_{X/Y}$.  Thus the pair $(\mc{I}_D,u_D)$ is a
flat family of $\pse$ ideal sheaves of $D/Y$ over $\text{Hilb}_{X/Y}$. 
\end{prop}

\begin{proof}
Associated to the Cartier divisor $D$ there is an injective homomorphism of
invertible sheaves 
$$
t':\OO_X(-D) \xrightarrow \OO_X.
$$  
This induces a
morphism of locally finitely presented, quasi-coherent sheaves
$$
t:\mc{I}\otimes_{\OO_X} \OO_X(-D) \rightarrow \mc{I}.
$$
The cokernel of $t$ is $\mc{I}_D$.  
By the local flatness criterion, \cite[Proposition 11.3.7]{EGA4}, to
prove that $t$ is injective and $\mc{I}_D$ is flat over
$\text{Hilb}_{X/Y}$, it suffices to prove that the ``fiber'' of $t$
over every point of $\text{Hilb}_{X/Y}$ is injective.  Thus, let
$\kappa$ be a field, let $y:\SP \kappa Y$ be a morphism, and let
$\mc{I}_y \xrightarrow \OO_{X_y}$ be an ideal sheaf.  Since $D$ is
$Y$-flat, the homomorphism of locally free sheaves
$$
t'_y:\OO_X(-D)\otimes_{\OO_Y} \kappa \rightarrow \OO_X\otimes_{\OO_Y}
\kappa
$$
is injective, thus a flat resolution of $\OO_D\otimes_{\OO_Y} \kappa$.
In particular, $\textit{Tor}_2^{\OO_{X_y}}(\OO_{X_y}/\mc{I}_y,
\OO_{D_y})$ equals zero because there is a flat resolution of
$\OO_{D_y}$ with amplitude $[-1,0]$.  By the long exact sequence
of Tor associated to the short exact sequence
$$
0 \rightarrow \mc{I}_y \rightarrow \OO_{X_y} \rightarrow
\OO_{X_y}/\mc{I}_y \rightarrow 0,
$$
there is an isomorphism
$$
\textit{Tor}_1^{\OO_{X_y}}(\mc{I}_y,
\OO_{D_y}) \cong
\textit{Tor}_2^{\OO_{X_y}}(\OO_{X_y}/\mc{I}_y,
\OO_{D_y}) = 0.
$$
But this Tor sheaf is precisely the kernel of
$$
t_y:\mc{I}_y\otimes_{\OO_X} \OO_X(-D) \rightarrow \mc{I}_y.
$$
Thus $t_y$ is injective, and so $\mc{I}_D$ is flat over
$\text{Hilb}_{X/Y}$.  
\end{proof}

\begin{notat} \label{notat-iotaD}
\marpar{notat-iotaD}
Denote by 
$$
\iota_D:\text{Hilb}_{X/Y} \rightarrow \pis{D/Y}
$$
the $1$-morphism associated to the flat family $(\mc{I}_D,u_D)$
of $\pse$ ideal sheaves of $D/Y$ over $\text{Hilb}_{X/Y}$.
This is the \emph{divisor restriction map}.  
\end{notat}

\mni
Since $\text{Hilb}_{X/Y}$ and $\pis{D/Y}$ are both locally finitely
presented over $Y$, $\iota_D$ is locally finitely presented.  Since
$\Hilb_{X/Y}$ is an algebraic space, $\iota_D$ is representable (by
morphisms of algebraic spaces).  Since the diagonal morphism of
$\pis{D/Y}$ over $Y$ is separated, and since $\text{Hilb}_{X/Y}$ is
separated over $Y$, $\iota_D$ is separated.  

\subsection{Infinitesimal study of the divisor restriction map} \label{ssec-inf}
\marpar{ssec-inf}

\mni
Let
$A'$ be a local Artin $\OO_Y$-algebra with maximal ideal
$\mathfrak{m}$ and residue field $\kappa$.  
And let 
$$
0 \rightarrow J \rightarrow A' \rightarrow A \rightarrow 0
$$
be an infinitesimal extension, i.e., $\mathfrak{m} J$ is zero.  Denote
by $X_{A'}$, resp. $X_A$, $X_\kappa$, the fiber product of
$X\rightarrow Y$ with $\SP A' \rightarrow Y$, resp. $\SP A \rightarrow
Y$, $\SP \kappa \rightarrow Y$.

\mni
Let
$(\mc{F}_{A'},u_{A'})$ be a $\pse$ ideal sheaf of $D/Y$ over $\SP A'$.
Denote by $(\mc{F}_A,u_A)$, resp. $(\mc{F}_\kappa,u_\kappa)$, the
restriction of $(\mc{F}_{A'},u_{A'})$ to $A$, resp. to $\kappa$. 
Let $\mc{I}_A$ be the ideal sheaf of a flat family $C_A$ of closed
subschemes of $X/Y$ over $\SP A$.  Denote by $\mc{I}_\kappa$,
resp. $C_\kappa$, the restriction of $\mc{I}_A$ to $\kappa$, resp. of
$C_A$ to $\kappa$. 
And assume that $\iota_D$ sends
$\mc{I}_A$ to $(\mc{F}_A,u_A)$.

\begin{prop} \label{prop-inf}
\marpar{prop-inf}
Let $n$ be a nonnegative integer.
Assume that $C_\kappa$ is a regular immersion of codimension $n$ in
$X_\kappa$, 
cf. \cite[D\'efinition 16.9.2]{EGA4} (since $X_\kappa$ is an algebraic
space, in that definition 
one must replace the Zariski covering by affine schemes 
by an \'etale covering by affine schemes).     
\begin{enumerate}
\item[(i)]
The morphism $\iota_D$ is \emph{locally unobstructed} at $C_\kappa$ in
the following sense.  For every \'etale morphism $\SP R_{A'}
\rightarrow X_{A'}$ such that the pullback $I_\kappa$ of $\mc{I}_\kappa$ in
$R_\kappa$ is generated by a regular sequence, there exists an ideal
$I_{A'}$ in $R_{A'}$ whose restriction $I_A$ to $R_A$ equals the
pullback of $\mc{I}_A$ and whose ``local $\pse$ ideal sheaf''
$(I_{A'}\otimes_{\OO_X} \OO_D,v)$ equals the pullback of
$(\mc{F}_{A'},u_{A'})$.   Moreover, the set of such ideals $I_{A'}$ is
naturally a torsor for the $R_\kappa$-submodule 
$$
J\otimes_\kappa \OO_X(-D)\cdot\text{Hom}_{R_\kappa}(I_\kappa,R_\kappa/I_\kappa)
$$
of
$$
J\otimes_\kappa \text{Hom}_{R_\kappa}(I_\kappa,R_\kappa/I_\kappa)
$$ 
(here
$\OO_X(-D)\cdot$ denote multiplication by the inverse image ideal of
$\OO_X(-D)$).  
\item[(ii)]
There exists an element $\omega$ in 
$$
J\otimes_\kappa H^1(C_\kappa,\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa}))
$$
which equals $0$ if and only if there exists a flat family $C_{A'}$ of
closed subschemes of $X/Y$ over $\SP A'$ whose restriction to $\SP A$
equals $C_A$ and whose image under $\iota_D$ equals
$(\mc{F}_{A'},u_{A'})$.  When it equals $0$, the set of such families
$C_{A'}$ is naturally a torsor for the $\kappa$-vector space
$$
J\otimes_\kappa H^0(C_\kappa,\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa})).
$$
\end{enumerate}
In particular, if $h^1(C_\kappa,\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa}))$
equals $0$, then $\iota_D$ is smooth at $[C_\kappa]$.  
\end{prop}

\begin{proof}
\textbf{(i)}  Let $v_{(A,1)},\dots,v_{(A,n)}$ be a regular sequence in
$R_A$ generating $I_A$.  Denote by $v_A$ the $R_A$-module homomorphism
$$
v_A:R_A^{\oplus n} \rightarrow R_A, \ \ v_A(\mathbf{e}_i) = v_{(A,i)}.
$$
Denote the associated $R_A$-module homomorphism by
$$
v'_A:R_A^{\oplus \binom{n}{2}} \rightarrow R_A^{\oplus n}, \ \
v'_A(\mathbf{e}_i\wedge \mathbf{e}_j) = v_{(A,i)}\mathbf{e}_j -
v_{(A,j)}\mathbf{e}_i.  
$$
Since $(v_{(A,1)},\dots,v_{(A,n)})$ is a regular sequence, the
following sequence is exact
$$
R_A^{\oplus \binom{n}{2}} \xrightarrow{v'_A} R_A^{\oplus n}
\rightarrow I_A \rightarrow 0.
$$
Thus there is an exact sequence
$$
(R_A\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} \xrightarrow{v'_A\otimes
  \text{Id}} (R_A\otimes_{\OO_X}\OO_D)^{\oplus n} \rightarrow
(I_A\otimes_{\OO_X} \OO_D) \rightarrow 0.
$$

\mni
Denote by $F_{A'}$ the $R_{A'}$-module whose associated quasi-coherent
sheaf is the pullback of $\mc{F}_{A'}$.  Since $\mc{F}_A$ equals
$\mc{I}_A\otimes_{\OO_X} \OO_D$, there is an isomorphism
$$
F_{A'}/JF_{A'} \cong I_A\otimes_{\OO_X} \OO_D
$$
compatible with the maps $u_{A'}$ and $u_A$.  Since
$(R_{A'}\otimes_{\OO_X} \OO_D)^{\oplus
  n}$ is a projective $(R_{A'}\otimes_{\OO_X}\OO_D)$-module, 
there exists an $(R_{A'}\otimes_{\OO_X}\OO_D)$-module
homomorphism
$$
a:(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus n} \rightarrow F_{A'}
$$
whose restriction to $A$ is the surjection above.  So, by Nakayama's
lemma, this map is also surjective.  Since both the source and target
of the surjection are $A'$-flat, also
the kernel is $A'$-flat.  Thus there is also a lifting of the set of
generators of the kernel, i.e., there is an exact sequence of
$(R_{A'}\otimes_{\OO_X}\OO_D)$-modules
$$
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} \xrightarrow{b}
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus n} \xrightarrow{a} 
F_{A'} \rightarrow 0
$$
whose restriction to $A$ is the short exact sequence above.  

\mni
The composition of the surjection with $u_{A'}$ defines an 
$(R_{A'}\otimes_{\OO_X}\OO_D)$-module
homomorphism
$$
w_{A'}:(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus n} \rightarrow
(R_{A'}\otimes_{\OO_X}\OO_D)
$$
whose restriction to $A$ equals $v_A\otimes \text{Id}$.  There is an
associated map
$$
w'_{A'}:(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} \rightarrow
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus n}, \ \
w'_{A'}(\mathbf{e}_i\wedge \mathbf{e}_j) =
w_{A'}(\mathbf{e}_i)\mathbf{e}_j - w_{A'}(\mathbf{e}_j)\mathbf{e}_i.
$$
Since
$(\mc{F}_{A'},u_{A'})$ is a $\pse$ ideal sheaf, the induced map
$u'_{A'}$ equals $0$.  Therefore the image of $w'_{A'}$ is contained
in the kernel of $a$.  Since $(R_{A'}\otimes_{\OO_X} \OO_D)^{\oplus
  \binom{n}{2}}$ is a free $(R_{A'}\otimes_{\OO_X} \OO_D)$-module,
there is a lifting
$$
c:(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} \rightarrow
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} 
$$
such that $w'_{A'} = b\circ c$.  In particular, the restriction of $c$
to $A$ is an isomorphism.  Thus, by Nakayama's lemma, $c$ is
surjective.  A surjection of free modules of the same finite rank is
automatically an isomorphism.  Thus there is a presentation
$$
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus \binom{n}{2}} \xrightarrow{w'_{A'}}
(R_{A'}\otimes_{\OO_X}\OO_D)^{\oplus n} \xrightarrow{a} 
F_{A'} \rightarrow 0.
$$

\mni
Because both $R_{A'}$ and $R_{A'}\otimes_{\OO_X}\OO_D$ are $A'$-flat,
there is a commutative diagram of exact sequences
$$
\begin{CD}
0 @>>> J\otimes_{\kappa} R_\kappa @>>> R_{A'} @>>> R_A @>>> 0 \\
& &  @VVV  @VVV  @VVV \\
0 @>>> J\otimes_\kappa R_\kappa \otimes_{\OO_X} \OO_D @>>>
R_{A'}\otimes_{\OO_X} \OO_D @>>> R_A \otimes_{\OO_X} \OO_D @>>> 0
\end{CD}
$$
where the vertical maps are each surjective.  By the snake lemma, the
induced map
$$
R_{A'}/(J\OO_X(-D)\cdot R_{A'}) \rightarrow R_A
\times_{(R_A\otimes_{\OO_X} \OO_D)} ( R_{A'}\otimes_{\OO_X} \OO_D )
$$
is an isomorphism.  
Thus, for every integer $i=1,\dots,n$, there exists an element
$v_{(A',i)}$ in $R_{A'}$ whose image in $R_A$ equals $v_{(A,i)}$ and
whose image in $R_{A'}\otimes_{\OO_X}\OO_D$ equals
$w_{A'}(\mathbf{e}_i)$.  Moreover, the set of all such elements is
naturally a torsor for $J\otimes_\kappa (\OO_X(-D)\cdot R_\kappa)$.
In other words, there is an $R_{A'}$-module homomorphism
$$
v_{A'}:R_{A'}^{\oplus n} \rightarrow R_{A'}
$$
whose restriction to $A$ equals $v_A$ and such that $v_{A'}\otimes
\text{Id}$ equals $w_{A'}$.  

\mni
Since $(v_{(A,1)},\dots,v_{(A,n)})$ is a
regular sequence in $R_A$, also $(v_{(A',1)},\dots,v_{(A',n)})$ is a
regular sequence in $R_{A'}$. One way to see this is to tensor the
Koszul complex $K^\bullet(R_{A'},v_{A'})$ of $v_{A'}$ with the short exact
sequence of $A'$-modules
$$
0 \rightarrow J \rightarrow A' \rightarrow A \rightarrow 0.
$$
Since the terms in the Koszul complex are free $R_{A'}$-modules, and
since $R_{A'}$ is $A'$-flat, the associated sequence of complexes is
exact
$$
0 \rightarrow J\otimes_A K^\bullet(R_A,v_A) \rightarrow
K^\bullet(R_{A'},v_{A'}) \rightarrow K^\bullet(R_A,v_A) \rightarrow 0.
$$
Thus there is a 
long exact sequence of Koszul cohomology
$$
\dots \rightarrow J\otimes_A H^n(K^\bullet(R_A,v_{A})) \rightarrow
H^n(K^\bullet(R_{A'}, v_{A'})) \rightarrow H^n(K^\bullet(R_A,v_A))
\rightarrow J\otimes_A H^{n+1}(K^\bullet(R_A,v_A)) \rightarrow \dots
$$
Since $v_A$ is regular, $H^{n-1}(K^\bullet(R_A,v_A))$ is zero, which
then implies $H^{n-1}(K^\bullet(R_{A'},v_{A'}))$ by the long exact
sequence above.  Thus also $v_{A'}$ is regular.  Moreover, this gives
a short exact sequence
$$
0 \rightarrow J\otimes_\kappa (R_A/I_A) \rightarrow
R_A'/\text{Image}(v_{A'}) \rightarrow R_A/I_A \rightarrow 0
$$
from which it follows that $R_{A'}/\text{Image}(v_{A'})$ is
$A'$-flat.  Denote by $I_{A'}$ the image of $v_{A'}$.  

\mni
Both the $\pse$ ideal $I_{A'}\otimes_{\OO_X}\OO_D$ and $F_{A'}$ equal
the cokernel of $w'_{A'}$.  Thus there is a unique isomorphism between
them compatible with $w'_{A'}$.  Moreover, since the compositions
$$
(R_{A'}\otimes_{\OO_X} \OO_D)^{\oplus n} \xrightarrow{a} F_A'
\xrightarrow{u_{A'}} (R_{A'}\otimes_{\OO_X} \OO_D)
$$
and
$$
(R_{A'}\otimes_{\OO_X} \OO_D)^{\oplus n} \xrightarrow{v_{A'}\otimes
  \text{Id}} I_A'\otimes_{\OO_X} \OO_D
\rightarrow (R_{A'}\otimes_{\OO_X} \OO_D)
$$ 
both equal $w_{A'}$, the isomorphism above is compatible with the maps
to $(R_{A'}\otimes_{\OO_X} \OO_D)$.  Thus it is an isomorphism of
$\pse$ ideal sheaves.  Therefore there exists
$I_{A'}$ satisfying all the conditions
in the proposition.

\mni
For every lift $I_{A'}$ of $I_A$ which is $A$-flat, the surjection
$v_A:R_A^{\oplus n} \rightarrow I_A$ lifts to a surjection
$R_{A'}^{\oplus n} \rightarrow  I_{A'}$.  Composing this surjection
with the injection $I_{A'} \hookrightarrow R_{A'}$, it follows that
every lift $I_{A'}$ arises from a lift $v_{A'}$ of $v_A$.  As
mentioned previously, the set of lifts $v_{A'}$ whose restriction to $A$
equals $v_A$ and with $v_{A'}\otimes \text{Id}$ equal to $w_{A'}$ is
naturally a torsor for 
$$
J\otimes_\kappa \text{Hom}_{R_{A'}}(R_{A'}^{\oplus n},\OO_X(-D)\cdot
R_\kappa).
$$
But the translate of a lift by a homomorphism with image in
$I_\kappa$ gives the same ideal $I_{A'}$ (just different surjections
from $R_{A'}^{\oplus n}$ to the ideal).  Thus, the set of lifts
$I_{A'}$ of $I_A$ whose $\pse$ ideal sheaf is the pullback of
$(\mc{F}_{A'},u_{A'})$ is naturally a torsor for
$$
J\otimes_\kappa \text{Hom}_{R_{A'}}(R_{A'}^{\oplus n},\OO_X(-D)\cdot
(R_\kappa/I_\kappa)) = J\otimes_\kappa
\text{Hom}_{R_\kappa}(I_\kappa,R_\kappa/I_\kappa).
$$

\mni
\textbf{(ii)}
Let $\Gamma$ be an indexing set and let $(\SP R^\gamma_{A'}
\rightarrow X_{A'})_{\gamma \in \Gamma}$ be an \'etale covering such
that for every $\gamma$, either $I^\gamma_\kappa \subset
R^\gamma_\kappa$ is generated by a regular sequence of length $n$ or
else equals $R^\gamma_\kappa$.  By the hypothesis that $C_\kappa$ is a
regular immersion of codimension $n$, there exists such a covering.  

\mni
Suppose first that $I^\gamma_\kappa$ equals $R^\gamma_\kappa$.  Then
also $(F^\gamma_\kappa,u^\gamma_\kappa)$ is isomorphic to $R^\gamma_\kappa
\otimes_{\OO_X} \OO_D \xrightarrow{=} R^\gamma_\kappa \otimes_{\OO_X}
\OO_D$.  It is straightforward to see that the only deformations over
$A'$
of
$R^\gamma_\kappa \xrightarrow{=} R^\gamma_\kappa$, resp. 
$R^\gamma_\kappa \otimes_{\OO_X} \OO_D \xrightarrow{=} R^\gamma_\kappa
\otimes_{\OO_X} 
\OO_D$, as $\pse$ ideals are $R^\gamma_{A'} \xrightarrow{=} R^\gamma_{A'}$,
resp. $R^\gamma_{A'} \otimes_{\OO_X} \OO_D \xrightarrow{=}
R^\gamma_{A'}\otimes_{\OO_X} \OO_D$.  Thus there is a lifting
$I^\gamma_{A'}$ of $I^\gamma_A$, and it is unique.

\mni
On the other hand, if $I^\gamma_\kappa$ is generated by a regular
sequence of length $n$, by (i) there exist liftings $I^\gamma_{A'}$
and the set of all liftings is a torsor for 
$$
J\otimes_\kappa
\OO_X(-D)\cdot\text{Hom}_{R^\gamma_\kappa}(I^\gamma_\kappa ,
R^\gamma_\kappa/I^\gamma_\kappa).
$$
For a collection of liftings $(I^\gamma_{A'})_{\gamma \in \Gamma}$,
for every $\gamma_1,\gamma_2 \in \Gamma$, the basechanges of
$I^{\gamma_1}_{A'}$ and $I^{\gamma_2}_{A'}$ to
$$
R^{\gamma_1,\gamma_2}_{A'}:= R^{\gamma_1}_{A'}\otimes_{\OO_X}
R^{\gamma_2}_{A'}
$$ 
differ by an element $\omega^{\gamma_1,\gamma_2}$ in 
$$
J\otimes_\kappa
\OO_X(-D)\cdot\text{Hom}_{R^{\gamma_1,\gamma_2}_\kappa}(I^{\gamma_1,\gamma_2}_\kappa
, R^{\gamma_1,\gamma_2}_\kappa/I^\gamma_\kappa).
$$
It is straightforward to see that
$(\omega^{\gamma_1,\gamma_2})_{\gamma_1,\gamma_2 \in \Gamma}$ is a
$1$-cocycle for 
$$
\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa})
$$
with respect to the given \'etale covering.  Moreover, changing the
collection of lifts $(I^\gamma_{A'})$ by translating by elements in 
$$
(J\otimes_\kappa
\OO_X(-D)\cdot\text{Hom}_{R^\gamma_\kappa}(I^\gamma_\kappa ,
R^\gamma_\kappa/I^\gamma_\kappa))_{\gamma \in \Gamma}
$$
precisely changes $(\omega^{\gamma_1,\gamma_2})_{\gamma_1,\gamma_2\in
  \Gamma}$ by a $1$-coboundary.  Therefore, the cohomology class
$$
\omega \in  J\otimes_\kappa H^1(C_\kappa,\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa}))
$$
is well-defined and equals $0$ if and only if there is a lifting
$\mc{I}_{A'}$ as in the proposition.  And in this case, the set of
liftings is a torsor for the set of compatible families
$(t^\gamma)_{\gamma\in \Gamma}$ of elements $t^\gamma$ in 
$$
J\otimes_\kappa
\OO_X(-D)\cdot\text{Hom}_{R^\gamma_\kappa}(I^\gamma_\kappa ,
R^\gamma_\kappa/I^\gamma_\kappa),
$$
i.e., it is a torsor for the set of elements $t$ in
$$
J\otimes_\kappa H^0(C_\kappa,\OO_X(-D)\cdot
\textit{Hom}_{\OO_{C_\kappa}}(\mc{I}_\kappa/\mc{I}_\kappa^2,\OO_{C_\kappa})).
$$
\end{proof}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Pseudo $R$-equivalence
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Pseudo $R$-equivalence} \label{sec-2q}
\marpar{sec-2q}

\mni
\textbf{Overview.}
The definition of $R$-equivalence in the introduction 
is valid over any field.  But
in applications to number theory and geometry, $R$-equivalence is most
often applied when the field is the fraction field of a Henselian DVR.
(Although it is interesting to see what can be proved over
other fields.)  For varieties over a DVR there is another
notion, ``$\pse$ $R$-equivalence'', which follows from
$R$-equivalence and which captures the ``continuous nature'' of the
known cohomological obstructions to weak approximation.  

\mni
Let $\whtOO$ be a DVR.  In later sections $\whtOO$ will
equal the complete local ring of a smooth, closed point of a curve,
i.e., $k\Sem{u}$ for an algebraically closed field $k$.
Let $\mcY$ be a flat, quasi-projective $\whtOO$-scheme whose generic
fiber is smooth.  Let $\whts$ and $\whtt$ be $\whtOO$-points of $\mcY$
and consider the corresponding $\Fr(\whtOO)$-points of the generic
fiber.  Since $\mcY$ is separated, $\whtt$ specializes to $\whts$ only
if $\whtt$ equals $\whts$.  However, 
if $\whts$ and $\whtt$ are $R$-equivalent then $\whts$ is the
specialization of a family of sections of $\mcY$ which agree with
$\whtt$ to arbitrary order.  We say that $\whtt$ ``pseudo specializes'' to
$\whts$.  The same holds with $\whts$ and $\whtt$ permuted.  Thus we say
that $\whts$ and $\whtt$ are ``pseudo $R$-equivalent''.
Pseudo specialization is invariant under modification of the
closed fiber of $\mcY$.  If $\whtOO$ is Henselian and if 
$\whtt$ pseudo specializes to $\whts$,
then both $\whtt$ and $\whts$ determine the same ``Brauer pairing'',
$$
\text{Br}(\mcY_{\Fr(\whtOO)}) \rightarrow \text{Br}(\Fr(\whtOO)).
$$
Finally, pseudo $R$-equivalence satisfies the fibration property from
the introduction which is not currently known for (usual) $R$-equivalence.
For
all these reasons we develop these ``pseudo'' notions in this
section.  Some of the arguments are technical.  The reader who wants to
skip this section may safely replace ``pseudo $R$-equivalence'' by
``$R$-equivalence'' wherever it occurs in the later sections.

\mni
\textbf{Definitions.}
As above, let $\whtOO$ be a DVR, and denote by $u$ a generator of the
maximal ideal.
Let $\mcY$ be a flat,
quasi-projective $\whtOO$-scheme whose generic fiber is smooth.
The \emph{bidisk}
$\Boo$ is the power series ring, 
$$
\Boo := \SP(\whtOO\Sem{v}).
$$ 
The \emph{punctured bidisk} is the scheme 
$$
\Boos := \Boo \setminus \{
\langle u,v \rangle \} = \SP(\whtOO\Sem{v}) \setminus \{ \langle u,v
\rangle \}.
$$
The \emph{$v$-divisor} $\Dv$
is
the closed subscheme of $\Boo$ with defining ideal
$\langle v \rangle$,
$$
\Dv := \SP(\whtOO\Sem{v}/\langle v \rangle).
$$
Similarly, define $\Dvs$ to be the intersection of $\Dv$ with $\Boos$.
Also
for every integer $N$ the \emph{$u^N$-divisor}
$\Du{N}$ is the
closed subscheme of $\Boo$ with defining ideal
$\langle u^N \rangle$,
$$
\Du{N} = \SP(\whtOO\Sem{v}/\langle u^N \rangle).
$$
Similarly, define $\Dus{N}$ to be the intersection of $\Du{N}$
with $\Boos$.

\mni
Observe that $\Dv$ equals $\SP
\whtOO$ and $\Dvs$ equals $\SP \Fr(\whtOO)$ as $\whtOO$-schemes.  
Also $\Du{N}$ equals $\SP(\whtOO/u^N)\Sem{v}$
and $\Dus{N}$ equals $\SP(\whtOO/u^N)\Semr{v}$ as $\whtOO$-schemes.  
In particular, the
restriction of $\whts$ to $\Fr(\whtOO)$ gives an $\whtOO$-morphism
from $\Dvs$ 
to $\mcY$.  And the restriction of $\whtt$ to $\whtOO/u^N$ gives an
$\whtOO$-morphism from $\Dus{N}$ to $\mcY$ (which happens to be
constant in $v$). 

\begin{defn} \label{defn-psiR}
\marpar{defn-psiR}
Let $\whts$ be a $\whtOO$-point of $\mcY$ and let $\whtt_N$ be a
$\whtOO/u^N$-point of $\mcY$.  An \emph{$N$-jet specialization} of
$\whtt_N$ to $\whts$ is an $\whtOO$-morphism
$$
r:\Boos \rightarrow \mcY
$$
such that $r|_{\Dvs}$ equals the restriction of $\whts$ to $\SP
\Fr(\whtOO)$ and $r|_{\Dus{N}}$ equals the base change of $\whtt_N$.  When
$t_N$ equals the restriction of $\whtt$ to $\whtOO/u^N$, we say that
$\whtt$ \emph{$N$-jet specializes} to $\whts$.

\mni
The $\whtOO$-point $\whtt$ \emph{directly $\pse$ specializes} to 
$\whts$ if for
every nonnegative integer $N$, $\whtt$ $N$-jet specializes to $\whts$.  
The
$\whtOO$-point 
$\whtt$ \emph{$\pse$ specializes} to $\whts$ if there exists a sequence of
$\whtOO$-points $\whtt = \whtt_0, \dots, \whtt_n = \whts$ such that for every
$i=1,\dots,n$, $\whtt_{i-1}$ directly $\pse$ specializes to $\whtt_i$.  
Elements $\whts$ and $\whtt$ are \emph{$\pse$ $R$-equivalent},
resp. \emph{directly $\pse$ $R$-equivalent}, if each element
$\pse$ specializes to the other, resp. directly $\pse$ specializes to
the other.
If every pair of $\whtOO$-points of $\mcY$ are $\pse$
$R$-equivalent, then $\mcY$ is \emph{$\pse$ $R$-connected}.
%% Finally, $\mcY$ is \emph{geometrically
%%   $\pse$ $R$-connected} if for every local homomorphism of complete DVRs
%% $\whtOO \rightarrow \whtOO'$ with $\whtOO'/\mf{m}_{\whtOO} \whtOO'$ an
%% algebraically closed field, the basechange of $\mcY$ to $\whtOO'$ is
%% $\pse$ $R$-connected.
\end{defn}

\mni
\textbf{Basic results.}
Although it is defined in terms of $\whtOO$-schemes, in fact
pseudo $R$-specialization depends only on the generic fiber, i.e., the
fiber over $\Fr(\whtOO)$.

\begin{lem} \label{lem-model}
\marpar{lem-model}
Let $f:\mcY' \rightarrow \mcY$ be a morphism of flat, quasi-projective
$\OO$-schemes.  Let $\whts'$ and $\whtt'$ be $\whtOO$-points of
$\mcY'$ mapping to $\whts$ and $\whtt$.  
\begin{enumerate}
\item[(i)]
Every $N$-jet specialization of $\whtt'$ to $\whts'$ in $\mcY'$ maps
to an $N$-jet specialization of $\whtt$ to $\whts$ in $\mcY$. 
Thus if $\whtt'$ directly $\pse$ specializes to $\whts'$ in $\mcY'$, then
$\whtt$ directly $\pse$ specializes to $\whts$ in $\mcY$. 
\item[(ii)]
Assume that $f$ is projective and is an isomorphism on
$\Fr(\whtOO)$-fibers.  Then there exists a nonnegative integer $c$
such that for every nonnegative integer $N$, every $(N+c)$-jet
specialization of $\whtt$ to $\whts$ in $\mcY$ is the image of an
$N$-jet specialization of $\whtt'$ to $\whts'$ in $\mcY'$.  Thus if
$\whtt$ directly $\pse$ specializes to $\whts$ in $\mcY$, 
then $\whtt'$ directly $\pse$ specializes to $\whts'$ in $\mcY'$.
\item[(iii)]
If $\whtt$ and $\whts$ are directly $R$-equivalent, then they are
directly $\pse$ $R$-equivalent.
\end{enumerate}
\end{lem}

\begin{proof}
Item (i) is obvious.  Next assume the hypotheses in Item (ii).  Let
$$
r:\Boos \rightarrow \mcY
$$
be an $M$-jet specialization of $\whtt$ to $\whts$.  By the valuative
criterion of properness there exists a unique $\whtOO$-morphism
$$
r':\Boos \rightarrow \mcY'
$$
such that $f\circ r'$ equals $r$.  Since $f$ is an isomorphism on
$\Fr(\whtOO)$-fibers and since $r|_{\Dvs}$ equals $\whts$, also
$r'|_{\Dvs}$ equals $\whts'$.  It remains to prove that there exists
an integer $c$ such that for every nonnegative integer $N$,
$r'|_{\Dus{N}}$ equals the restriction of $\whtt'$ if $r|_{\Dus{N+c}}$
equals the restriction of $\whtt$.

\mni
This is easiest to see in coordinates.  Let $V$ be an open affine in
$\mcY$ containing $\whtt$ and let $V'$ be an open affine in
$f^{-1}(V)$ containing $\whtt'$.  Let $y_1,\dots,y_l$ be generators
for $\Gamma(V',\OO_{V'})$ as a $\whtOO$-algebra such that the ideal of
$\whtt'$ is $\langle y_1,\dots,y_l \rangle$.  
There exists an
integer $c$ such that $u^cy_i$ is in $\Gamma(V,\OO_V)$ for every
$i=1,\dots,l$.  And then $u^cy_i$ is in the ideal of $\whtt$ for every
$i$.  Let $\tau$ be a $\whtOO$-point of $V'$.  If $(f\circ\tau)^*(u^cy_i)$ is
divisible by $u^{N+c}$, then $\tau^*(y_i)$ is divisible by $u^{N+c}$.
Thus if $f\circ \tau$ agrees with $\whtt$ to order $N+c$, then $\tau$
agrees with $\whtt'$ to order $N$.  Applying this to the germ of $r'$
at $\Dus{N}$ shows that $r'|_{\Dus{N}}$ equals the restriction of
$\whtt'$ if $r|_{\Dus{N+c}}$ equals the restriction of $\whtt$. 

\mni
For (iii), first consider the special case when $\mcY$ equals
$\PP^1_{\whtOO}$, i.e., $\text{Proj}\ \whtOO[Y_0,Y_1]$.  Also  
assume that $\whts$, resp. $\whtt$, equals $[1,0]$,
resp. $[0,1]$.  For every nonnegative integer $N$, consider the
morphism
$$
r:\Boos \rightarrow \PP^1_{\whtOO}, \ \ r^*[Y_0,Y_1] = [v,u^N].
$$
This is clearly an $N$-jet specialization of $\whtt$ to $\whts$.  Thus
$\whtt$ directly $\pse$ specializes to $\whts$.  By symmetry, $\whtt$
and $\whts$ are directly $\pse$ $R$-equivalent.

\mni
Next consider the case when $\mcY$ is an $\whtOO$-curve $P$ whose
generic fiber $P_{\Fr(\whtOO)}$ is 
isomorphic to
$\PP^1_{\Fr(\whtOO)}$.  
Any two
distinct rational points of $\PP^1_{\Fr(\whtOO)}$ are projectively
equivalent to $[1,0]$ and $[0,1]$.  So by the previous paragraph and
by (ii), any two
$\whtOO$-points of $P$ are directly $\pse$ $R$-equivalent.

\mni
Finally consider the case of general $\mcY$.  Let $\whts$ and $\whtt$
be $\whtOO$-points which are directly $R$-equivalent, i.e., which are
the images under a $\whtOO$-morphism
$$
f:P \rightarrow \mcY
$$
of $\whtOO$-points of $P$,
where $P$ is a
$\whtOO$-curve as in the previous paragraph.  By the previous
paragraph, every pair of $\whtOO$-points of $P$ are directly $\pse$
$R$-equivalent.  Thus, by (i), $\whts$ and $\whtt$ are directly $\pse$
$R$-equivalent in $\mcY$.
\end{proof}

\mni
To prove the relation between pseudo specialization and Brauer
equivalence, it is useful to prove a lemma about extending Brauer
classes.  
We
believe this is well-known; we include the proof to illustrate the
technique.  

\begin{lem} \label{lem-unramified}
\marpar{lem-unramified}
Let $\whtt$ be an $\whtOO$-point of $\mcY$.
Let $\alpha$ be a Brauer class on the generic fiber
$\mcY_{\Fr(\whtOO)}$ such that the pullback $\whtt^*\alpha$ on $\Fr(\whtOO)$
extends to all of $\whtOO$.  Then
there
exists a projective, birational morphism $f:\mcY' \rightarrow \mcY$, an
$\whtOO$-point $\whtt'$ of $\mcY'$, and an open subscheme $U$ of $\mcY'$
containing the generic fiber and containing 
$\text{Image}(\whtt')$, such that $\alpha$ extends to all of $U$.
\end{lem}

\begin{proof}
There is a first reduction.  Denote by $p:\mcY \rightarrow \SP \whtOO$
the ``structure morphism''.  Consider $\beta = \alpha -
p^*\whtt^*\alpha$.  Since $p^*\whtt^*\alpha$ is defined on all of
$\mcY$, for a morphism $f$ and open set $U$ as above, $f^*\alpha$
extends to all of $U$ if and only if $f^*\beta$ extends to all of
$U$.  So, up to replacing $\alpha$ by $\beta$, we may assume that
$\whtt^*\alpha$ equals $0$.

\mni
Since $\mcY_{\Fr(\whtOO)}$ is smooth and quasi-projective, there
exists an integer $n$ and a Brauer-Severi variety
$$
\pi:\mc{P} \rightarrow \mcY_{\Fr(\whtOO)}
$$
with geometric fibers $\PP^{n-1}$ such that the associated Brauer
class is $\alpha$.  Since $\whtt^*\alpha$ equals $0$, $\whtt^*\mc{P}$
is isomorphic to $\PP^{n-1}_{\Fr(\whtOO)}$.  

\mni
Denote by $\omega_\pi$
the relative dualizing sheaf.  The associated sheaf $\omega_\pi^\vee$
is $\pi$-relatively very ample and $\pi_*(\omega_\pi^\vee)$ is locally free
of rank $\binom{2n}{n}$.  
Since Brauer classes on a regular scheme satisfy glueing for
Zariski open covers, it suffices to prove the existence of $f$ and $U$
after replacing $\mcY$ by any open affine neighborhood of $\whtt$.  So
without loss of generality assume that $\mcY$ is an affine
neighborhood on which $\pi_*(\omega_\pi^\vee)$ is trivialized.  More
precisely, choose a free basis $s_1,\dots,s_M$ for
$H^0(\PP^{n-1}_\ZZ,\omega_{\PP^{n-1}_\ZZ}^\vee)$ and then choose a
trivialization
$$
\phi:\OO_{\mcY_{\Fr(\whtOO)}}^{\oplus M} \rightarrow 
\pi_*(\omega_{\pi}^\vee)
$$
whose pullback by $\whtt$ has components $(s_1,\dots,s_M)$.  Since
$\omega_\pi^\vee$ is $\pi$-relatively very ample, this trivialization
induces a closed immersion of $\mcY$-schemes,
$$
\iota:\mc{P} \hookrightarrow \PP^{M-1}_{\mcY_{\Fr(\whtOO)}}.
$$

\mni
Denote by $h(d)$ the numerical polynomial
$$
h(d) := \binom{n(d+1)}{n}.
$$
The closed immersion $\iota$ determines a $\whtOO$-morphism to the
Hilbert scheme
$$
\widetilde{\iota}:\mcY_{\Fr(\whtOO)} \rightarrow
\text{Hilb}^{h(d)}_{\PP^{M-1}_{\mcY}/\mcY}.
$$
As $\text{Hilb}^{h(d)}_{\PP^{M-1}_{\mcY}/\mcY}$ is projective over
$\mcY$, there exists a projective, birational morphism
$$
f:\mcY' \rightarrow \mcY
$$
such that the pullback of $\mc{P}$ over $\mcY'$ extends to a
$\mcY'$-flat closed subscheme
$$
\mc{P}' \hookrightarrow \PP^{M-1}_{\mcY'}.
$$

\mni
There exists an open subscheme $U$ of $\mcY'$ such that for every
$\mcY'$-scheme $T$, the pullback of $\mc{P}'$ to $T$ is smooth over
$T$ if and only if $T$ factors through $U$; for the projection
$\pi':\mc{P}' \rightarrow \mcY'$, $U$ is the complement of the image
under $\pi'$ of the singular locus of $\pi'$.  
By construction,
$\widetilde{\iota}\circ \whtt|_{\Fr(\whtOO)}$ 
is the constant morphism corresponding
to the anticanonical embedding of $\PP^{n-1}$ in $\PP^{M-1}$ by the
basis $[s_1,\dots,s_M]$.  Thus the restriction of $\mc{P}'$ to the
strict transform $\whtt'$ of $\whtt$ in $\mcY'$ equals the constant
family $[s_1,\dots,s_M](\PP^{n-1})$ over the dense open subscheme $\SP
\Fr(\whtOO)$ of $\SP \whtOO$.  By separatedness of the Hilbert scheme,
the restriction of $\mc{P'}$ over all of $\whtt'(\SP \whtOO)$ equals
the constant family.
In particular, $\whtt'(\SP \whtOO)$ is contained
in $U$.  

\mni
The restriction of $\mc{P}'$ over $U$ is a Brauer-Severi scheme whose
associated Brauer class extends $\alpha$.  
\end{proof}

\mni
As promised, pseudo specialization implies Brauer equivalence.

\begin{prop} \label{prop-Brauereq}
\marpar{prop-Brauereq}
Assume that $\whtOO$ is Henselian.  If $\whtt$ $\pse$ specializes to
$\whts$, then the pullback maps,
$$
\whtt^*, \whts^*:\text{Br}(\mcY_{\Fr(\whtOO)}) \rightarrow
\text{Br}(\Fr(\whtOO))
$$
are equal.
\end{prop}

\begin{proof}
Of course it suffices to verify this when $\whtt$ directly $\pse$
specializes to $\whts$.  The goal is to prove that the difference
$\whtt^* - \whts^*$ equals the zero map, i.e., the kernel is the
entire Brauer group.  Of course the kernel contains the pullback
$p^*\text{Br}(\Fr(\whtOO))$ since $\whtt^*p^* = (p\circ \whtt)^* =
\text{Id}^*$ and likewise for $\whts^*p^*$.  And the sum of
$p^*\text{Br}(\Fr(\whtOO))$ and $\text{Ker}(\whtt^*)$ is the entire
Brauer group.  Thus it suffices to prove that the kernel of $\whtt^*$
is contained in the kernel of $\whts^*$.

\mni
To this end, let $\alpha$ be a Brauer class on $\mcY_{\Fr(\whtOO)}$
such that $\whtt^*\alpha$ equals $0$.  By Lemma ~\ref{lem-unramified},
there exists a projective, birational morphism $f:\mcY' \rightarrow
\mcY$ and an open subscheme $U$ of $\mcY'$ containing both the generic
fiber and the strict transform $\whtt'$ of $\whtt$ such that $\alpha$
extends to $U$.  Denote by $\whts'$ the strict transform of $\whts$.
By Lemma ~\ref{lem-model}(ii), $\whtt'$ directly $\pse$ specializes to
$\whts'$.  In particular, there exists a $1$-jet specialization
$$
r:\Boos \rightarrow \mcY'
$$
of $\whtt'$ to $\whts'$.  Since $U$ contains both the generic fiber
and the image of $\whtt'$, $U$ also contains the image of $r$.
Consider $r^*\alpha$.  By purity for Brauer classes,
cf. \cite[Th\'eor\`eme 6.1(b), Corollaire 6.2]{BrauerIII}, $r^*\alpha$
extends to a Brauer class $\gamma$ defined on all of $\Boo$.  

\mni
Consider first the restriction
of $\gamma$ to $\Du{1} = \SP (\whtOO/u)\Sem{v}$.  Since
$(\whtOO/u)\Sem{v}$ is Henselian, every Brauer class $\gamma$ on
$(\whtOO/u)\Sem{v}$ is the pullback of a Brauer class $\gamma_0$ on
$\whtOO/u$.  But since $r|_{\Dus{1}}$ equals the restriction of
$\whtt'$, and since $\alpha$ is trivial on $\whtt'$, the restriction
of $\gamma$ to $(\whtOO/u)\Semr{v}$ equals $0$.  Therefore $\gamma_0$
equals $0$.  

\mni
Consider next the restriction of $\gamma$ to $\Dv = \SP(\whtOO)$.
The further restriction to the closed point is $\gamma_0$, which
equals $0$ by the previous paragraph.  Since $\whtOO$ is Henselian, a
Brauer class on $\whtOO$ which is trivial on the closed point is
trivial on all of $\whtOO$.  Thus the restriction of $\gamma$ to $\Dv$
equals $0$.  In other words, $\whts^*(\alpha)$ equals $0$.  Therefore
the kernel of $\whtt*$ is contained in the kernel of $\whts^*$.
\end{proof}

\mni
\textbf{Relation to weak approximation.}  
Let $k$ be an
algebraically closed field.  Let $\pi:\mcX \rightarrow B$ be a flat,
proper $k$-morphism from an algebraic space $\mcX$ to a smooth,
projective $k$-curve $B$. 
Let $b$ be a $k$-point of $B$.  Denote by
$\whtOO$ the completion of the stalk of $\OO_B$ at $b$.  And denote by
$\mcX_{\whtOO}$ the base change of $\mcX$ to $\whtOO$.  A
$\whtOO$-point $\whts$ of $\mcX_{\whtOO}$ is \emph{approximable} if
for every integer $N$ there exists a section $s_N$ of $\pi$ such that
$s_N \cong \whts$ modulo $\mfm^N$.  If each $s_N$ intersects the very
free locus of $\pi$, then $\whts$ is \emph{approximable by sections
  intersecting the very free locus}.  

\begin{thm} \label{thm-second}
\marpar{thm-second}
Assume that $\mcX_{\whtOO}$ is regular.  Let $\whts$ and $\whtt$ be
$\whtOO$-points of $\mcX_{\whtOO}$.  If $\whtt$ $\pse$ specializes to
$\whts$, and if $\whts$ is approximable by sections intersecting the
very free locus, then also $\whtt$ is approximable by sections
intersecting the very free locus.
\end{thm}

\mni
Of course one can deduce an analogue using a larger number of closed
points by the Weil restriction technique of de Jong, cf. \cite{GHS}.
Sections of $\pi$ are parameterized by an algebraic space which is
locally of finite type over $k$.  But $\whtOO$-points of
$\mcX_{\whtOO}$ are not parameterized by an object which is locally of
finite type over $k$.  An important step in the proof of Theorem
~\ref{thm-second}, and hence Theorem ~\ref{thm-first}, is the
introduction of a new parameter space which is locally of finite type
over $k$ and which serves a similar role to the set of $\whtOO$-points
of $\mcX_{\whtOO}$.  This is done in the next two sections.

\mni
[CONTINUE HERE:  From this point on remains to be written / revised.]
% Let $k$ be an algebraically closed field.  And let
% $\pi:\mcX\rightarrow B$ be a surjective $k$-morphism of smooth,
% projective $k$-schemes with $B$ a curve.
% It is
% convenient to formulate weak approximation as a ``fiber-by-fiber''
% condition.  

% \begin{defn} \label{defn-int}
% \marpar{defn-int}
% Let $b_0$ be an element in $B(k)$.  The morphism $\pi$
% satisfies \emph{conditional approximation at $b_0$} 
% if for every finite
% collection of $(b_1,\dots,b_m)$ of distinct elements of $B(k)\setminus\{b_0\}$,
% for every section $s$ of $\pi$, for every element
% $\whts_0$ in $\mcX(\SP \whtOO_{B,b_0})$, and for every integer $N\geq 0$,
% there exists a section $t$ of $\pi$ such that $t\cong s$ modulo
% $\mfm_{B,b_i}^{N+1}$ for every $i=1,\dots,m$, and $t\cong \whts_0$
% modulo $\mfm_{B,b_0}^{N+1}$.  

% \mni
% The morphism satisfies \emph{geometric conditional approximation at
%   $b_0$} if for every algebraically closed extension $\kappa/k$, the
% basechange to $\kappa$ satisfies conditional approximation at $b_0$.    
% \end{defn}

% \mni
% The morphism $\pi$ satisfies weak approximation if and only if both
% $\pi$ has a section and $\pi$ satisfies conditional approximation at
% every point $b_0$ in $B(k)$.  

% \begin{ques}[The Hassett-Tschinkel question] \label{ques-wa}
% \marpar{ques-wa}
% Assuming the geometric generic fiber of $\pi$ is smooth and separably
% rationally connected, does $\pi$ satisfy conditional approximation at
% every element $b_0$ in $B(k)$?
% \end{ques}

% \mni
% Next let $\OO$ be a discrete valuation ring which is essentially of finite
% type over the coefficient field $k$. 
% Let $\pi_Y:\mcY \rightarrow \SP \OO$ 
% be a proper, flat $\OO$-scheme which is regular.  
% Denote by $K$ the fraction field of the completion, $K
% = \Fr(\whtOO)$.  

% \begin{defn} \label{defn-Requiv}
% \marpar{defn-Requiv}
% Elements $\whts$ and $\whtt$ of $\mcY(K)$ 
% are 
% \emph{directly $R$-equivalent} if
% there exists a $K$-morphism 
% $$
% f:\PP^1_{K} 
% \rightarrow \mcY\otimes_{\whtOO} K
% $$
% such that 
% $f(0)$
% equals $\whts$ and $f(\infty)$ equals $\whtt$.  
% Elements $\whts$ and $\whtt$ of $\mcY(K)$ are 
% \emph{$R$-equivalent} if there exists a finite
% sequence 
% $\whts=\whts_0,\whts_1,\dots,\whts_n = \whtt$ of elements $\whts_i$ of $\mcY(K)$
% such that for every $i=1,\dots,n$, $\whts_{i-1}$ is directly
% $R$-equivalent to $\whts_i$.

% \mni
% The $K$-scheme $\mcY\otimes_{\OO} K$ is 
% \emph{$R$-connected} 
% if every pair $\whts,\whtt$ of elements of
% $\mcY(K)$ is
% $R$-equivalent.  
% Finally, the $\OO$-scheme $\mcY$ is \emph{geometrically
% $R$-connected} if for every algebraically closed field extension
% $\kappa/k$, the basechange $\mcY\otimes_{\OO} \Fr(\wht{\OO\otimes_k
%   \kappa})$ is an $R$-connected scheme over $\Fr(\wht{\OO\otimes_k
%   \kappa})$.  
% \end{defn}

% \mni
% Theorem ~\ref{thm-main} deducing weak approximation from
% $R$-connectedness passes through the weaker notion of $\pse$
% $R$-connectedness.  The key notions are as follows.

% \begin{defn} \label{defn-psiR}
% \marpar{defn-psiR}
% Let $\whts$ and $\whtt$ be $\whtOO$-points of $\mcY$. 
% For every nonnegative integer $N$,
% $\whtt$ \emph{$N$-jet specializes} to $\whts$ if 
% there exists a complete DVR $A$ with coefficient field $k$
% and an $A$-flat closed subscheme
% $$
% \wht{\mc{C}} \subset \mcY \otimes_{\OO}(A\otimes_k \whtOO)
% $$
% whose restriction over $(A/\mf{m}_A)\otimes_k \Fr(\whtOO)$
% equals $\whts$ and whose restriction over $\Fr(A)\otimes_k
% (\whtOO/\mf{m}^{N+1})$ 
% equals the basechange from $k$ to $\Fr{A}$ of 
% the restriction of $\whtt$ modulo $\mf{m}^{N+1}$.   

% \mni
% The $\whtOO$-point $\whtt$ \emph{directly $\ps$-specializes} to $\whts$ if for
% every nonnegative integer $N$, $\whtt$ $N$-jet specializes to $\whts$.  The
% $\whtOO$-points 
% $\whtt$ \emph{$\ps$-specializes} to $\whts$ if there exists a sequence of
% $\whtOO$-points $\whtt = \whtt_0, \dots, \whtt_r = \whts$ such that for every
% $i=1,\dots,r$, $\whtt_{i-1}$ directly $\ps$-specializes to $\whtt_i$.  
% Elements $\whts$ and $\whtt$ are \emph{$\pse$ $R$-equivalent},
% resp. \emph{directly $\pse$ $R$-equivalent}, if each element
% $\ps$-specializes to the other, resp. directly $\ps$-specializes to
% the other.
% If every pair of $\whtOO$-points of $\mcY$ are $\pse$
% $R$-equivalent, then $\mcY$ is \emph{$\pse$ $R$-connected}.
% Finally, $\mcY$ is \emph{geometrically
%   $\pse$ $R$-connected} if for every algebraically closed field extension
% $\kappa/k$, the basechange $\mcY\otimes_{\OO} K(\wht{\OO\otimes_k
%   \kappa})$ is a $\pse$ $R$-connected scheme over $K(\wht{\OO\otimes_k
%   \kappa})$.   
% \end{defn}

% \mni
% An observation of Hassett shows that $R$-equivalence implies $\pse$
% $R$-equivalence.  

% \begin{lem}[Hassett] \label{lem-H}
% \marpar{lem-H}
% If $\whts$ and $\whtt$ are directly $R$-equivalent, then they are directly
% $\pse$ $R$-equivalent.  Thus $R$-equivalent points are $\pse$
% $R$-equivalent.  And if $\mcY$ is $R$-connected, resp. geometrically
% $R$-connected, then also $\mcY$ is $\pse$ $R$-connected,
% resp. geometrically $\pse$ $R$-connected.
% \end{lem}

% \mni
% The heart of this article is the following deformation theory result.

% \begin{prop} \label{prop-main}
% \marpar{prop-main}
% Let $\pi:\mcX \rightarrow B$ be a surjective $k$-morphism from a smooth,
% projective $k$-scheme $\mcX$ to a smooth, projective $k$-curve $B$.
% Denote by $\OO$ the local ring $\OO_{B,b}$. 
% Let $A$ be a
% complete DVR with coefficient field $k$.  
% Let 
% $$
% \wht{\mc{C}} \subset \mcX \otimes_{\OO_B}(A\otimes_k \whtOO)
% $$     
% be an $A$-flat closed subscheme 
% whose restriction over $(A/\mf{m}_A)\otimes_k \Fr(\whtOO)$
% equals the restriction over $\Fr(\whtOO)$ of a
% $\whtOO$-point $\whts$ of $\mcX$.  Denote by $N_0$ the smallest
% nonnegative integer such that the restriction of $\wht{\mc{C}}$ over
% $(A/\mf{m}_A)\otimes_k \whtOO$ is contained in the closed subscheme 
% $\wht(s) \cup N_0 \pi^{-1}(b)$.

% \mni
% Let $N\geq N_0$ be a nonnegative integer and let $s$ be a section of
% $\pi$ whose $N$-jet at $b$ agrees with $\whts$ and which maps the
% geometric generic point $\ol{\eta}$ of $B$ into a very free curve
% inside $(\mcX_{\ol{\eta}})_{\text{smooth}}$.  
% For every nonnegative integer $L$, and
% for every
% collection
% $b_1,\dots,b_m$ of distinct elements of $B(k)\setminus
% \{b_0\}$
% there exists an $A$-flat closed
% subscheme
% $$
% \mc{C} \subset \mcX \otimes_{\OO_B}(A\otimes_k \OO_B)
% $$
% such that
% \begin{enumerate}
% \item[(i)]
% the restriction of $\mc{C}$ over $(A/\mf{m}_A) \otimes_k \Fr(\OO_B)$
% equals the generic fiber of $s$,
% \item[(ii)]
% the restriction of $\mc{C}$ over $\Fr(A)\otimes_k
% (\whtOO/\mf{m}_{\whtOO}^{N+1})$ 
% equals the restriction of $\wht{\mc{C}}$ over $\Fr(A)\otimes_k
% (\whtOO/\mf{m}_{\whtOO}^{N+1})$, and 
% \item[(iii)]
% for every $i=1,\dots,m$, 
% the restriction of $\mc{C}$ over $\Fr(A)\otimes_k
% (\OO_{B,b_i}/\mf{m}_{b_i}^{L+1})$ 
% equals the basechange of
% $s$ modulo $\mf{m}_{b_i}^{L+1}$.
% \end{enumerate}
% \end{prop}

% \mni
% In particular, for a section $s$ of $\pi$ mapping $\ol{\eta}$ into a
% very free curve in $(\mcX_{\ol{\eta}})_{\text{smooth}}$, 
% for every integer $N$, for every $\whtOO$-point $\whtt$
% of $\mcX$ which $N$-jet specializes to the completion $\whts$ of $s$, 
% for every integer $L$,
% and for every collection of distinct elements $b_1,\dots,b_m$ of
% $B(k)\setminus \{ b_0 \}$, 
% there
% exists a section $t$ of $\pi$ mapping the geometric generic point
% $\ol{\eta}$ into a very free curve in
% $(\mcX_{\ol{\eta}})_{\text{smooth}}$, whose $N$-jet at $b$ equals the
% $N$-jet of $\whtt$, and whose $L$-jet at $b_i$ equals the $L$-jet of
% $s$ for every $i=1,\dots,m$.  

% \begin{cor} \label{cor-main}
% \marpar{cor-main}
% Let $\pi:\mcX \rightarrow B$ be a surjective $k$-morphism from a
% smooth, projective $k$-scheme $\mcX$ to a smooth, projective $k$-curve
% $B$.  If the geometric generic fiber $\mcX_{\ol{\eta}}$ is smooth and
% separably rationally connected, and if the Laurent fiber at $b$ is
% $\pse$ $R$-connected, then $\pi$ satisfies conditional approximation
% at $b$.  
% \end{cor}

% \mni
% A result of Koll\'ar is that two points of $Y$ are directly
% $R$-equivalent if their $N$-jets agree for sufficiently large $N$.

% \begin{lem}\cite{KollarSpec} \label{lem-K}
% \marpar{lem-K}
% Let $s$ be an $\whtOO$-point of $\mcY$.  Assume that $s(K)$ is
% contained in a very free curve inside the smooth locus
% $(\mcY_K)_{\text{smooth}}$.  Then there exists an integer $N$ such
% that for every $\whtOO$-point $t$ of $\mcY$ with $s\cong t$ modulo
% $\mf{m}_{\whtOO}^{N+1}$, $t$ and $s$ are directly $R$-equivalent (and
% thus also directly $\pse$ $R$-equivalent).
% \end{lem}

% \mni
% Together with Proposition ~\ref{prop-main}, this implies the
% following.  

% \begin{cor} \label{cor-K}
% \marpar{cor-K}
% Let $s$ and $t$ be $\whtOO$-points of $\mcY$.  Assume that $s(K)$
% and $t(K)$ are each contained in a very free curve (not necessarily
% the same one) inside the smooth locus of $(\mcY_K)_{\text{smooth}}$.
% Let $N_s$ and $N_t$ be as in Lemma ~\ref{lem-K} and denote
% $\max(N_s,N_t)$ by $N$.  Assume that $s$ $N$-jet specializes to an
% $\whtOO$-point $t'$ with the same $N$-jet as $t$.  Then $s$
% $\ps$-specializes to $t$.  
% \end{cor}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%
% %% Subection: The main results
% %% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \subsection{The main results} \label{ssec-sor}
% \marpar{ssec-sor}

% \mni
% Denote by $\mcX\psm{\pi}$ the maximal open subscheme of $\mcX$ on
% which $\pi$ is smooth.
% Our main result is the following.  

% \begin{thm} \label{thm-Requiv2wa}
% \marpar{thm-Requiv2wa}
% Let $b$ be an element of $B(k)$.  If the
% $\ol{k(B)}$-scheme $\mcX\psm{\pi}\otimes_{\OO_B}\ol{k(B)}$ 
% is strongly separably
% rationally connected and if the $\OO_{B,b}$-scheme 
% $\mcX\psm{\pi}\otimes_{\OO_B} \OO_{B,b}$
% is geometrically $R$-connected, then the morphism $\pi$ satisfies
% geometric conditional approximation at $b$.

% \mni
% In particular, if $\mcX\otimes_{\OO_B}
% \ol{k(B)}$ is smooth and separably rationally connected, and if 
% every ``Laurent fiber'' $\mcX\otimes_{\OO_B} \OO_{B,b}$
% is geometrically $R$-connected,
% then the $k(B)$-scheme $\mcX\otimes_{\OO_B} k(B)$ 
% satisfies weak approximation
% (also geometrically, i.e., after base change from $k$ to any
% algebraically closed extension).
% \end{thm}

% \mni
% This is a consequence of the following deformation result.   
% For a section $s$ of $\pi$, 
% a \emph{comb-like curve with handle $s$} is a connected, 
% closed, regularly
% embedded curve $C$ in $\mcX$ such that for some dense open subset $W$
% of $B$, $C\cap \pi^{-1}(W)$ equals $s(W)$.  

% \mni
% Let $E$ be
% an effective divisor in $B$, and let 
% $$
% s_E:E \rightarrow \mcX
% $$
% be a section of $\pi$ over $E$.  A comb-like curve $C$
% \emph{deforms to sections agreeing
%   with $s_E$ near $E$} if there exists 
% a DVR $A$ with coefficient field $k$, an open neighborhood $W$ of $\SP
% A \times_{\SP k} E$ in $\SP A \times_{\SP k} B$, 
% and 
% an $A$-flat closed subscheme
% $$
% \mc{C} \subset \mcX \times_B W
% $$
% whose closed fiber $\SP k \times_{\SP A} \mc{C}$ equals $C\times_B W_0$ 
% and whose
% generic fiber is the graph of a $B$-morphism
% $$
% t:W_\eta \rightarrow \mcX
% $$
% whose restriction to $\SP K(A)\times_k E$ equals the base change of
% $s_E$.  

% \mni
% Similarly, for a connected component $N\cdot \ul{b}$ of 
% $E$, $C$
% \emph{formally deforms to sections agreeing with
%   $s_E$ near $b$} if there exists a DVR $A_b$ with coefficient field
% $k$ and an $A$-flat closed
% subscheme 
% $$
% \mc{C}_b \subset \SP A \times_{\SP k}(\SP \whtOO_{B,b} \times_B
% \mcX)
% $$
% whose closed fiber $\SP k\times_{\SP A} \mc{C}_b$ equals $C\times_B
% \SP \whtOO_{B,b}$ and whose generic fiber is the graph of a
% $B$-morphism
% $$
% t_b:\SP K(A) \times_k \SP \whtOO_{B,b} \rightarrow \mcX
% $$
% whose restriction to $\SP K(A)\times_k \SP
% \whtOO_{B,b}/\mfm_{B,b}^{N+1}$ equals the base change of
% $s_E$.  

% \begin{thm} \label{thm-main}
% \marpar{thm-main}
% Let $s$ be a section of $\pi$ which 
% sends the geometric generic point
% of $B$ into the very free locus of $\mcX\psm{\pi}\otimes_{\OO_B}
% \ol{K(B)}$.  Let $E$ be an effective Cartier divisor
% in $B\setminus S$.  
% Let $s_E:E\rightarrow U$ be a section of $\pi$ over
% $E$.  And let $C$ be a 
% comb-like curve with handle $s$. 

% \mni
% If for every point $b$ in $\text{Supp}(E)$, $C$ formally deforms to
% sections agreeing with $s_E$ near $b$, then $C$ deforms to
% sections
% agreeing with $s_E$ near $E$.  
% \end{thm}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%
% %% Subection: Consequences
% %% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \subsection{Consequences} \label{ssec-or}
% \marpar{ssec-or}

% \mni
% There is a connection between $R$-connectedness and the existence of formal
% deformations of a comb-like curve to section curves.  This comes 
% from the
% following criterion whose original version is due to Brendan Hassett.  

% \begin{lem} \label{lem-Hassett}
% \marpar{lem-Hassett}
% Let $s$ be a
% section of $\pi$.  Let $b$ be an element in
% $B(k)$.  Let $\whts$ be an element of $\mcX(\SP
% \whtOO_{B,b})$.  Assume that 
% there exists a closed subscheme $P$ of
% $\mcX\otimes_{\OO_B} \whtOO_{B,b}$ which is proper and regular, which
% contains $s(\SP \whtOO_{B,b})$ and $\whts(\SP \whtOO_{B,b})$, and with
% $P\otimes_{\whtOO_{B,b}} K(\whtOO_{B,b}) \cong
% \PP^1_{K(\whtOO_{B,b})}$ as $K(\whtOO_{B,b})$-schemes.
% Then for every integer $N\geq 0$ there
% exists a comb-like curve $C_N$ in $\mcX$ with handle $s$
% which 
% formally deforms to sections agreeing with $\whts|_{N\cdot b}$ near
% $b$.  
% Moreover, for every integer $N\geq 0$ 
% there exists an integer $M\geq N$ such that every comb-like curve $C'$
% agreeing with $C_N$ modulo $\mfm_{B,b}^{M+1}$ also deforms to sections
% agreeing with $\whts|_{N\cdot b}$ near $b$.
% \end{lem}

% \mni
% The hypothesis that $P$ is regular is annoying.  Typically $P$ is
% defined to be the closure in $\mcX\otimes_{\OO_B} \whtOO_{B,b}$
% of a rational curve in $\mcX\otimes_{\OO_B}
% K(\whtOO_{B,b})$, which may fail to be regular.
% Using
% resolution of singularities of surfaces, e.g., as presented in
% \cite[Chapter 7]{Cutkosky}, this hypothesis can be removed up to
% replacing $\mcX$ by $\mcX \times_k (\PP^1_k)^a$ for some integer
% $a\geq 0$. For the applications, in particular for Theorem
% ~\ref{thm-Requiv2wa}, this replacement is harmless.

% \begin{prop} \label{prop-Hassett}  
% \marpar{prop-Hassett}
% Let $s$ be a
% section of $\pi$.  Let $b$ be an element in
% $B(k)$.  Let $\whts$ be an element of $\mcX(\SP
% \whtOO_{B,b})$.  Assume that there exists a morphism of
% $K(\whtOO_{B,b})$-schemes, 
% $$
% f:\PP^1_{K(\whtOO_{B,b})} \rightarrow \mcX\otimes_{\whtOO_{B,b}}
% K(\whtOO_{B,b}), 
% $$
% sending $0$ to 
% $s(\SP K(\whtOO_{B,b}))$ and sending $\infty$ to $\whts(\SP
% K(\whtOO_{B,b}))$.  Then there exists an integer $a\geq 0$ such that
% for every integer $N\geq 0$ there exists 
% a comb-like curve $C_N$ in $\mcX\times_k
% (\PP^1_k)^a$ with handle $s\times(0,\dots,0)$ which formally deforms
% to sections agreeing with $\whts|_{N\cdot b}\times
% (\infty,\dots,\infty)$. 
% Moreover there exists an integer $M\geq N$ such that every comb-like
% curve $C'$ agreeing with $C_N$ modulo $\mfm_{B,b}^{M+1}$ also deforms
% to sections agreeing with $\whts|_{N\cdot b}\times
% (\infty,\dots,\infty)$.  
% \end{prop}

% \mni
% There are some simple criteria for $R$-connectedness of a Laurent
% fiber.  

% \begin{lem} \label{lem-Rtriv-sorites}
% \marpar{lem-Rtriv-sorites}
% Let $(k,\OO,\mcY)$ be as in Definition ~\ref{defn-Requiv}.  
% \begin{enumerate}
% \item[(i)]
% A pair $s$ and $t$ of elements in $\mcY(\whtOO)$ are 
% directly
% $R$-equivalent if the closed points $s_0$ and $t_0$ are contained in a
% very free rational curve in $(\mcY_0)\sm$.  
% \item[(ii)]
% Let $\mcY'$ also be a proper, flat $\OO$-scheme which is regular.  
% If
% $\mcY\otimes_{\whtOO} K$ and $\mcY'\otimes_{\whtOO} K$ are
% birational as $K$-varieties, then $\mcY$ is $R$-connected,
% resp. geometrically $R$-connected, if
% and only if $\mcY'$ is $R$-connected, resp. geometrically
% $R$-connected. In particular, if $\mcY\otimes_{\whtOO} K$ is
% $K$-rational, then $\mcY$ is geometrically $R$-connected.
% \item[(iii)]
% Let $\mcY'$ be a projective, flat $\OO$-scheme which is
% regular and let
% $$
% f:\mcY' \rightarrow \mcY
% $$
% be a dominant morphism of $\OO$-schemes.  Let $U$ be the maximal open
% subscheme of $\mcY'$ on which $f$ is smooth and let $V$ be $f(U)$.  If 
% $\mcY'$ is $R$-connected, then every pair of elements in $V(\whtOO)$
% are $R$-connected in $\mcY$.  
% \item[(iv)]
% [HERE]
% Finally, as above, let $\mcY'$ be a proper, flat $\OO$-scheme
% which is 
% regular and let
% $$
% f:\mcY'\otimes_{\whtOO} K \rightarrow \mcY\otimes_{\whtOO}K
% $$
% be a dominant morphism of $K$-schemes whose geometric generic
% fiber is normal and separably rationally connected in the sense of
% \cite{dJS}.  
% If $\mcY'$ is $R$-connected, resp. geometrically
% $R$-connected, then also $\mcY$ is $R$-connected,
% resp. geometrically $R$-connected.   
% \end{enumerate}
% \end{lem}

% \mni
% There are also some criteria for weak approximation.

% \begin{lem} \label{lem-wa-sorites}
% \marpar{lem-wa-sorites}
% [HERE]
% \end{lem}

% \mni
% Combined with Theorem ~\ref{thm-main}, this gives a second proof of a
% result whose original version was first proved by Hassett and
% Tschinkel.  

% \begin{cor}\cite{HT06} \label{cor-HT}
% \marpar{cor-HT}
% Let $(k,B,\mcX,U,S)$ be as above.  Let $b$ be
% an element of $B(k)\setminus S$.  If $U_b$ is strongly separably
% rationally connected, then $(k,B,\mcX,U,S)$ satisfies geometric 
% conditional approximation at $b$.  In particular, if
% every fiber
% $(\mcX\psm{\pi})_b$ is strongly separably rationally connected, then
% $(k,B,\mcX)$ satisfies geometric weak approximation.
% \end{cor}

% \mni



% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %% %%
% %% %% Subsection: Statement of results
% %% %% 
% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %% \subsection{Statement of results} \label{sec-sor}
% %% \marpar{sec-sor}

% %% \mni
% %% Let $k$ be an algebraically closed field.  Let $B$ be a smooth,
% %% projective, connected curve over $k$.  

% %% \begin{notat} \label{notat-rings}
% %% \marpar{notat-rings}
% %% For every closed point $b$ of
% %% $B$, denote the 
% %% Henselian local ring of $B$ at $b$ by $\OO_{B,b}^h$ and
% %% denote its fraction field $K(\OO_{B,b}^h)$ by $K_{B,b}^h$.  Denote the
% %% completion of the local ring by $\whtOO_{B,b}$, and denote its
% %% fraction field by $\wht{K}_{B,b}$.  

% %% \mni
% %% More generally, for an effective Cartier
% %% divisor $E$ in $B$ with $\text{Supp}(E) =\{b_1,\dots,b_N\}$, denote by
% %% $\OO_{B,E}$ the semilocal ring of $B$ at $\{b_1,\dots,b_N\}$, denote
% %% by $\OO_{B,E}^h$ the product 
% %% $\OO_{B,b_1}^h \times \dots \times \OO_{B,b_N}^h$, and
% %% denote its total ring of fractions of $\OO_{B,E}^h$ by $K_{B,E}^h =
% %% K_{B,b_1}^h \times \dots \times K_{B,b_N}^h$.  Denote by
% %% $\whtOO_{B,E}$ the product $\whtOO_{B,b_1} \times \dots
% %% \times \whtOO_{B,b_N}$, and denote by $\wht{K}_{B,E}$ its
% %% total ring of fractions $\wht{K}_{B,E} \times \dots \times
% %% \wht{K}_{B,E}$.  
% %% \end{notat}

% %% \mni
% %% For each of the $B$-algebras, $\OO = \OO_{B,E}$,
% %% resp., $\OO = \OO_{B,E}^h$, $\OO = \whtOO_{B,E}$, the morphism
% %% $$
% %% \SP \OO \times_B E \rightarrow E
% %% $$
% %% is an isomorphism.  Thus we identify $E$ with its inverse image in
% %% each of these affine $B$-schemes.    

% %% \mni
% %% Let $X$ be a normal, 
% %% projective, connected $k$-scheme and let $\pi:X\rightarrow B$ be a
% %% surjective $k$-morphism whose geometric generic fiber is normal.
% %% %% Assume that the geometric generic fiber of
% %% %% $\pi$ is normal and contains a very free rational curve inside its
% %% %% smooth locus.  

% %% \begin{defn} \label{defn-comblike}
% %% \marpar{defn-comblike}
% %% Let $s:B\rightarrow X$ be a section of $\pi$. 
% %% A \emph{comb-like curve
% %%   with handle $s$ in $X$} is a connected, closed curve $C$ in $X$ such
% %% that $C\rightarrow X$ is a regular immersion
% %% and such that the intersection of $C$ with the generic fiber of $B$
% %% equals $s(\eta_B)$.
% %% In other words, $s(B)$ is the only
% %% irreducible component of $C$ dominating $B$, and $C$ is reduced at the
% %% generic point of $s(B)$.  
% %% \end{defn}

% %% \begin{rmk} \label{rmk-comblike}
% %% \marpar{rmk-comblike}
% %% Among all comb-like curves are the \emph{combs} with handle $s$ as defined by
% %% Koll\'ar-Miyaoka-Mori, and the \emph{combs with broken teeth} as defined by
% %% Hassett-Tschinkel.  There are many comb-like curves which are neither
% %% combs nor combs with broken teeth, most of which are
% %% uninteresting.  We are interested only in those comb-like curves
% %% which deform to smooth, connected curves.
% %% This imposes many local constraints which we have not attempted to
% %% characterize.  
% %% \end{rmk}

% %% \begin{defn} \label{defn-jet}
% %% For an effective Cartier divisor $E$ in $B$, an \emph{$E$-section
% %%   of $\pi$}, or just \emph{$E$-section}, 
% %% is a $B$-morphism $s_E:E\rightarrow X$.  
% %% A comb-like curve $C$ in $X$
% %% \emph{deforms to section curves agreeing with $s_E$ Zariski locally},
% %% resp. \emph{\'etale locally}, \emph{formally locally},
% %% if for
% %% $B' = \SP \OO_{B,E}$, resp. for $B'=\SP \OO_{B,E}^h$, $B'=\SP
% %% \whtOO_{B,E}$, 
% %% there exists a DVR $(R,\mf{n})$
% %% with coefficient field $k$ 
% %% and an $R$-flat, closed subscheme $\mc{C}\subset \SP R \times_{\SP k}
% %% B' \times_B X$ 
% %% such that
% %% \begin{enumerate}
% %% \item[(i)]
% %% the fiber of $\mc{C}$ over the residue field
% %% $\SP R/\mf{n}$ equals the base change of $C$ to
% %% $B' \times_B X$, and
% %% \item[(ii)]
% %% the fiber of $\mc{C}$ over the fraction field $\SP K(R)$ equals the image
% %% of a section $s_\eta:\SP K(R) \times_{\SP k} B' \rightarrow
% %% \SP K(R) \times_{\SP k} B' \times_B X$ whose restriction to 
% %% $\SP K(R) \times_{\SP k} E$ equals the
% %% base change of $s_E$.  
% %% \end{enumerate}
% %% In other words, on a Zariski neighborhood, resp. 
% %% \'etale neighborhood, formal neighborhood, of the fiber $\pi^{-1}(E)$
% %% in $X$, $C'$ deforms to section curves all of which contain $s_E(E)$. 
% %% \end{defn}

% %% \begin{rmk} \label{rmk-jet}
% %% \marpar{rmk-jet}
% %% Let $E = E_1 \sqcup \dots \sqcup E_N$ be the connected components of
% %% $E$.  Then $C$ deforms to section curves agreeing with $s_E$ \'etale
% %% locally, resp. formally locally,  
% %% if and only if for every $i=1,\dots,N$, $C$ deforms to section
% %% curves agreeing with $s_E|_{E_i}$ \'etale locally, resp. formally
% %% locally.  
% %% Thus the \'etale and formal local conditions
% %% can each be studied fiber-by-fiber.
% %% \end{rmk}

% %% \begin{thm} \label{thm-main}
% %% \marpar{thm-main}
% %% Let $s:B\rightarrow X$ be a section of $\pi$ mapping the geometric
% %% generic point of $B$ into the very free locus of the geometric generic
% %% fiber of $\pi$. (In particular, the very free locus is nonempty and
% %% $\pi$ is a separably rationally connected fibration.)  
% %% Let $E$ be an effective Cartier divisor in $B$.  Let
% %% $s_E$ be an $E$-section of $\pi$.  Let $C$ be a comb-like curve with
% %% handle $s$.  Assume $C$ deforms to section curves agreeing with $s_E$
% %% \'etale locally or formally locally. 
% %% Then $C$ deforms to section curves agreeing with
% %% $s_E$ Zariski locally.  To be precise,
% %% after attaching to $C$
% %% sufficiently many very free teeth in fibers of $\pi$ at general points
% %% of $s(B)$ and with general normal directions, $C$ deforms to comb-like
% %% curves all of which are section
% %% curves agreeing with $s_E$ over a Zariski open neighborhood of $E$ in $B$.    
% %% \end{thm}

% %% \mni
% %% This reduces weak approximation to the problem of finding comb-like
% %% curves $C$ with handle $s$ deforming to section curves agreeing with $s_E$
% %% either \'etale locally or formally locally.  There is a criterion
% %% due to Hassett guaranteeing existence of such curves.  
% %% Originally formulated as a
% %% criterion in characteristic $0$, the extension to positive
% %% characteristic poses some minor issue in characteristics $2$, $3$ and
% %% $5$.  In order to avoid these issues, we use a slight trick.

% %% \begin{defn} \label{defn-conic}
% %% \marpar{defn-conic}
% %% Let $B$ be a smooth, connected, projective $k$-curve and let $E$ be an
% %% effective Cartier divisor in $B$.  An \emph{$E$-conic} is a
% %% smooth, projective $k$-surface $S$ together with a birational morphism
% %% $\nu:P\rightarrow \PP^1\times_k B$ such that the restriction
% %% $$
% %% \nu:\nu{-1}(\PP^1\times_k(B-E)) \rightarrow \PP^1 \times_k (B-E)
% %% $$
% %% is an isomorphism.  The \emph{zero section} of $P$ is the unique
% %% $B$-morphism
% %% $$
% %% 0_P:B\rightarrow P
% %% $$
% %% whose composition with $\nu$ is the zero section of $\PP^1 \times_k
% %% B$.  
% %% \end{defn}

% %% \begin{rmk} \label{rmk-conic}
% %% \textbf{(i)}
% %% Given a section $s$ and an $E$-section $s_E$, suppose there exists an
% %% $E$-conic $\nu:P\rightarrow X\times_B \PP^1 $ and a comb-like curve
% %% $C$ in $X\times_B \PP^1$ with handle $(s,0_P)$ which \'etale locally
% %% or formally locally deforms to section curves agreeing with a given
% %% $E$-section $(s_E,t_E)$.  Then Theorem ~\ref{thm-main} applies to
% %% produce a section in $X\times_B \PP^1$ agreeing with $(s_E,t_E)$.
% %% Thus the projection into $X$ is a section of $\pi$ agreeing with
% %% $s_E$.  Therefore, to prove the existence of sections of $\pi$
% %% agreeing with $s_E$, it suffices to prove the local condition for
% %% $X\times_B \PP^1$

% %% \mni
% %% \textbf{(ii)}
% %% Also,for a divisor $E$ with support $\{b_1,\dots,b_N\}$, 
% %% the modification $\nu:P \rightarrow \PP^1\times_k B$ and the 
% %% comb-like curve $C$ can both be defined in the neighborhood of every
% %% fiber $\pi^{-1}(b_i)$ and then these local pieces can be assembled
% %% into a global modification, resp. a global comb-like curve.  Thus the
% %% existence of $\nu$ and $C$ with the required properties can be
% %% verified near each fiber $\pi^{-1}(b_i)$ separately.
% %% \end{rmk}

% %% \begin{prop}[Hassett] \label{prop-Hassett}
% %% \marpar{prop-Hassett}
% %% Let $B$ be a smooth, connected, projective $k$-curve, let $X$ be a
% %% normal, projective, connected $k$-scheme, and let $\pi:X\rightarrow B$
% %% be a surjective $k$-morphism.  Let $s:B\rightarrow X$ be a section of
% %% $\pi$ whose image $s(B)$ is contained in the smooth locus of $\pi$.
% %% Let $E$ be an effective Cartier divisor in $B$ supported at a single
% %% point $b$.  Let $\OO$ denote $\OO_{B,b}^h$, resp. $\whtOO_{B,b}$,
% %% and let $K$ denote $K_{B,b}^h$, resp. $\wht{K}_{B,b}$.  Let $s_E$ be an
% %% $E$-section of $\pi$. 

% %% \mni
% %% Roughly stated, 
% %% if the \'etale, resp. formal, germ of $s$ over $b$ gives a
% %% $K$-point of $X(K)$ which is $R$-equivalent to the $K$-point of a germ
% %% agreeing with $s_E$, then for some $E$-conic $P$, there exists a
% %% comb-like curve $C$ in $X\times_B P$ with handle $(s,0_P)$ which
% %% \'etale locally, resp. formally locally, deforms to section curves
% %% agreeing with an $E$-section $(s_E,t_E)$.  

% %% \mni
% %% Following is one precise formulation (certainly not the most general).
% %% Assume there exists a morphism of $K$-schemes
% %% $$
% %% f:\PP^1_{K} \rightarrow \SP K \times_B X
% %% $$
% %% whose closure $S$ in $\SP \OO \times_B X$ intersects the closed
% %% fiber $\pi^{-1}(b)$ only at smooth points of $X$ and sending the
% %% $K$-points $0$, resp. $\infty$, of $\PP^1_{K}(K)$ to the $K$-point of
% %% the germ of $s$, 
% %% resp. 
% %% to a $K$-point whose unique extension
% %% $$
% %% s':\SP \OO \rightarrow \SP \OO \times_B X
% %% $$
% %% agrees with $s_E$.  Then there exists an $E$-conic $P$ and a comb-like
% %% curve $C$ in $X\times_B P$ with handle $(s,0_P)$
% %% which \'etale locally, resp. formally locally, deforms 
% %% to section curves agreeing with an $E$-section $(s_E,t_E)$.

% %% \mni
% %% Moreover, if $f$ is an immersion and if the closed fiber of $S$ is
% %% smooth, then there exists a comb-like curve $C$ in $X$ with handle $s$
% %% which \'etale locally, resp. formally locally, deforms to section
% %% curves agreeing with $s_E$ (i.e., there is no need for the $E$-conic).
% %% \end{prop}

% %% \mni
% %% When combined with Proposition ~\ref{prop-Hassett}, Theorem
% %% ~\ref{thm-main} unifies and extends several known results regarding
% %% weak approximation.  Most of these follow immediately; we content
% %% ourselves with the following.

% %% \begin{cor}\cite{HT06} \label{cor-HT}
% %% \marpar{cor-HT}
% %% With hypotheses as in Theorem ~\ref{thm-main}, assume also that $s(B)$ is
% %% contained in the smooth locus of $X$.  Let $E$ be an effective Cartier
% %% divisor in $B$ and let $s_E$ be an $E$-section of $\pi$.  If the
% %% relative dimension of $\pi$ is $\geq 3$ and if every fiber
% %% $\pi^{-1}(b)$ in $\pi^{-1}(E)$ is smooth and separably rationally connected,
% %% then there exists a comb-like curve $C$
% %% with handle $s$ which Zariski locally deforms to
% %% section curves agreeing with $s_E$.  If the relative dimension is
% %% $\leq 2$ and if every fiber $\pi^{-1}(b)$ in $\pi^{-1}(E)$ is smooth
% %% and rationally connected, then for the $E$-conic $P = B\times_k \PP^1$
% %% there exists a comb-like curve $C$ in $X\times_B P$ with
% %% handle $(s,0_P)$ which Zariski locally deforms to section curves
% %% agreeing with an $E$-section $(s_E,t_E)$.  

% %% \mni
% %% More generally, the two conclusions above
% %% hold if for every point $b$ in $\text{Supp}(E)$ there exists an
% %% immersed, resp. not necessarily immersed, very
% %% free 
% %% rational curve in the smooth locus of $\pi^{-1}(b)$ containing both
% %% $s(b)$ and $s_E(b)$ (possibly both the same point).  
% %% \end{cor}

% \begin{cor} \label{cor-rational}
% \marpar{cor-rational}
% With hypotheses as in Theorem ~\ref{thm-main}, assume also that $s(B)$
% is in the smooth locus of $X$.  Let
% $E$ be an effective Cartier divisor in $B$ such that $X$ is smooth
% along $\pi^{-1}(E)$.  Assume that for each $b$ in $\text{Supp}(E)$, either
% $\SP \wht{K}_{B,b} \times_B X$ is
% $\wht{K}_{B,b}$-rational or $\wht{K}_{B,b}$-stably rational. 
% %% or
% %% $\wht{K}_{B,b}$-stably birational to a conic bundle over a
% %% rational variety. 
% Then for
% every $E$-section $s_E$ there exists an $E$-conic $P$ over $B$ and a 
% comb-like curve $C$ in $X\times_B P$ with
% handle $(s,0_P)$ which Zariski locally deforms to section curves
% agreeing with an $E$-section $(s_E,t_E)$.  
% \end{cor}

% \mni
% In particular, for a smooth cubic surface over a local field
% specializing to a cubic surface with only rational double points and
% no more than $3$ of these, the cubic surface is rational.  

% \begin{cor}\cite{HT07} \label{cor-cubic}
% \marpar{cor-cubic}
% Let $S$ be a smooth
% cubic surface over $\SP k\Semr{t}$ extending to a flat, proper
% scheme over $\SP k\Sem{t}$ whose closed fiber is a cubic surface
% with only rational double
% points, and at most $3$ of these.  Then $S$ is $k\Semr{t}$-rational.   

% \mni
% Assume $X$ is smooth and the geometric generic fiber of $\pi$ is a smooth
% cubic surface.  Let $s$ be a section of $\pi$.  Let $E$ be a Cartier
% divisor in $B$ such that for every point $b$ in $\text{Supp}(E)$ the
% basechange $\SP \wht{K}_{B,b}\times_B X$ extends to a flat, proper
% scheme over $\SP \whtOO_{B,b}$ whose closed fiber is a cubic
% surface with only rational double points, and at most $3$ of these.
% Then for every $E$-section $s_E$, there exists an $E$-conic $P$
% and a comb-like curve $C$ in $X\times_B P$ 
% with handle $(s,0_P)$ which Zariski locally deforms 
% to section curves agreeing with an $E$-section $(s_E,t_E)$.  
% \end{cor}    

% \begin{rmk} \label{rmk-cubic}
% \marpar{rmk-cubic}
% In fact Hassett and Tschinkel prove this result for families of cubic
% surfaces whose singular fibers have rational double points, but
% possibly more than $3$ such double points.  There is only one cubic
% surface with $4$ rational double points, the zeroset of
% $X_0X_1X_2+X_0X_1X_3 + X_0X_2X_3+X_1X_2X_3$ which has $4$ ordinary
% double points at the vertices of the coordinate tetrahedron.  The
% method in ~\cite{HT07} uses \emph{strong rational connectedness},
% i.e., proving the smooth locus of each singular fiber is rationally
% connected, combined with Corollary ~\ref{cor-HT}.  For the unique
% cubic surface with $4$ ordinary double points, one can explicitly
% construct rational curves in the smooth locus connecting any pair of
% points.     
% \end{rmk}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%
% %% Section: Deforming comb-like curves
% %% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \section{Deforming comb-like curves} \label{sec-def}
% \marpar{sec-def}

% \mni
% Let $k$ be an algebraically closed field.  Let $B$ be a smooth,
% projective, connected curve over $k$.  Let $X$ be a normal, 
% projective, connected $k$-scheme.  And let $\pi:X\rightarrow B$ be a
% surjective $k$-morphism of relative dimension $d$.  
% Assume that the geometric generic fiber of
% $\pi$ is normal and contains a very free rational curve inside its
% smooth locus.  

% \begin{lem} \label{lem-comblike}
% \marpar{lem-comblike}
% There is an open subscheme of $\text{Hilb}_{X/k}$ precisely
% parametrizing the comb-like curves.
% \end{lem}

% \begin{proof}
% There is an open subscheme of $\text{Hilb}_{X/k}$ parametrizing
% regular immersions of codimension $d-1$.  The degree of such a curve
% over $B$ is a locally constant integer-valued function.  Thus the open
% subscheme of comb-like curves is precisely the open subset on which
% the degree equals $1$.
% \end{proof}

% \begin{prop} \label{prop-comblike}
% \marpar{prop-comblike}
% Let $s:B\rightarrow X$ be a section of $\pi$ 
% mapping the geometric generic point of $B$ into the very
% free locus of the geometric generic fiber of $\pi$.  
% Let $E$ be an effective Cartier divisor in $B$, and denote by $D$ the
% inverse image 
% Cartier divisor $\pi^{-1}(E)$ in $X$.  Let $C$ be a comb-like curve
% with handle $s$ which is contained in the scheme-theoretic union of
% $s(B)$ and $D$.  There exists an integer $N$ with the following
% property.  For every comb-like curve $C'$ obtained by attaching at
% least $N$ very free curves in fibers of $\pi$ to $C$ at general points
% of $s(B)$ and with general normal directions to $s(B)$, the divisor 
% restriction map, $\iota_D:\text{Hilb}_{X/k} \rightarrow \pis{D/k}$, is
% smooth at $[C']$.  
% \end{prop}

% \begin{proof}
% Since $s$ maps the geometric generic point of $B$ into the very free
% locus of $\pi$, with the exception of finitely many points, for every
% $k$-point $b$ of $B$, $\pi$ is smooth at $s(b)$ and for every normal
% direction to $s(B)$ in $X$ at $s(b)$, there exists a very free curve in
% $\pi^{-1}(b)$ containing $s(b)$ and having the specified normal
% direction.   
% So long as the attachment points $s(b)$ is a smooth point of $C$
% (which also holds with finitely many exceptions), the curve obtained
% by attaching this very free curve to $C$ is again comb-like.  The
% final exception is that we only attach very free curves at points $b$
% not contained in $E$.  

% \mni
% Let $C'$ be a curve obtained from $C$ by attaching 
% a collection of very free rational curves in fibers of $\pi$ at points
% of $s(B)$ which are smooth points of $C$. 
% Denote by $C''$ the subcurve of $C'$ consisting of $s(B)$ and all the
% attached very free curves in fibers of $\pi$.  
% Since
% $C'\rightarrow X$ is a regular immersion, the sheaf
% $\mc{I}_{C'/X}/\mc{I}_{C'/X}^2$ is a locally free $\OO_{C'}$-module,
% where $\mc{I}_{C'/X}$ is the ideal sheaf of $C'$ in $X$.  Denote by
% $N_{C'/X}$ the dual sheaf, 
% $$
% N_{C'/X} := \textit{Hom}_{\OO_{C'}}( \mc{I}_{C'/X}/\mc{I}_{C'/X}^2 ,
% \OO_{C'} ).
% $$
% The subsheaf $\OO_X(-D)\cdot N_{C'/X}$ is supported on $C''$ since $C$
% is contained in the scheme theoretic union of $s(B)$ and $D$.
% Moreover, $\OO_X(-D)\cdot N_{C'/X}|_{s(B)}$ is obtained from
% $\OO_X(-D)\cdot N_{C/X}|_{s(B)}$ by the construction from \cite[Lemma
% 2.6]{GHS}.  Thus, by the argument from \cite[p. 62]{GHS}, if $C'$ is
% obtained from $C$ by attaching sufficiently many very free curves in
% fibers of $\pi$ to general points of $s(B)$ with general normal
% directions, then $h^1(C'', \OO_X(-D)\cdot N_{C'/X})$ equals $0$.
% Thus, by Proposition ~\ref{prop-inf}, the divisor restriction map is
% smooth at $[C']$.  
% \end{proof}

% % \begin{defn} \label{defn-jet}
% % \marpar{defn}
% % Let $F$ be an effective Cartier divisor in $B$.  A \emph{section
% % of $\pi$ over $F$} is a $B$-morphism $s_F:F\rightarrow X$.  Let $E$ be an
% % effective Cartier divisor in $B$ containing $F$ and let $D$ denote
% % $\pi^{-1}(E)$.  Denote by $E_F$ the open subscheme of $E$ whose
% % support equals the support of $F$, and denote $\pi^{-1}(E_F)$ by
% % $D_F$.  
% % A $\pse$ ideal sheaf $(\mc{F},u)$ of $D$
% % (over $k$) \emph{agrees with $s_F$} if its restriction to $D_F$ is the
% % ideal sheaf of the image $s_{E_F}(E_F)$ of a section over $E_F$ whose
% % restriction to $F$ equals $s_F$.  A $\pse$ ideal sheaf $(\mc{F},u)$ of
% % $D$ \emph{deforms to $\pse$ ideal sheaves agreeing with $s_F$} if
% % there exists a connected, normal $k$-curve $T$, a closed point $t$ of
% % $T$, and a family of $\pse$ ideal sheaves of $D$ over $T$ whose fiber
% % over $t$ equals $(\mc{F},u)$ and whose fiber over every other closed
% % point of $T$ agrees with $s_F$.  
% % \end{defn}

% \begin{cor} \label{cor-comblike}
% \marpar{cor-comblike}
% With the same hypotheses as in Proposition ~\ref{prop-comblike},
% let $F$ be a subdivisor of $E$, let $s_F$ be a section of $\pi$ over $F$,
% assume that the $\pse$ ideal sheaf $\iota_D([C])$ deforms to
% $\pse$ ideal sheaves agreeing with $s_F$.  Then there exists a section
% $s_\infty$ 
% of $\pi$ whose restriction to $F$ equals $s_F$.  
% \end{cor}

% \begin{proof}
% Since none of the very free curves attached to $C$ intersect $D$,
% $\iota_D([C])$ equals $\iota_D([C'])$.  Let $(T,t)$ be as in
% Definition [REFERENCE].  Form the pullback morphism
% $$
% \iota_{D,T}:T\times_{\psi{D/k},\iota_D} \text{Hilb}_{X/k} \rightarrow
% T.
% $$
% By Proposition ~\ref{prop-comblike}, this morphism is smooth at
% $(t,[C'])$.  By Lemma
% ~\ref{lem-comblike}, there is an open neighborhood $U$ of $(t,[C'])$
% precisely parametrizing pairs for which the closed subscheme is a
% comb-like curve.  The restriction of $\iota_D$ to this open subscheme
% is also smooth at $(t,[C'])$.  Thus it is dominant.  So there exists a
% $k$-point $(t_\infty,[C_\infty])$
% in this open subscheme such that $t_\infty \neq t$.  Therefore the
% $\pse$ ideal sheaf of $C_\infty$ agrees with $s_F$.  Let $s_\infty$ be
% the unique section of $\pi$ such that $s_\infty(B)$ is an irreducible
% component of $C_\infty$.  Then the ideal sheaf of $s_\infty(E_F)$ is a
% subsheaf of the $\pse$ ideal sheaf of $C_\infty$ over $E_F$.   But two
% ideal sheaves of sections can be contained one in the other if and
% only if they are equal.  Therefore the restriction of $s_\infty$ to
% $E_F$ is the given section over $E_F$.  In particular, the restriction
% of $s_\infty$ to $F$ equals $s_F$.1
% \end{proof}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%
% %% Section: Proofs of the main theorems
% %% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \section{Proofs of the main theorems} \label{sec-proofs}
% \marpar{sec-proofs}

% \begin{proof}[Proof of Theorem ~\ref{thm-main}]
% Let $E'$ be an effective divisor in $B$ containing $E$ and such that
% for the inverse image Cartier divisor $D=\pi^{-1}(E')$, $C$ is
% contained in the scheme-theoretic union $s(B) \cup D$.  The \'etale
% local deformation of $C$ to section curves agreeing with $s_E$ in
% particularly restricts on $D$ to a deformation of the $\pse$ ideal
% sheaf of $C$ to ideal sheaves of section curves agreeing with $s_E$.
% Thus the theorem follows from Corollary ~\ref{cor-comblike}.  
% \end{proof}

% \begin{proof}[Proof of Proposition ~\ref{prop-Hassett}]
% First we consider the case when the $\OO$-morphism
% $$
% f:\PP^1_K \rightarrow \SP \OO \times_B X
% $$
% does not necessarily extend to all of $\PP^1_\OO$.  

% \begin{claim} \label{claim-Hassett1}
% \marpar{claim-Hassett1}
% There exists an
% $E$-conic $\nu:P\rightarrow \PP^1\times_k B$ such that the composition
% $$
% f\circ \nu: \SP K \times_B P \rightarrow \PP^1_K \rightarrow \SP \OO
% \times_B X
% $$
% extends to an $\OO$-morphism
% $$
% f_P: \SP \OO\times_B P \rightarrow \SP \OO \times_B X.
% $$
% \end{claim}

% \mni
% To prove this, first form the closure $\Sigma$
% of the graph of $f$ in $\PP^1_\OO
% \times_B X$.  This gives a possibly singular, $2$-dimensional scheme
% whose projection onto $\PP^1_\OO$ is projective and birational.  By
% resolution of 
% singularities for $2$-dimensional schemes, e.g. as in \cite{Lipman},
% there exists 
% a projective desingularization of $\Sigma$.  The composition with
% projection onto $\PP^1_\OO$ is a projective, birational morphism, thus
% equals the blowing up of $\PP^1_\OO$
% along a coherent ideal sheaf $\mc{I}_\OO$
% supported on the closed fiber.  Since
% every such ideal sheaf $\mc{I}_\OO$ equals the basechange of a coherent ideal
% sheaf $\mc{I}$ on $\PP^1\times_k B$ supported on $\PP^1\times_k \{b\}$, the
% blowing up of $\PP^1_\OO$ along $\mc{I}_\OO$
% equals $\SP \OO \times_B P$ where $P$ is the
% $E$-conic obtained by blowing up $\PP^1\times_k B$ along $\mc{I}$.
% This proves Claim ~\ref{claim-Hassett1}.

% \begin{claim} \label{claim-Hassett2}
% \marpar{claim-Hassett2}
% Denote the graph of $f_P$ by
% $$
% \Gamma_{f_P}:\SP \OO \times_B P \rightarrow \SP \OO \times_B
% (X\times_B P).
% $$
% Each comb-like curve $C$ in $P$ which equals $s(B)$ over
% $B-\{b\}$ equals the isomorphic projection of a unique 
% comb-like curve $C_\Gamma$ in $X\times_B P$ 
% whose basechange to $\SP \OO \times_B
% (X\times_B P)$ equals $\Gamma_{f_P}(\SP \OO\times_B C)$.
% \end{claim}

% \mni
% There exists a nonnegative integer $r$ such that $C$ is contained in
% the scheme-theoretic union of $s(B)$ and $(r\cdot b)\times_B P$.
% Denote by $Z$ the scheme-theoretic union of $s(B)$ and $(r\cdot
% b)\times_B (X\times_B P)$.  The pullback of $\OO_Z$ to $\SP \OO
% \times_B (X\times_B P)$ equals the scheme-theoretic union of the germ
% of $s(B)$ and $(r\cdot b)\times_B (X\times_B P)$.  The ideal sheaf
% $\mc{I}$
% of $\Gamma_{f_P}(\SP \OO\times_B C)$ in $\SP \OO \times_B Z$ is a
% coherent subsheaf supported entirely on $(r\cdot b)\times_B (X\times_B
% P)$.  Thus there exists a unique coherent ideal sheaf
% $\mc{I}_{C_\Gamma}$ in $\OO_Z$ supported entirely on $(r\cdot
% b)\times_B (X\times_B P)$ whose pullback to $\SP \OO \times_B
% (X\times_B P)$ equals $\mc{I}$.  The corresponding closed subscheme
% $C_\Gamma$ of $Z$ is the unique closed subscheme whose basechange to
% $\SP \OO \times_B (X\times_B P)$ equals $\Gamma_{f_P}(\SP \OO\times_B
% C)$.  Moreover, because the projection of $\Gamma_{f_P}(\SP \OO
% \times_B C)$ into $\SP \OO\times_B P$ equals $\SP \OO \times_B C$,
% also the projection of $C_\Gamma$ into $P$ equals $C$.

% \mni
% Finally, since $\SP \OO\times_B X$ is smooth along $S$, also $\SP \OO
% \times_B (X\times_B P)$ is smooth along $\Gamma_{f_P}(\SP \OO \times_B
% P)$.  Thus $\Gamma_{f_P}$ is a regular immersion.  Since $\SP
% \OO\times_B C$ is a Cartier divisor in $\SP \OO \times_B P$, also
% $\Gamma_{f_P}(C)$ is regularly immersed in $\SP \OO \times_B P$.  Thus
% $C_\Gamma$ is regularly immersed \'etale locally, resp. formally
% locally, near $\{b\} \times_B (X\times_B P)$.  Now the property of
% being a regular immersion is a formal local property since the Koszul
% cohomology relative to the complete local ring of the images of a
% minimal set of generators of the stalk of the ideal equals the
% completion of the Koszul cohomology relative to the local ring of the
% minimal set of generators.  Thus $C_\Gamma$ is a regular immersion at
% every point of $C_\Gamma$ in $\{b\}\times_B (X\times_B P)$.  Since
% also
% $X\times_B P$ is smooth along $(s,0_P)$, $C_\Gamma$ is regularly
% immersed at every point of $C_\Gamma$, i.e., $C_\Gamma$ is a comb-like
% curve in $X\times_B P$.  This proves Claim ~\ref{claim-Hassett2}. 

% \begin{claim} \label{claim-Hassett3}
% \marpar{claim-Hassett3}
% If there exists a comb-like curve $C$ in $P$ with handle $0_P$ which
% \'etale locally deforms to section curves agreeing with the
% restriction $\infty_P|_E$, then there exists a comb-like curve $C$ in
% $X\times_B P$ with handle $(s,0_P)$ which \'etale locally deforms to
% section curves agreeing with an $E$-section $(s_E,t_E)$.
% \end{claim}

% \mni
% There is a unique $E$-section of $X\times_B P$ such that
% $\Gamma_{f_P}\circ \infty_P$ equals $(s_E,t_E)$.  
% By Claim ~\ref{claim-Hassett2}, there exists a comb-like curve
% $C_\Gamma$ in $X\times_B P$ with handle $(s,0_P)$ whose basechange to
% $\SP \OO \times_B (X\times_B P)$ equals $\Gamma_{f_P}(\SP \OO \times_B
% C)$.  If $\SP \OO \times_B C$ deforms to a family of section curves $D_t$ in
% $\SP \OO \times_B P$ agreeing with $\infty_P|_E$, then
% $\Gamma_{f_P}(\SP \OO \times_B C)$ deforms to the family of section
% curves $\Gamma_{f_P}(D_t)$ in $\SP \OO \times_B (X\times_B P)$
% agreeing with $\Gamma_{f_P}\circ \infty_P = (s_E,t_E)$.  This proves
% Claim ~\ref{claim-Hassett3}.

% \begin{claim} \label{claim-Hassett4}
% \marpar{claim-Hassett4}
% There exists a comb-like curve $C$ in $P$ with handle $0_P$ which
% Zariski locally deforms to section curves in $P$ agreeing with $\infty_P|_E$.  
% \end{claim}

% \mni
% On $P$
% consider the invertible sheaf $\mc{L} = \OO_P(\infty_P(B) - 0_P(B) -
% \pi^*E)$. Since the restriction to the generic fiber of $P\rightarrow
% B$ is a degree $0$ invertible sheaf on $\PP^1$, it has a
% $1$-dimensional space of global sections.  Thus the pushforward of
% $\mc{L}$ to $B$ is a torsion-free, coherent $\OO_B$-module of rank
% $1$, i.e., it is an invertible sheaf.  Thus there exists an effective
% divisor $\Delta$ in $B$, not intersecting $E$ such that
% $\OO_P(\infty_P(B) - 0_P(B) +\pi^* \Delta - \pi^* E)$ has a global section.
% In other words, there is an effective divisor $B$ in $P$, necessarily
% vertical, such that
% $$
% \infty_P(B) + \pi^* \Delta = 0_P(B) + \pi^* E + B.
% $$
% Define $C$ to be $0_P(B) + \pi^* E + B$ and let the curves $D_t$ be
% the members of the pencil spanned by $C$ and $\infty_P(B) +
% \pi^*\Delta$.  Since $\infty_P(B)+\pi^*\Delta$ is a section curve over
% $E$, all but finitely many members of this pencil are section curves
% over $E$.  Since the base locus of the pencil contains
% $\infty_P(B)\cap \pi^* E$, these section curves agree with
% $\infty_P(B)$ over $E$.  This proves Claim ~\ref{claim-Hassett4},
% completing the proof of the proposition.  

% \mni
% Of course if $f$ extends to a closed immersion
% $$
% f_\OO:\SP \OO\times_k \PP^1 \rightarrow \SP \OO \times_B X,
% $$
% then one can take $P$ to be $\PP^1\times_k B$ and one can work with
% $f_\OO$ instead of $\Gamma_{f_P}$.  In this way one produces a
% comb-like curve in $X$ which \'etale locally, resp. formally locally,
% deforms to section curves in $X$ agreeing with $s_E$.  
% \end{proof}

% \begin{proof}[Proof of Corollary ~\ref{cor-HT}]
% First of all, the results of the first paragraph follow from the
% results of the second, since every pair of points in a smooth,
% projective, separably rationally connected variety are connected by a
% very free rational curve, which may also be taken to be immersed if
% the dimension is $\geq 3$.  Next, if there is a very free rational
% curve in $\pi^{-1}(b)$
% containing $s(b)$ and $s_E(b)$, then deformations of the
% rational curve in fibers of $\pi$ and intersecting both $s$ and $s_E$
% are unobstructed.  Thus there exist both \'etale local and formal
% local morphisms
% $$
% f_\OO:\SP \OO \times_k \PP^1 \rightarrow \SP \OO \times_B X
% $$
% whose closed fiber is the given very free rational curve in
% $\pi^{-1}(b)$ and whose $0$-section, resp. $\infty$-section, is the
% basechange of $s$, resp. agrees with $s_E$.  If the original very free
% rational curve in $\pi^{-1}(b)$ is an immersion, then also $f_\OO$ is an
% immersion.  So the corollary follows from Theorem ~\ref{thm-main} and
% Proposition ~\ref{prop-Hassett}. 
% \end{proof}

% \begin{proof}[Proof of Corollary ~\ref{cor-rational}]
% Let $U$ be a dense Zariski open subset of $\PP^n_{\SP
%   \wht{K}_{B,b}}$ 
% which is isomorphic to a dense Zariski open subset of
% $\SP \wht{K}_{B,b} \times_B( X\times_k \PP^r_k)$.  Possibly after
% deforming $s$, we may assume the $\wht{K}_{B,b}$-point of $\SP
%   \wht{K}_{B,b}\times_B X$ determined by 
% $s$ is
% the image of a $\wht{K}_{B,b}$-point of $\SP \wht{K}_{B,b}
%   \times_B( X\times_k \PP^r_k)$ which is
% contained in $V$.
% Similarly, there exists a formal section of $\SP \whtOO_{B,b}
%   \times_B X$ agreeing with $s_E$ whose $\wht{K}_{B,b}$-point is
% the image of a
% $\wht{K}_{B,b}$-point of $V$.  The isomorphism of $U$ and $V$ sends
% these two $\wht{K}_{B,b}$-points of $V$ two $\wht{K}_{B,b}$
% points of $U$.  Every pair of $\wht{K}_{B,b}$-points of
%   $\PP^n_{\SP \wht{K}_{B,b}}$ are contained in a line which is
%   isomorphic to $\PP^1_{\SP \wht{K}_{B,b}}$.  The composition with
%   the isomorphism from $U$ to $V$ determines a $\wht{K}_{B,b}$-rational map
% $$
% f_K:\PP^1_{\SP \wht{K}_{B,b}} \dashrightarrow 
% \SP \wht{K}_{B,b} \times_B( X\times_k \PP^r_k)
% $$
% which extends to all of $\PP^1_{\SP \wht{K}_{B,b}}$ by the
% valuative criterion of properness.  Now apply Theorem ~\ref{thm-main}
% and Proposition ~\ref{prop-Hassett}.  
% \end{proof}

% \begin{proof}[Proof of Corollary ~\ref{cor-cubic}]
% Let $b$ be a point of $B$ for which $X_b$ is a cubic surface with
% rational double point singularities.  The Galois group of
% $\wht{K}_{B,b}$ is topologically cyclic.  The action on the
% N\'eron-Severi lattice of the geometric generic fiber
% $\SP \overline{\wht{K}}_{B,b} \times_B X$ sends a topological
% generator to an element in the automorphism group of the
% N\'eron-Severi lattice, namely $\mc{W}(E_6)$.  The conjugacy class of
% this element in $\mc{W}(E_6)$ is well-defined independent of the
% choice of topological generator.  We call it the \emph{monodromy class}.

% \mni
% The possibilities for the monodromy class
% are listed in
% \cite[Table 1, p. 176]{CubicForms}.  As explained in \cite[31.1(2),
% p. 174]{CubicForms}, the cubic surface $\SP \wht{K}_{B,b} \times_B
% X$ is minimal if and only if the monodromy class is in one of the
% first $5$ rows of the table, and the minimal model is a Del Pezzo of
% degree $\geq 5$ unless it is one of the first $11$ rows.  Since every
% smooth Del Pezzo of degree $\geq 5$ is rational, Corollary
% ~\ref{cor-rational} applies unless the monodromy class is in one of
% the first 11 rows.

% \begin{claim} \label{claim-cubic1}
% \marpar{claim-cubic1}
% If the cubic surface $X_b$ has only rational double points, and no
% greater than $3$ of these, then the monodromy class is not in one of
% the first $11$ rows.
% \end{claim}

% \mni
% The $8^\text{th}$ column of each row lists the orbit decomposition of
% the $27$ lines.  Each orbit specializes to a line in the singular fiber
% whose multiplicity is at least as large as the size of the orbit.
% More precisely, the multiplicity of each line in the singular fiber
% equals the sum of the
% lengths of the orbits specializing to that line.

% \mni
% Every line in the singular fiber not containing a singular point
% necessarily has multiplicity $1$ since the normal bundle of a line in
% the nonsingular locus of a cubic surface is isomorphic to 
% the nonspecial invertible
% sheaf $\OO_{\PP^1}(-1)$. 
% Thus it remains to consider the multiplicities
% of lines containing a rational double point.  For each double point,
% the
% length of the scheme
% of lines in the cubic surface containing that double point is always
% $6$.  The multiplicity of a line joining two or more 
% double points equals the sum of the contributions from each double
% point it contains.
% Thus, if
% there are $m$ rational double points, then the sum of the multiplicities of
% the lines containing at least one double point equals $6m$.  Since all
% other lines have multiplicity $1$, the
% number of orbits of size $1$ must be at least $27-6m$.  Since every
% orbit type in the first 11 rows begins with $1^e$ for $e\leq 3$, such
% a monodromy class may occur only if the cubic surface has $4$ double
% points.  This proves Claim ~\ref{claim-cubic1}.  Finally apply
% Corollary ~\ref{cor-rational} to complete the proof.
% \end{proof}







% %% \mni
% %% Typically weak approximation is defined for algebraic spaces defined
% %% over the function field $F=k(B)$.  Let $U_F$ be a smooth, separated,
% %% quasi-compact algebraic space over $F$.  A \emph{model} for $U_F$ is a
% %% pair $(\pi:\mcX\rightarrow B,U\subset \mcX)$ as above together
% %% with an isomorphism of $U\otimes_{\OO_B} F$ with $U_F$ as algebraic
% %% spaces over $F$.  

% %% \begin{defn} \label{defn-wa}
% %% \marpar{defn-wa}
% %% The algebraic space $U_F$ over $F$ satisfies \emph{weak approximation}
% %% if every model $(k,B,\mcX,U)$ for $U_F$ satisfies weak
% %% approximation.  The algebraic space $U_F$ over $F$ satisfies
% %% \emph{geometric weak approximation} if for every algebraically closed
% %% extension field $\kappa/k$, the algebraic space 
% %% $U_F \otimes_F (F\ast\kappa)$ over $F\ast \kappa$ satisfies weak
% %% approximation, where $F\ast \kappa$ is the compositum, i.e., $F\ast
% %% \kappa = \kappa(B)$.  
% %% \end{defn}

% %% \mni
% %% This is not precisely the same as definitions of weak approximation
% %% appearing elsewhere.  The following lemma explains the relationship
% %% between the different definitions.  It will be proved in [REFERENCE].

% %% \begin{lem} \label{lem-wa}
% %% \marpar{lem-wa}
% %% \begin{enumerate}
% %% \item[(i)]
% %% Let $\nu:\mcX'\rightarrow \mcX$ be a proper, birational morphism
% %% and define $U' = \nu^{-1}(U)$.  Assume that $U'$ is regular at every
% %% point in the exceptional set of $\nu$.  Then (geometric)
% %% weak approximation holds
% %% for $(k,B,\mcX,U)$, resp. (geometric) strong approximation holds for
% %% $(k,B,\mcX,U,S)$, if and only if it holds for $(k,B,\mcX',U')$,
% %% resp. for $(k,B,\mcX',U',S)$.  
% %% \item[(ii)]
% %% Let $U_F$ be a smooth, separated, quasi-compact algebraic space over
% %% $F$ and let $V_F$ be a dense open subset of $U_F$.  Then (geometric)
% %% weak
% %% approximation holds for $U_F$ if and only if it holds for $V_F$.
% %% Therefore weak approximation and geometric weak approximation are both 
% %% birational properties among smooth, separated, quasi-compact algebraic
% %% spaces over $F$.
% %% \item[(iii)]
% %% Let $\mcX$ be a proper, flat algebraic space over $B$.  
% %% Let $U$ be the
% %% maximal open subset of $\mcX$ which is smooth over $B$.
% %% If $\mcX$ is
% %% regular, then (geometric) weak approximation holds for $\mcX_F$
% %% if and only if (geometric) weak
% %% approximation holds for $(k,B,\mcX,U)$. 
% %% \end{enumerate}
% %% \end{lem}

% %% \mni
% %% An important question is which $F$-varieties satisfy weak
% %% approximation.  One direction is straightforward.
% %% Let $X_F$ be a nonempty, proper algebraic space over $F$ and let $U_F$
% %% be a dense open subset which is smooth over $F$.

% %% \begin{lem}[Hassett] \label{lem-converse}
% %% \marpar{lem-converse}
% %% If $U_F$ satisfies geometric weak approximation, then $X_F\otimes_F
% %% \ol{F}$ is integral and rationally chain connected.  In particular, if
% %% also $\text{char}(k)$ equals $0$, then $X_F\otimes_F \ol{F}$ is
% %% separably rationally connected.
% %% \end{lem}

% %% \mni
% %% So, at least in characteristic $0$, weak approximation implies
% %% $X_F\otimes_F \ol{F}$ is rationally connected.  Since resolution of
% %% singularities exists in characteristic $0$, we may also assume $X_F$
% %% is smooth.  This leads to the following question.

% %% \begin{ques}\cite{HT06} \label{ques-wa}
% %% \marpar{ques-was}
% %% If $X_F\otimes_F \ol{F}$ is integral, smooth, projective and separably
% %% rationally connected, does $X_F$ satisfy geometric weak approximation?
% %% \end{ques}

% %% \mni
% %% This question is open, but there are positive answer in many special
% %% cases
% .  

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