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\title
{Restriction of sections for families of Abelian varieties}

\author[Graber]{Tom Graber}
\address{Department of Mathematics \\
California Institute of Technology \\
Pasadena, CA 91125}
\email{graber@caltech.edu}

\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}
Given a family of Abelian varieties over a positive-dimensional base,
we prove that for a sufficiently general curve in the base, every
rational section of the family over the curve is contained in a unique
rational section over the entire base.  
\end{abstract}

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\maketitle

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\section{Main results} \label{sec-intro}
\marpar{sec-intro}

\mni
The starting point for this article is the following theorem.

\begin{thm}\cite[Theorem 6.2]{GHMS} \label{thm-GHMS}
\marpar{thm-GHMS}
Let $B$ be a smooth, quasi-projective, complex variety, let
$A\rightarrow B$ be an Abelian scheme over $B$ (i.e. a family of
Abelian varieties over $B$), and let $\pi:T\rightarrow B$ be a torsor
for $A\rightarrow B$.  Then $\pi$ is a trivial torsor if and only if
for every curve $C\subset B$, the restriction $T_C\rightarrow C$ is a
trivial torsor for $C\times_B A\rightarrow C$.  Stated more
succinctly, the following map of \'etale cohomology groups is
injective
$$
H^1_\text{\'et}(B,A) \rightarrow \prod_{C\subset B}
H^1_\text{\'et}(C, C\times_B A).
$$
\end{thm}

\mni
The main result of this article is an analogue for
$H^0_\text{\'et}(B,A)$, together with an extension to positive
characteristic.

\mni
Let $k$ be an algebraically closed field.  To simplify statements,
assume $k$ is uncountable.  We remind the reader that a subset of a
scheme is \emph{general}, resp. \emph{very general}, if it contains a
dense, open subset, resp. the intersection of a countable collection
of dense, open subsets.  A property of points in a scheme holds at a
\emph{general point}, resp. a \emph{very general point}, if the set of
points where it holds is general, resp. very general.

\begin{thm} \label{thm-main} 
\marpar{thm-main}
Let $B$ be a integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $A$ be an Abelian scheme over $B$.
\begin{enumerate}
\item[(i)]
  For every nontrivial $A$-torsor $T$ over $B$, for $C$ a very general
  triangle curve in $B$, resp. a very general cubic curve in $B$, the
  restriction $C\times_B T$ is a nontrivial $C\times_B A$-torsor over
  $C$.
\item[(ii)]
  For a very general triangle curve $C$ in $B$, the map
$$
H^0_\text{\'et}(B,A) \rightarrow H^0_\text{\'et}(C,C\times_B A)
$$
is a bijection.  If $\text{char}(k)=0$, this also holds for a very
general cubic curve $C$ in $B$.
\end{enumerate}
Both parts also hold with $C$ replaced by a very general planar
surface $\Pi$ in $B$.
\end{thm}

\mni
The terms ``triangle curve'', ``planar surface'' and ``cubic curve''
depend on the choice of a quasi-finite, generically unramified
morphism $f:B\rightarrow \PP^b_k$.

\begin{defn} \label{defn-tria}
\marpar{defn-tria}
An \emph{$f$-linear curve} is an integral, smooth curve $C$ in $B$ of
the form $B\times_{\PP^b_k} L$ for some line $L \subset \PP^b_k$.

\mni
An \emph{$f$-triangle map} is a pair $(C,h)$ of
\begin{enumerate}
\item[(i)]
  a reducible, nodal $k$-curve $C = C_1\cup C_2 \cup C_3$ whose
  irreducible components each intersect pairwise in a single node,
  $C_i\cap C_j = \{q_{i,j}\}$,
\item[(ii)]
  together with a $k$-morphism $h:C\rightarrow B$ whose restriction to
  every component, $h|_{C_i}:C_i \rightarrow B$, is a closed immersion
  to an $f$-linear curve in $B$.
\end{enumerate}

\mni
An \emph{$f$-triangle curve} is the image of an $f$-triangle map.
Equivalently, it is a union of three $f$-linear curves in $B$
intersecting pairwise in three distinct points (together with other,
possibly concurrent, intersections).  Stated a third way, it is a
curve in $B$ of the form $C= B\times_{\PP^b_k} T$ where $T$ is a
triangle of lines in $\PP^b_k$ (and satisfying the intersection
condition above).

\mni
An \emph{$f$-cubic curve} is an integral, smooth curve $C$ in $B$ of
the form $B\times_{\PP^b_k} E$ where $E$ is a smooth, plane, cubic
curve in $\PP^b_k$.

\mni
Finally, an \emph{$f$-planar surface} is an integral, normal surface
$S$ in $B$ of the form $B\times_{\PP^b_k} \Pi$ where $\Pi$ is a linear
$2$-plane in $\PP^b_k$.

\mni
When there is no likelihood of confusion, we drop the prefix $f$.
\end{defn}

\begin{rmk} \label{rmk-main}
\marpar{rmk-main}
There are several remarks.
\begin{enumerate}
\item[(i)]
  When $k=\CC$, Theorem \ref{thm-main} (i) is simply Theorem
  ~\ref{thm-GHMS}.  By a straightforward descent argument, part (i)
  follows from part (ii).  Thus, the new content of Theorem
  ~\ref{thm-main} is in part (ii).
\item[(ii)]
  The group $H^1_\text{\'et}(B,A)$ is a torsion group.  The subgroup
$$
H^1_\text{\'et}(B,A)' \subset H^1_\text{\'et}(B,A)
$$
of elements whose order is not divisible by $\text{char}(k)$ is
countable (it is the whole group if $\text{char}(k)=0$).  Therefore,
Theorem ~\ref{thm-main} (i) implies that for $C$ a very general
triangle curve, resp. a very general cubic curve, the restriction map
$$
H^1_\text{\'et}(B,A)' \rightarrow H^1_\text{\'et}(C,C\times_B A)'
$$
is injective.
\item[(iii)]
  There is nothing special about triangle curves, resp. cubic curves.
  The proof works for any type of curve which is at least as
  ``complex'' as triangle curves, resp. cubic curves.
\item[(iv)]  
  The proof of (ii) uses a partial compactification of the pair
  $(B,A)$ to a N\'eron model $(\widetilde{B},\widetilde{A})$.  A
  general triangle curve, resp. cubic curve, $C$ in $\widetilde{B}$ is
  projective.  Thus, using the Hilbert scheme for instance, the
  sections of $C\times_{\widetilde{B}} \widetilde{A}$ over $C$ are the
  $k$-points of a naturally defined group $k$-scheme $\Sigma_C$.  If
  $\text{char}(k)=0$, this is a reduced group scheme.  But if
  $\text{char}(k) > 0$, this group scheme is sometimes nonreduced, cf.
  \cite[Propostion 3]{MBpinceauxVIII}.  This is the reason for the
  $\text{char}(k)=0$ hypothesis for cubic curves in part (ii).  To
  extend our argument to positive characteristic, one would need to
  prove there exists a homomorphism of group schemes over $B$
$$
\Sigma_C \times_k \widetilde{B} \rightarrow \widetilde{A}
$$
splitting the restriction map.  We do not know whether such a
splitting always exists.
\end{enumerate}   
\end{rmk}

\mni
We also give some examples related to the theorem.

\begin{prop} \label{prop-neg}
\marpar{prop-neg}
\begin{enumerate}
\item[(i)]
  There exist $B$ and $A$ such that for every cubic curve $C$ in $B$
  the map
$$
H^1_\text{\'et}(B,A) \rightarrow H^1_\text{\'et}(C,A)
$$
is not surjective.  
\item[(ii)]
  If $\text{char}(k) = p$ is positive, there exist $B$ and $A$ such
  that for every triangle curve $C$, resp. cubic curve $C$, the map
$$
H^1_\text{\'et}(B,A) \rightarrow H^1_\text{\'et}(C,A)
$$
is not injective.  More precisely, there exists an $A$-torsor $T$ over
$B$ (depending on $C$) whose order equals $p$ and whose restriction
$C\times_B T$ is a trivial $C\times_B A$-torsor over $C$.
\item[(iii)]
  There exists $B$ and $A$ such that every dense open subset of the
  parameter space of cubic curves contains a cubic curve $C$ in $B$
  for which
$$
H^0_\text{\'et}(B,A) \rightarrow H^0_\text{\'et}(C,C\times_B A)
$$
is not surjective.
\end{enumerate}
\end{prop}

\mni
\textbf{Acknowledgments.}  We are very grateful to A. J. de Jong who
pointed out that a purely inseparable Bertini result as in Corollary
~\ref{cor-B} must exist, who recommended replacing an earlier argument
by an application of the N\'eron extension property, and who explained
to us Moret-Bailly's counterexamples from \cite{MBpinceauxVIII}.

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\section{A Bertini theorem} \label{sec-Bertini}
\marpar{sec-Bertini}

\mni 
One consequence of the classical Bertini theorem is that sections of a
finite, separable cover of a quasi-projective scheme $B$ of dimension
$\geq 2$ are detected by the restriction of the cover to a general
hyperplane section of $B$.  In this section we recall this and extend
the result to covers which may not be separable.

\begin{thm} \cite[Th\'eor\`eme 4.10, 6.10]{Jou} \label{thm-B}
\marpar{thm-B}
Let $B$ be an integral scheme and let $f:B\rightarrow \PP^N_k$ be a
finite type morphism.  If $f$ is generically unramified, then for a
general hyperplane $H$, $B\times_{\PP^N_k}H$ is geometrically reduced.
If $\text{dim}(f(B)) \geq 2$, then for a general hyperplane $H$,
$B\times_{\PP^N_k}H$ is geometrically irreducible.
\end{thm}

\mni
A straightforward consequence is the separable case of the following
result.  We also explain the inseparable case.

\begin{cor} \label{cor-B}
\marpar{cor-B}
Let $B$ be an integral scheme of dimension $\geq 2$ and let
$f:B\rightarrow \PP^N_k$ be a generically unramified, finite type
morphism.  
Let $g:X\rightarrow B$ be a generically finite morphism.
For a general hyperplane $H\subset \PP^N_k$, the restriction map from
the set of rational sections of $g$ over $B$ to the set of rational
sections of 
$$
g_H:X\times_{\PP^N_k} H
$$ 
over $B\times_{\PP^N_k} H$ is a bijection.
\end{cor}

\begin{proof}
  If $g$ has a rational section, its restriction to $B\times_{\PP^N_k}
  H$ is a rational section of $g_H$ for general $H$.  By Noetherian
  induction, it suffices to consider the case that $X$ is irreducible.
  If $g$ is not dominant, the result is clear.  If $g$ has a rational
  section, then $g$ is birational and again the result is clear.  Thus
  assume $g$ is dominant and has no rational section.  To prove the
  corollary, we must prove that also $g_H$ has no rational sections
  for $H$ a general hyperplane.

\mni
Because $g$ is dominant, there is an extension of fraction fields
$$
g^*:k(B) \rightarrow k(X).
$$
Because $g$ is generically finite, this is a finite field extension.
Because there is no rational section, it is a nontrivial field
extension, i.e., it has degree $d>1$.  Let $E/k(B)$ be a nontrivial
extension intermediate to $k(X)/k(B)$ such that there is no field
strictly intermediate between $E$ and $k(B)$.  Denote by $X_E$ the
integral closure of $B$ in $E$ and denote by $g_E:X_E\rightarrow B$
the induced morphism.  Since every rational section of $g_H$ maps to a
rational section of $(g_E)_H$, it suffices to prove $(g_E)_H$ has no
rational section for general $H$.  Thus, assume finally that
$k(X)/k(B)$ has no intermediate subfields.

\mni
\textbf{Case I. $k(X)/k(B)$ is separable.}  There are two cases
depending on whether or not $k(X)/k(B)$ is separable.  If $k(X)/k(B)$
is separable, then the composition $g\circ f:X\rightarrow \PP^N_k$ is
generically unramified.  By Theorem ~\ref{thm-B} applied to $f\circ
g$, for a general $H$, $X\times_{\PP^N_k}H$ is integral.  By generic
flatness, the morphism
$$
g_H: X\times_{\PP^N_k} H \rightarrow B\times_{\PP^N_k} H
$$
is generically finite and flat of degree $d$.  Thus
$k(B\times_{\PP^N_k} H) \rightarrow k(X\times_{\PP^N_k} H)$ is a
finite field extension of degree $d>1$.  Since the extension is not
degree $1$, it has no splitting.  Thus $g_H$ has no rational section.

\mni
\textbf{Case II. $k(X)/k(B)$ is not separable.}
Let $p$ denote the characteristic of $k$.  Because $k(X)/k(B)$ has no
intermediate subfields, for any $z\in k(X)\setminus k(B)$, $k(X) =
k(B)[z]$ and the minimal polynomial of $z$ is of the form $z^p-u$ for
some $u\in k(B)\setminus k(B)^p$.  Thus $k(X)=k(B)[z]/\langle z^p - u
\rangle$.  Replacing $B$ and $X$ by dense open subsets if necessary,
assume $B$ and $X$ are affine and normal, assume $f$ is unramified and
factors through $\AAA^N_k\subset \PP^N_k$, assume $u\in k[B]$, and
assume $k[X] = k[B][z]/\langle z^p-u \rangle.$

\mni
For general $H$, by Theorem ~\ref{thm-B}, $B\times_{\AAA^N_k} H$ is
integral and normal.  In this case $X\times_{\AAA^N_k} H$ is integral
and $g_H$ has no rational section precisely if the image of $u$
satisfies
$$
\overline{u} \not \in k[B\times_{\AAA^N_k} H]^p
$$
which is the same as the condition for $X\times_{\AAA^N_k} H$ to be
generically reduced.  If $X\times_{\AAA^N_k} H$ is generically
nonreduced, then for a general codimension $b-1$, linear subspace of
$\AAA^N_k$ contained in $H$, $\Lambda \subset H$, also
$X\times_{\AAA^N_k} \Lambda$ is nonreduced.  Inside the parameter
space for subvarieties $H$, resp. $\Lambda$, the set of $H$, resp.
$\Lambda$, such that $X\times_{\AAA^N_k} H$, resp. $X\times_{\AAA^N_k}
H$, is generically reduced is open, [REFERENCE, Jouanolou Theorem 4.10
approx.]  Thus, to prove $X\times_{\AAA^N_k} H$ is generically
nonreduced for all $H$ in a dense open subset, it suffices to find a
single $\Lambda$ such that $X\times_{\AAA^N_k} \Lambda$ is generically
nonreduced.  Equivalently, it suffices to find a single $\Lambda$ such
that for every minimal prime $\mf{q}$ of
$k[B]\otimes_{k[\AAA^N_k]}k[\Lambda]$, the image of $u$ in
$(k[B]\otimes_{k[\AAA^N_k]} k[\Lambda])_\mf{q}$ is not a $p^\text{th}$
power.  This is what we will actually prove.

\mni
Denote $b=\text{dim}(B)$.  For a general linear projection $p:\AAA^N_k
\rightarrow \AAA^b_k$, the induced morphism $p\circ f:B\rightarrow
\AAA^b_k$ is dominant and \'etale.  Because the inverse image in
$\AAA^N_k$ of a linear subspace of $\AAA^b_k$ of codimension $b-1$ is
again a linear subspace of codimension $b-1$, it suffices to prove the
result with $f$ replaced by $p\circ f$.  Thus, without loss of
generality, assume $N=b = \text{dim}(B)$.

\mni 
Denote by
$$
f^*:k[y_1,\dots,y_b] \rightarrow k[B]
$$
the pullback map on coordinate rings associated to $f$.  Extend this
to a morphism of $k[y_1,\dots,y_b]$-algebras,
$$
\psi:k[x,y_1,\dots,y_b] \rightarrow k[B]
$$
by $\psi(x) = u$.  Denote $k[A] = \text{Image}(\psi)$.  Because $A$ is
between $k[y_1,\dots,y_b]$ and $k[B]$
$$
b=\text{tr. deg}_k(k(y_1,\dots,y_b)) \leq \text{tr. deg}_k(k(A)) \leq
\text{tr. deg}_k(k(B)) = \text{dim}(B) = b.
$$
Thus $\text{dim}(k[A]) = \text{tr. deg}_k(k(A)) = b$.  So
$\text{Ker}(\psi)$ is a height $1$ prime ideal.  Because
$k[x,y_1,\dots,y_b]$ is a UFD, $\text{Ker}(\psi) = \langle g \rangle$
for a nonconstant, irreducible element $g\in k[x,y_1,\dots,y_b]$,
i.e.,
$$
k[A] = k[x,y_1,\dots,y_b]/\langle g \rangle.
$$

\mni
The $k$-algebra homomorphism 
$$
k[x]\rightarrow k[B],\ \ x \mapsto u
$$
determines a field extension $k(x) \rightarrow k(B)$.  Because $u$ is
not in $k(B)^p$, this is a separably generated field extension.  Since
$k(A)/k(x)$ is a subextension, it is also separably generated.  By the
Jacobian criterion, there exists an integer $i=1,\dots,b$ such that
the ideal $\langle g, \partial g/\partial y_i \rangle$ has height $2$.
Permuting indices if necessary, assume $i=1$.

\mni 
For a $(b-1)$-tuple of elements $\textbf{c}=(c_2,\dots,c_n)$ in $k$,
denote $k[\Lambda]=k[y_1]$ and denote by
$$
\chi_\textbf{c}:k[y_1,\dots,y_b] \rightarrow k[\Lambda]
$$
the unique $k[y_1]$-algebra homomorphism with $\chi_\textbf{c}(y_i) =
c_i$ for $i=2,\dots,b$.  The closed scheme of $\AAA^N_k$ associated to
$$
\text{Ker}(\chi_\textbf{c}) = 
\langle y_2-c_2,\dots,y_b-c_b\rangle
$$
is a codimension $b-1$ linear subspace.  For every
$k[y_1,\dots,y_b]$-algebra $R$, denote
$$
R_\Lambda = R\otimes_{k[y_1,\dots,y_b]} k[\Lambda].
$$

\mni
Because $\langle g, \partial g/\partial x_1\rangle \in
k[x,y_1,\dots,y_b]$ has height $2$, by standard results of dimension
theory (cf. \cite[Exercise II.3.22]{H}), for a general $\textbf{c}$,
the ideal
$$
\langle g(x,y_1,c_2,\dots,c_b), \partial g/\partial
x_1(x,y_1,c_2,\dots,c_b) \rangle \subset k[x,y_1]
$$
also has height $2$.  Then by the Jacobian criterion, the homomorphism
$$
k[x] \mapsto k[A]_\Lambda
$$
is generically smooth.  Because $k[A]\rightarrow k[B]$ is generically
\'etale, for general $\textbf{c}$ also
$$
k[A]_\Lambda \rightarrow k[B]_\Lambda
$$
is generically \'etale.  Therefore the composition
$$
k[x] \mapsto k[B]_\Lambda
$$
is generically smooth, i.e., for every minimal prime $\mf{q}$ of
$k[B]_\Lambda$, the field extension
$$
k(x) \rightarrow \kappa(\mf{q}):=(k[B]_\Lambda)_\mf{q}
$$
is separably generated.  This precisely means that the image
$$
\overline{u} \in (k[B]\times_{k[\AAA^n_k]} k[\Lambda])_\mf{q}
$$
is not a $p^\text{th}$ power for every minimal prime $\mf{q}$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: A triangle lemma
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A triangle lemma} \label{sec-triang}
\marpar{sec-triang}

\mni
Let $B$ be an integral, normal, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $f:B\hookrightarrow \PP^b_k$ be a
quasi-finite, generically unramified morphism.  Let $D\subset
\text{Grass}(1,\PP^b_k)$ be the dense, Zariski open subset of the
Grassmannian of lines in $\PP^b_k$ parametrizing lines $L$ for which
$B\times_{\PP^b_k} L$ is an integral, smooth curve.  This is the
parameter space for $f$-linear curves in $B$.

\mni 
Also denote by $C_D \subset B\times_k D$ the family of $f$-linear
curves in $B$ parametrized by $D$.  Denote by $\rho_D:C_D\rightarrow
D$ and $h_D:C_D \rightarrow D$ the obvious projections.

% Let $D \subset \text{Grass}(N-b+1,\PP^N)$ be the dense, Zariski open
% subset
% parametrizing codimension $b-1$
% linear subvarieties $\Lambda$ of $\PP^N$ such that
% $B\times_{\PP^N} \Lambda$ is smooth, one-dimensional and irreducible.  
% Let $\rho_D:C_D \rightarrow D$ denote the
% universal family of intersections $C = B \times_{\PP^N} \Lambda$ and let
% $\rho_D^{(2)}:C_D^{(2)} \rightarrow D$ denote the fiber product of $C_D$
% with itself over $D$.  Denote by $h_D:C_D \rightarrow B$ the obvious
% map.

% \begin{defn} \cite[Definition 3.5]{GHMS} \label{defn-tri}
% \marpar{defn-tri}
% A \textbf{triangle in} $D$, or more briefly a \textbf{triangle}, is a stable
% map $h:C \rightarrow X$ such that 
% \begin{enumerate}
% \item
% $C$ has three irreducible components
% $C_1$, $C_2$, and $C_3$,
% \item
% $C$ has three nodes $q_{1,2} = C_1 \cap C_2$, $q_{1,3} = C_1 \cap
% C_3$, and $q_{2,3} = C_2 \cap C_3$, and
% \item
% for each $i=1,2,3$ the map $h|_{C_i}:C_i \rightarrow X$ is a closed immersion
% whose image is parametrized by $D$.
% \end{enumerate}

% \mni
% The \textbf{space of triangles in} $D$, or simply the \textbf{space of
% triangles}, is the locally closed subvariety $T(D) \subset C_D^{(2)} \times
% C_D^{(2)} \times C_D^{(2)}$ 
% which is the subset of triples
% $$
% ( (C_1,q_{1,3},q_{1,2}),
% (C_2,q_{2,1},q_{2,3}), (C_3,q_{3,2},q_{3,1}) ) \in C_D^{(2)} \times
% C_D^{(2)} \times C_D^{(2)}
% $$ 
% such that $q_{i,j}=q_{j,i}$ for $1 \leq i < j \leq 3$, and such that
% $q_{i,j}
% \neq q_{i,k}$ for each triple of distinct integers $1\leq i,j,k \leq 3$.
% The universal family of triangles over $T(D)$ is the family
% of curves $C=C_1 \cup C_2 \cup C_3$
% obtained by identifying the points $q_{i,j}$ as above.  
% \end{defn}

% \begin{defn} \label{defn-tri2}
% Let $B$ be an integral, normal, quasi-projective $k$-scheme of
% dimension $b\geq 2$.  Let $f:B\rightarrow \PP^b_k$ be a generically
% unramified, quasi-finite $k$-morphism.  A \emph{deformation of an
%   $f$-triangle curve} is an integral, smooth curve $C$ of the form
% $E\times_{\PP^b_k} B$ for a smooth, plane cubic curve $E$ in
% $\PP^b_k$.
% \end{defn}

\mni
The following ``triangle lemma'' is essentially the same as in
~\cite{GHMS}.  We need to use the result in two slightly different
contexts.  Rather than repeat the proof twice, we find it convenient
to introduce the following terminology which encompasses both
contexts.

\begin{defn} \label{defn-sec}
\marpar{defn-sec}
Let $q:\SP \kappa \rightarrow B$ be a point.  A \emph{family of
  sections of $\pi$ over $f$-linear curves containing $q$} is a pair
$(\tau,\sigma)$ of a locally finite type morphism
$$
\tau:M\rightarrow h_D^{-1}(q) = C_D\times_B \SP \kappa
$$
and a morphism of $B$-schemes
$$
\sigma:M\times_{\rho_D\circ \tau,D,\rho_D} C_D \rightarrow X,
$$
where $M\times_D C_D$ is considered as a $B$-scheme via
$$
M\times_D C_D \xrightarrow{\text{pr}_{C_D}} C_D \xrightarrow{h_D} B
$$
(rather than via $h_D\circ \tau\circ \text{pr}_M$).  The family is
\emph{locally generically finite} if for every quasi-compact open $U$,
the restriction
$$
\tau|_U:U \rightarrow h_D^{-1}(q)
$$
is generically finite.  The family \emph{avoids rational curves over
  $q$} if for a very general curve $[C,q]$ in $h_D^{-1}(q)$, for every
$m$ in $\tau^{-1}([C,q])$, the point $\sigma_m(q) \in \pi^{-1}(q)$ is
contained in no rational curve in $\pi^{-1}(q)$.

\mni
Let $M$ be a family of sections of $\pi$ over $f$-linear curves
containing $q$ which avoids rational curves.  Let $C$ be a triangle
such that $q_{1,2}=q$.  A stable section of $X_C\rightarrow C$ is
\emph{in $M$ over $C_1$ and $C_2$} if for $i=1,2$ the associated
section of $X_{C_i}\rightarrow C_i$ is the section $s_m$ associated to
a point $m\in r^{-1}([C_i,q])$.
\end{defn}

\begin{lem} \cite[Lemma 3.7]{GHMS} \label{lem-tri}
\marpar{lem-tri}
Let $B$ be an integral, normal, quasi-projective $k$-scheme of
dimension $b \geq 2$.  Let $f:B\rightarrow \PP^b$ be a quasi-finite,
generically morphism.  Let $\pi:X\rightarrow B$ be a projective
morphism. Let $q\in B$ be a smooth point.  Let
$$
(\tau:M\rightarrow h_D^{-1}(q),\sigma:M\times_D C_D \rightarrow X)
$$
be a locally generically finite family of sections of $\pi$ over
$f$-linear curves containing $q$ which avoids rational curves over
$q$, cf. Definition \ref{defn-sec}.  Let $C$ be a very general
triangle curve in $B$ with $q_{1,2}=q$.
\begin{enumerate}
\item[(i)]
  Every stable section of $X_C\rightarrow C$ in $M$ over $C_1$ and
  $C_2$ contains a section of $X_C\rightarrow C$ in $M$ over $C_1$ and
  $C_2$.
\item[(ii)]
  For every section $s$ of $X_C\rightarrow C$ in $M$ over $C_1$ and
  $C_2$, there exists a unique irreducible closed subvariety $Z\subset
  X$ containing $s(C)$ such that $\pi|_Z:Z\rightarrow B$ is an
  isomorphism over $q$ and such that the restriction of $Z$ over a
  very general linear curve $L$ in $B$ containing $q$ is a section of
  $X_L \rightarrow L$ in $M$.
\end{enumerate}
\end{lem}

\begin{proof}
  We may replace $B$ by the integral closure $B^\text{new}$ of $\PP^b$
  in the fraction field of $B$, which contains $B$ as a dense, Zariski
  open subset.  And we may replace $X$ by some projective morphism
  $X^\text{new} \rightarrow B^\text{new}$ whose restriction over $B$
  equals $X$.  Thus, without loss of generality, assume that $B$ is
  projective.  Because $g$ is proper, every rational section of $X$
  over a normal curve in $B$ extends uniquely to a regular section (by
  the valuative criterion of properness).

\mni
The subset $\Omega$ of $X$ consisting of the images of all sections
parametrized by $M$ is a countable union of irreducible, locally
closed subvarieties $\Omega_i^\text{pre}$ of $X$, $i\in \ZZ$.  For
each $i$, denote by $\Omega_i$ the normalization of the closure in $X$
of $\Omega_i^\text{pre}$.  Observe that if $Z$ is a subvariety as in
(ii), then $Z$ is contained in $\Omega_i$ for some $i$.  Thus, to
prove the uniqueness in (ii), it suffices to consider subvarieties $Z$
contained in some $\Omega_i$.  And the next claim implies each such
$Z$ will equal $\Omega_i$.

\begin{claim} \label{claim-1}
\marpar{claim-1}
For every $i\in \ZZ$, the morphism
$$
\pi_{\Omega_i}:\Omega_i \rightarrow B
$$
is generically finite.  For a very general member ${C_3}$ of $D$, for
every $i$, $\Omega_i|_{C_3} \rightarrow {C_3}$ admits a section only
if $\Omega_i\rightarrow B$ is birational.  If $\Omega_i \rightarrow B$
is birational, then it is an isomorphism over $q$ and its restriction
to every very general linear curve $L$ in $B$ is a section of
$X_L\rightarrow L$ in $M$.
\end{claim}

\mni
First of all, $\Omega_i\rightarrow B$ is generically finite if and
only if $\Omega_i^\text{pre} \rightarrow B$ is generically finite.
Let $M_i$ be a quasi-compact locally closed subset of $M$ mapping to
$\Omega_i^\text{pre}$.  For every $q'\not\in f^{-1}(f(q))$, there is a
unique line $\Lambda$ in $\PP^b_k$ containing $f(q)$ and $f(q')$.
Thus, for $q'$ general, there is a unique $f$-linear curve $C$
containing $q$ and $q'$.  Because $\tau$ is locally generically
finite, $M_i \rightarrow h_D^{-1}(q)$ is generically finite.  Thus,
for $q'$ general, there are only finitely many points $m$ in $M_i$
mapping to $[C,q]$.  Thus there are only finitely many images $s_m(q')
\in \Omega_i^\text{pre}\cap \pi^{-1}(q')$.  Therefore $\Omega_i
\rightarrow B$ is generically finite.

\mni
If $\Omega_i$ does not dominate $B$, then a general $f$-linear curve
$C$ is not contained in the image of $\Omega_i$.  Thus
$\Omega_i|_{C_3} \rightarrow {C_3}$ certainly admits no section.  Thus
assume $\Omega_i$ dominates $B$.  By Theorem ~\ref{cor-B}, the
restriction of $\Omega_i$ over a general member $C$ of $D$ has a
section if and only if $\Omega_i$ has a rational section over $B$,
i.e., if and only if $\Omega_i \rightarrow B$ is birational.  Finally,
by \cite[Theorem VI.1.9.3]{K}, $\Omega_i \rightarrow B$ is an
isomorphism over $q$ since otherwise $\Omega_i\cap \pi^{-1}(q)$ is not
uniruled (in fact there are no rational curves in $\pi^{-1}(q)$
intersecting $\Omega_i^\text{pre}$).  This proves Claim~\ref{claim-1}.

\begin{claim} \label{claim-2}
\marpar{claim-2}
For a very general $f$-linear curve ${C_3}$, for every pair $r,s$ of
very general points of ${C_3}$, for every $i$ and $j$, there exists a
section of $X_{C_3}\rightarrow {C_3}$ intersecting both $\Omega_i\cap
\pi^{-1}(r)$ and $\Omega_j \cap \pi^{-1}(s)$ only if $i=j$ and
$\Omega_i \rightarrow B$ is birational.  In this case, the image of
the section is contained in $\Omega_i$.
\end{claim}

\mni
Because every $\Omega_i \rightarrow B$ is generically finite, for a
very general point $r$ of $B$ the subsets $\Omega_i \cap \pi^{-1}(r)$
are all finite and disjoint subsets of $\pi^{-1}(r)$.  Because $M$
avoids rational curves in $\pi^{-1}(q)$, for a very general $f$-linear
curve $C_1$ containing $q$, for a very general point $r$ of $C_1$,
every section in $M$ over $[C_1,q]$ maps $r$ to a point contained in
no rational curve in $\pi^{-1}(r)$.  In other words, $\Omega_i \cap
\pi^{-1}(r)$ intersects no rational curve in $\pi^{-1}(r)$.  Thus, by
the rigidity lemma, for every $i$, every section of
$X_{C_3}\rightarrow {C_3}$ intersecting $\Omega_i \cap \pi^{-1}(r)$ is
rigid, as a section intersecting $\Omega_i\cap \pi^{-1}(r)$.  Thus,
there are at most countably many of these rigid sections.

\mni
For every section and for every $j$, either the section is contained in
$\Omega_j$ or else the section intersects $\Omega_j$ in finitely many
closed points.  Since there are countably many indices $i$ and $j$,
there is a countable set $S$ of closed points of $X$ such that for
every section of $X_{C_3} \rightarrow {C_3}$ intersecting $\Omega_i
\cap \pi^{-1}(r)$ and not contained in $\Omega_j$, the intersection of
the section and $\Omega_j$ is contained in $S$.  The image $\pi(S)$ is
a countable set of closed points of ${C_3}$.  Choosing $s$ in the
complement of this set, every section of $X_{C_3}\rightarrow {C_3}$
intersecting $\Omega_i\cap \pi^{-1}(r)$ and $\Omega_j\cap\pi^{-1}(s)$
is contained in $\Omega_j$.  In this case, firstly, it intersects
$\Omega_j\cap \pi^{-1}(r)$.  Since $\Omega_i\cap \pi^{-1}(r)$ and
$\Omega_j\cap \pi^{-1}(r)$ are disjoint for $i\neq j$, it follows that
$i$ equals $j$.  Secondly, since the section of $X_{C_3} \rightarrow
{C_3}$ is actually a section of $\Omega_j = \Omega_i$ over $C_3$,
Claim ~\ref{claim-1} implies that $\Omega_i \rightarrow B$ is
birational.  Finally, the image of the section is contained in
$\Omega_j$, which equals $\Omega_i$.  This proves Claim
~\ref{claim-2}.

\mni  
Let $C_1$, resp. $C_2$, be a general $f$-linear curve containing both
$q$ and $r$, resp. both $q$ and $s$.  Form the $f$-triangle map
$C=C_0\cup C_1 \cup C_2$ with vertices $q$, $r$ and $s$.

\begin{claim} \label{claim-3}
\marpar{claim-3}
Every stable section of $X_C\rightarrow C$ in $M$ over $C_1$ and $C_2$
is regular over $q$, $r$ and $s$, thus contains a section of
$X_C\rightarrow C$ in $M$ over $C_1$ and $C_2$.  Every section of
$X_C\rightarrow C$ in $M$ over $C_1$ and $C_2$ is contained in an
irreducible closed subset $\Omega_i$ such that $\Omega_i \rightarrow
B$ is birational.
\end{claim}
 
\mni
Given a stable section of $X_C \rightarrow C$, remove all vertical,
rational components not contained in $\pi^{-1}(q)$, $\pi^{-1}(r)$ or
$\pi^{-1}(s)$.  The remaining components give a stable section of
$X_C\rightarrow C$ whose vertical, rational components, if any, occur
over $q$, $r$ and $s$.  By hypothesis, the images of the sections over
$C_1$, resp. $C_2$, intersect no rational curves in $\pi^{-1}(q)$, and
then, because $r$ and $s$ are very general, they also intersect no
rational curve in $\pi^{-1}(r)$, resp. in $\pi^{-1}(s)$.  Thus the
stable section has no vertical, rational components over $q$, $r$ and
$s$, i.e., the stable section is a section.  By hypothesis, the image
of this section over $C_1$, resp. over $C_2$, is contained in a subset
$\Omega_i$, resp. $\Omega_j$.  The restriction of this section to
${C_3}$ is a section of $X_{C_3} \rightarrow {C_3}$ intersecting both
$\Omega_i\cap \pi^{-1}(r)$ and $\Omega_j\cap \pi^{-1}(s)$.  By Claim
~\ref{claim-2}, $i=j$, $\Omega_i\rightarrow B$ is birational, and the
section over ${C_3}$ is contained in $\Omega_i$.  Thus the entire
section is contained in $\Omega_i$.  This proves Claim ~\ref{claim-3},
and thus the lemma.
\end{proof}

\mni
We will use Lemma~\ref{lem-tri} via the following two corollaries.

\begin{cor} \label{cor-tri1}
\marpar{cor-tri1}
Let $B$ be an integral, normal, quasi-projective $k$-scheme of
dimension $b \geq 2$.  Let $f:B\rightarrow \PP^b$ be a generically
unramified, quasi-finite morphism.  Let $\pi:X\rightarrow B$ be a
projective morphism.  Let $q\in B$ be a smooth point and let $r\in
\pi^{-1}(q)$ be a point contained in no rational curve in
$\pi^{-1}(q)$.  Let $C$ be a very general $f$-triangle curve in $B$
with $q_{1,2}=q$.
\begin{enumerate}
\item[(i)]
  Every stable section of $X_C\rightarrow C$ mapping $q_{1,2}$ to $r$
  contains a section of $X_C\rightarrow C$ mapping $q_{1,2}$ to $r$.
\item[(ii)]
  For every section of $X_C\rightarrow C$ mapping $q_{1,2}$ to $r$,
  there exists a unique irreducible closed subvariety $Z\subset X$
  containing the image of the section and such that
  $\pi|_Z:Z\rightarrow B$ is an isomorphism over $q$.
\end{enumerate}
\end{cor}

\begin{proof}
  Let $(\tau:M\rightarrow h_D^{-1}(q),\sigma)$ be the family of all
  sections over $f$-linear curves $C$ containing $q$ which map $q$ to
  $r$.  Since $r$ is contained in no rational curve in $\pi^{-1}(q)$,
  this family avoids rational curves over $q$.  And by the rigidity
  lemma, since there is no rational curve in $\pi^{-1}(q)$ containing
  $r$, every section over $C$ mapping $q$ to $r$ is rigid (as a
  section mapping $q$ to $r$).  Thus the family is locally generically
  finite.  So the family satisfies the hypotheses of Lemma
  ~\ref{lem-tri}, which gives the corollary.
\end{proof}

\begin{cor} \label{cor-tri2}
\marpar{cor-tri2}
Let $B$ be an integral, normal, quasi-projective $k$-scheme of
dimension $b \geq 2$.  Let $f:B\rightarrow \PP^b$ be a generically
unramified, quasi-finite morphism.  Let $\pi:X\rightarrow B$ be a
projective morphism.  Let $q\in B$ be a smooth point.  Assume that for
a very general $f$-linear curve ${C'}$ containing $q$, every section
of $X_{C'}\rightarrow {C'}$ is rigid and maps $q$ to a point contained
in no rational curve in $\pi^{-1}(q)$.  Let $C$ be a very general
triangle curve in $B$ with $q_{1,2}=q$.
\begin{enumerate}
\item[(i)]
  Every stable section of $X_C\rightarrow C$ contains a section of
  $X_C\rightarrow C$.
\item[(ii)]
  For every section of $X_C\rightarrow C$, there exists a unique
  irreducible closed subvariety $Z\subset X$ containing the image of
  the section and such that $\pi|_Z:Z\rightarrow B$ is birational.
\end{enumerate}
\end{cor}

\begin{proof}
  As in the proof of Lemma ~\ref{lem-tri}, we may reduce to the case
  where $B$ is projective.  Then, by the existence of the Chow
  variety, etc., there exists a universal family $(\tau_U:U\rightarrow
  h_D^{-1}(q),\sigma_U)$ of sections over $f$-linear curves containing
  $q$.  Moreover, $U$ has countably many connected components, each of
  which is quasi-projective.  Denote by $M$ the closed subvariety of
  $U$ which is the union of all irreducible components $U_i$ such that
  $\tau_U(U_i)$ is dense in $h_D^{-1}(q)$.  In particular, for a very
  general $D$-curve $C'$ containing $q$, every irreducible component
  of $M$ intersects $\tau^{-1}([C',q])$.  By hypothesis, every section
  of $X_{C'}\rightarrow {C'}$ is rigid and avoids rational curves in
  $\pi^{-1}(q)$.  Therefore the family
$$
(\tau_U|_M:M\rightarrow h_D^{-1}(q),\sigma_U|_M)
$$
is locally generically finite and avoids rational curves over $q$.  So
the family satisfies the hypotheses of Lemma ~\ref{lem-tri}.  By
definition of $M$, for a very general triangle $C$ with $q_{1,2}=q$,
since both $C_1$ and $C_2$ are very general $f$-linear curves
containing $q$, every section of $X_{C_1}\rightarrow C_1$, resp. of
$X_{C_2}\rightarrow C_2$, is in $M$.  Thus Lemma ~\ref{lem-tri} gives
the corollary.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Elementary reductions
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Elementary reductions} \label{sec-red}
\marpar{sec-red}

\mni
In this section, we prove some elementary lemmas useful in the proof
of Theorem ~\ref{thm-main}.  The next definition simplifies the
statements of the lemmas.  As always, let $k$ be an algebraically
closed field.

\begin{defn} \label{defn-P}
\marpar{defn-P}
Let $B$ be an integral, smooth, quasi-projective scheme over an
algebraically closed field $k$.  Let $f:B\rightarrow \PP^b_k$ be a
quasi-finite, generically unramified morphism.  Let $A$ be an Abelian
scheme over $B$.  We say \emph{property $\mc{P}(B,f,A)$ holds} for
triangle curves, resp. for cubic curves, if
$$
H^0_\text{\'et}(B,A) \rightarrow H^0_\text{\'et}(C,
C\times_B A)
$$
is a bijection for a very general $f$-triangle curve, resp. for a very
general $f$-cubic curve.
\end{defn}

\begin{lem} \label{lem-red0}
\marpar{lem-red0}
For every dense, open immersion $U\rightarrow B$,
$\mc{P}(U,f|_U,U\times_B A)$ holds if and only if $\mc{P}(B,f,A)$
holds.
\end{lem}

\begin{proof}
\textbf{(i)}
By Weil's extension theorem, cf. \cite[Theorem 1, p.109]{Ner} and
\cite[Theorem VI.1.9.3]{K}, sections of $U\times_B A$ over $U$ extend
uniquely to sections of $A$ over $B$.  Assuming $C$ is smooth at every
point of $B\setminus U$, also sections of $U\times_B C\times_B A$ over
$U\times_B C$ extend uniquely to sections of $C\times_B A$ over $C$.
\end{proof}

\begin{lem} \label{lem-red1}
\marpar{lem-red1}
Let $B'$ be an integral scheme and let $h:B'\rightarrow B$ be a
finite, \'etale, Galois morphism with Galois group $G$.  Denote
$f'=f\circ h$ and $A'=A\times_B B'$.  If $\mc{P}(B',f',A')$ holds,
then $\mc{P}(B,f,A)$ holds.
\end{lem}

\begin{proof}
  Let $C$ be a very general triangle curve in $B$, resp. cubic curve
  in $B$.  The restriction map
$$
H^0_\text{\'et}(B',A') \rightarrow H^0_\text{\'et}(B'\times_B C,
(B'\times_B C)\times_{B'} A')
$$
is a homomorphism of Galois modules.  By \'etale descent, the induced
map of Galois invariants is canonically isomorphic to the restriction
map
$$
H^0_\text{\'et}(B,A) \rightarrow H^0_\text{\'et}(C,C\times_B A).
$$
Thus if the first map is an isomorphism, also the second map is an
isomorphism.
\end{proof}

\mni
The next reduction uses part of the theory of Chow's $K/k$-trace.  We
review the part that we need.

\begin{defn} \label{defn-Chow}
\marpar{defn-Chow}
Let $B$ be an integral, smooth, quasi-projective scheme over an
algebraically closed field $k$.  Let $A$ be an Abelian scheme over
$B$.  A \emph{Chow $B/k$-trace of $A$} is an initial pair
$(\text{Tr}_{B/k}(A),u)$ of an Abelian $k$-scheme $\text{Tr}_{B/k}(A)$
and a morphism of Abelian schemes over $B$,
$$
u:B\times_k \text{Tr}_{B/k}(A) \rightarrow A,
$$
i.e., for every pair $(A_0,v)$ of an Abelian $k$-scheme $A_0$ and a
morphism of Abelian schemes over $B$,
$$
v:B\times_k A_0 \rightarrow A,
$$
there exists a unique morphism of Abelian $k$-schemes
$$
w:A_0 \rightarrow \text{Tr}_{B/k}(A)
$$
such that $v=u\circ (\text{Id}_B\times w)$.   
\end{defn}

\mni
The basic result concerning the Chow trace is the following.

\begin{thm} \cite[\S VIII.3]{LangAV}, \cite{ConradKk} \label{thm-Chow}
\marpar{thm-Chow}
\begin{enumerate}
\item[(i)]
  For every integral, smooth, quasi-projective $k$-scheme $B$ and
  every Abelian scheme $A$ over $B$, there exists a Chow $B/k$-trace
  of $A$.
\item[(ii)]
  Let $E$ be an integral, smooth, quasi-projective $k$-scheme and let
  $E\rightarrow B$ be a dominant $k$-morphism such that $k(B)$ is
  separably closed in $k(E)$.  The induced morphism of Abelian
  $k$-schemes
$$
w:\text{Tr}_{B/k}(A) \rightarrow \text{Tr}_{E/k}(E\times_B A)
$$
is an isomorphism. 
\end{enumerate}
\end{thm}

\begin{rmk} \label{rmk-Chow}
\marpar{rmk-Chow}
A dense open immersion $U\rightarrow B$ satisfies the condition in
(ii).  Taking the limit over all dense open subsets of $B$, the Chow
trace depends only on the field extension $k(B)/k$ and the Abelian
$k(B)$-scheme $A\otimes_{\OO_B} k(B)$.  Usually the Chow trace
is formulated for pairs $(K/k,A_K)$ of a field extension $K/k$ and an
Abelian $K$-scheme.  It is more useful for us to formulate it as above.
\end{rmk}

\mni 
The Chow trace is closely related to the property of isotriviality of
an Abelian scheme.

\begin{defn} \label{defn-noniso}
\marpar{defn-noniso}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme.  An
Abelian scheme $A$ over $B$ is \emph{non-isotrivial}, resp.
\emph{strongly non-isotrivial}, if for the geometric generic point of
$B\times_k B$,
$$
(p,q):\SP \kappa \rightarrow B\times_k B
$$
the Abelian $\kappa$-schemes
$$
A_p := \SP \kappa \times_{p,B} A, \ \ A_q := \SP \kappa \times_{q,B} A
$$
are not isomorphic, resp. there is no nonzero morphism of Abelian
$\kappa$-schemes
$$
A_p \rightarrow A_q.
$$
\end{defn}

\begin{lem} \label{lem-Chow2}
\marpar{lem-Chow2}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme.  An
Abelian scheme $A$ over $B$ is strongly non-isotrivial if and only if
$\text{Tr}_{k(B)^\text{sep}/k}(A\otimes_{\OO_B}k(B)^\text{sep})$ is
zero.
\end{lem}

\begin{proof}
  First of all, dual to the morphism of Abelian varieties
$$
u:\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})\otimes_k
k(B)^\text{sep} \rightarrow A_{k(B)^\text{sep}},
$$
there is a morphism
$$
v:
A_{k(B)^\text{sep}} \rightarrow
\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})\otimes_k
k(B)^\text{sep}
$$
such that $v\circ u$ is multiplication by some positive integer $N$.
Thus, if $\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})$ is
nonzero, the composition
$$
A_p \xrightarrow{p^*v}
\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})\otimes_k \kappa
\xrightarrow{q^*u} A_q
$$
is a nonzero homomorphism.  

\mni 
Denote by $e_p,e_q:k(B)^\text{sep} \rightarrow \kappa$ the field
monomorphism associated to projection $p$, resp. $q$.  Then,
conversely, every nonzero homomorphism
$$
A_p \rightarrow A_q
$$
factors through
$$
\text{Tr}_{e_p}(A_q)\otimes_{k(B)^\text{sep},e_p}
\kappa \rightarrow A_q.
$$
By Theorem ~\ref{thm-Chow}, this second map is precisely
$$
e_q^*u:\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})\otimes_k
\kappa \rightarrow A_q.
$$
So if there is a nonzero homomorphism $A_p \rightarrow A_q$ (or
symmetrically $A_q\rightarrow A_p$), then
$\text{Tr}_{k(B)^\text{sep}/k}(A_{k(B)^\text{sep}})$ is nonzero.
\end{proof}

\begin{lem} \label{lem-rigid}
\marpar{lem-rigid}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 1$.  Let $Q$ be an Abelian scheme over $B$.  If
$\text{Tr}_{B/k}(Q) = 0$, then there are at most countably many
sections of $Q$ over $B$.
\end{lem}

\begin{proof}
  Let $\overline{B}$ be a normal, projective scheme containing $B$ as
  a dense, open subscheme.  Let $\overline{Q}\rightarrow \overline{B}$
  be a projective morphism whose restriction over $B$ equals $Q$.
  There is a Chow variety parametrizing cycles in $\overline{Q}$.  The
  Chow variety has countably many irreducible components, for the
  usual reason (countably many Hilbert polynomials, etc.).  The claim
  is that every cycle $Z_0\subset \overline{Q}$ for which
  $Z\rightarrow B$ is birational gives an isolated point of the Chow
  variety.

\mni
Towards this claim, let $T$ be an irreducible, quasi-projective curve
and let $Z\subset \overline{Q}\times_k T$ be a cycle such that
$Z\rightarrow \overline{B}\times_k T$ is birational.  Replacing $T$ by
a dense open subset if necessary, assume $T$ is smooth.  Then $Z\cap
B\times_k T$ is the graph of a $B$-rational transformation,
$$
F:B\times_k T \dashrightarrow Q.
$$
By \cite[Theorem VI.1.9.3]{K}, this rational transformation is
regular.  Fix a point $t_0\in T$ and denote
$$
G:B\times_k T \rightarrow Q, \ \ F^\text{new}(q,t) = F(q,t)
- F(q,t_0).
$$
Denote by
$$
e:T\rightarrow \text{Alb}(T)
$$
the Albanese morphism sending $t_0$ to $0$.  Then, by the universal
property of the Albanese, the morphism $G$ factors through
$$
\text{Id}_B\times e:B\times_k T \rightarrow B\times_k \text{Alb}(T).
$$
Because $\text{Tr}_{B/k}(Q)$ is trivial, by the universal property of
the trace, the induced homomorphism of Abelian schemes over $B$,
$$
\widetilde{G}:B\times_k \text{Alb}(T) \rightarrow Q
$$
is the zero homomorphism.  Thus $G$ is the zero map, i.e.,
$$
F(q,t) = F(q,t_0)
$$
for every $(q,t)\in B\times_k T$.  Thus $Z\cap (B\times_k T)$ is
independent of $t\in T$.  Since $Z$ is the closure of $Z\cap
(B\times_k T)$, the same holds for $Z$, i.e., $Z = Z_0\times_k T$ for
a cycle $Z_0\subset \overline{Q}$.  Therefore every $k$-morphism from
an irreducible, quasi-projective curve $T$ to the Chow variety
parametrizing rational transformations $Z$ is constant.  In other
words, every cycle $Z\subset \overline{Q}$ with $Z\rightarrow
\overline{B}$ birational gives an isolated point of the Chow variety.
\end{proof}

\begin{lem} \label{lem-trick} 
\marpar{lem-trick}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme and let
$A$ be an Abelian scheme over $B$.  Let $A_0$ be an Abelian
$k$-variety and let $v_0:A\rightarrow B\times_k A_0$ be a surjective
homomorphism of Abelian schemes.  Let $C$ be a curve in $B$ and let
$q$ be a closed point in $C$.  To prove that every section $s$ of
$C\times_B A$ over $C$ is the restriction of a unique section $S$ of
$A$ over $B$, it is necessary and sufficient to prove this for
sections $s$ with $v_0(s(q)) = (0,q)$.
\end{lem}

\begin{proof}
  Necessity is obvious.  To prove sufficiency, assume the result holds
  for sections with $v_0(s(q))=(0,q)$.  By existence of polarizations,
  dual homomorphisms of Abelian schemes, etc., there exists a
  homomorphism of Abelian schemes
$$
u_0:B\times_k A_0 \rightarrow A
$$
such that $v_0\circ u_0$ is multiplication by $N$ for some positive
integer $N$.  Let $s$ be a section of $C\times_B A$, not necessarily
satisfying the condition.  Let $a_0\in A_0$ be the unique closed point
with $v_0(s(q))=(a_0,q)$.  There exists an element $a_1\in A_0$ such
that $a_0 = N\cdot a_1$.  Denote by $\tilde{a_1}:B\rightarrow
B\times_k A_0$ the section whose projection to $A_0$ is the constant
morphism with value $a_1$.  Then $s^\text{new} = s - u_0\circ
\tilde{a_1}|_C$ is a section of $C\times_B A$ over $C$ such that
$v_0(s^\text{new}(q)) = (0,q)$.  By hypothesis, there exists a unique
section $S^\text{new}$ of $A$ over $B$ whose restriction to $C$ equals
$s^\text{new}$.  Then $S = S^\text{new} + u_0\circ \tilde{a_1}$ is the
unique section of $A$ over $B$ whose restriction to $C$ is $s$.
\end{proof}

\begin{cor} \label{cor-Chow2}
\marpar{cor-Chow2}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme and let
$A$ be an Abelian scheme over $B$.  There exists a dense open
subscheme $U\subset B$, a finite, \'etale, Galois morphism
$U'\rightarrow U$, an Abelian $k$-scheme $A_0$, an Abelian scheme $Q$
over $U'$ and an isogeny of Abelian schemes over $U'$,
$$
u = u_0\oplus u_Q:(U'\times_k A_0)\times_{U'} Q \rightarrow U'\times_B A,
$$
with the following properties.
\begin{enumerate}
\item[(i)]
The pair $(A_0,u_0)$ is a Chow $U'/k$-trace of $U'\times_B A$.  
\item[(ii)]
  For every dense open subset $V$ of $U$ and every finite, \'etale,
  Galois morphism $V'\rightarrow V\times_U U'$, the induced map of
  Abelian $k$-schemes
$$
A_0 = \text{Tr}_{U'/k}(U'\times_B A) \rightarrow \text{Tr}_{V'/k}(V'\times_B
A)
$$
is an isomorphism.
\item[(iii)]
The Abelian scheme $Q$ is strongly non-isotrivial.  
\item[(iv)]
  The quotient of $U'\times_B A$ by $U'\times_k A_0$ is isomorphic to
  $Q$ in such a way that the composition
$$
Q \xrightarrow{u_Q} U'\times_B A \xrightarrow{\text{quotient}} Q
$$
is the multiplication by $N$ isogeny for some positive integer $N$.
\end{enumerate}
\end{cor}

\begin{lem} \label{lem-Chow3}
\marpar{lem-Chow3}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $f:B\rightarrow \PP^b_k$ be a quasi-finite,
generically unramified morphism.  Let $A_0$ be an Abelian $k$-scheme,
let $A$ and $Q$ be Abelian schemes over $B$, and let
$$
u=u_0\oplus u_Q:(B\times_k A_0)\times_B Q \rightarrow A
$$
be an isogeny of Abelian schemes over $B$.  Assuming
$\text{Tr}_{B/k}(Q)$ is zero, $\mc{P}(B,f,A)$ holds for triangle
curves, resp. cubic curves, if and only if both $\mc{P}(B,f,B\times_k
A_0)$ and $\mc{P}(B,f,Q)$ hold for triangle curves, resp. cubic
curves.
\end{lem}

\begin{proof}
Denote
$$
A' := (B\times_k A_0) \times_B Q.
$$
Then $\mc{P}(B,f,A')$ holds if and only if both $\mc{P}(B,f,B\times_k
A_0)$ and $\mc{P}(B,f,Q)$ hold.  There exists an isogeny
$$
v = (v_0,v_Q):A \rightarrow A' = (B\times_k A_0) \times_B Q
$$
such that both $u\circ v$ and $v\circ u$ are multiplication by a
positive integer $N$.

\mni
Let $q$ be a very general $k$-point of $B$.  Let $C$ be a very general
triangle curve, resp. cubic curve, containing $q$.  Property
$\mc{P}(B,f,A)$ says that every section $s$ of $C\times_B A$ over $C$
is the restriction of a unique section $S$ of $A$ over $B$.  Let $S$,
resp. $s$, be a section of $A$ over $B$, resp. of $C\times_B A$ over
$C$.  By Lemma ~\ref{lem-trick}, it suffices to prove this for
sections $s$ with $v_0(s(q)) = (0,q)$.

\mni
Consider the collection of all sections $S'$ of $A'$ over $B$ with
$\text{pr}_{A_0}(S'(q)) = 0$.  This is the same as the collection of
pairs $(S'_0,S'_Q)$ of a section $S'_Q$ of $Q$ over $B$ and a morphism
$S'_0:B\rightarrow A_0$ such that $S'_0(q) = 0$.  By the rigidity
lemma, every morphism $S'_0$ as above is rigid, and thus there are
only countably many such morphisms.  By Lemma ~\ref{lem-rigid}, also
there are only countably many sections $S'_Q$ of $Q$ over $B$.
Altogether, there are only countably many sections $S'$ of $A'$ over
$B$.

\mni
For each section $S'$ of $A'$ over $B$, consider the scheme
$$
X = B \times_{S',A',v} A.
$$
The projection $X\rightarrow A$ is a finite, flat morphism; in fact a
torsor for $\text{Ker}(v)$.  By Corollary ~\ref{cor-B}, for $C$ a
general curve, the restriction map from the set of sections of $X$
over $B$ to the set of sections of $C\times_B X$ over $C$ is a
bijection.  To spell this out a bit, inductively applying Corollary
~\ref{cor-B} implies the result for a general $f$-linear curve $C_3$
in $B$. Then the result also holds for a triangle curve with
$q_{1,2}=q$ whose component $C_3$ is a general $f$-linear curve as
above.  Finally the result follows for cubic curves since cubic curves
specialize to triangle curves.

\mni
Said differently, the conclusion is that the restriction map from the
$$
\{S:B\rightarrow A| S \text{ a section, } v\circ S = S'\} \rightarrow
\{s:C\rightarrow C\times_B A| s \text{ a section, } v\circ s = S'|_C\}
$$
is a bijection if $C$ is general.
By the argument above, there
are at most countably many sections $S'$ with $\text{pr}_{A_0}(S'(q))
= 0$.  Therefore, for $C$ very general, for every section $S'$ with
$\text{pr}_{A_0}(S'(q))=0$, the restriction map is a bijection.

\mni
Let $s$ be a section of $C\times_B A$ over $C$ with $v_0(s(q)) = 0$.
Assuming $\mc{P}(B,f,A')$, there exists a unique section $S'$ of $A'$
over $B$ such that $S'|_C = v\circ s$.  In particular, for every
section $S$ of $A$ over $B$ with $S|_C =s$ (assuming such exist), the
composition $v\circ S$ must equal $S'$.  Since $v_0(s(q)) = 0$, also
$\text{pr}_{A_0}(S'(q)) = 0$.  By the last paragraph there exists a
unique section $S$ of $A$ over $B$ such that $S|_C = s$.  Thus
$\mc{P}(B,f,A)$ follows from $\mc{P}(B,f,A')$.

\mni
The argument in the reverse direction is similar.  Again by Lemma
~\ref{lem-trick}, to prove $\mc{P}(B,f,A')$, it suffices to prove
every section $s'$ of $C\times_B A'$ over $C$ such that
$\text{pr}_{A_0}(s'(q)) = 0$ is the restriction of a unique section
$S'$ of $A'$ over $B$.  Assuming $\mc{P}(B,f,A)$, there exists a
unique section $S$ of $A$ over $B$ such that $S|_C = u\circ s'$.  In
particular, $v_0(S(q)) = 0$.  By the argument above, there are
countably many sections $S$ of $A$ over $B$ with $v_0(S(q)) = 0$.
Thus, assuming $C$ is very general, the argument above proves the
restriction map from sections $S'$ of $A'$ over $B$ for which $u\circ
S'=S$ to sections $s'$ of $C\times_B A'$ over $C$ for which $u\circ s'
= S|_C$ is a bijection.  Therefore there exists a unique sections $S'$
such that $S'|_C = s'$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Proof of the theorem
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of the theorem} \label{sec-proof}
\marpar{sec-proof}

\mni
We are ready to prove the theorem.  We begin by proving the theorem in
the special cases of a constant family of Abelian varieties and of a
strongly non-isotrivial family of Abelian varieties.

\begin{lem} \label{lem-const}
\marpar{lem-const}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $A_0$ be an Abelian $k$-variety.  For a very
general triangle curve $C$ in $B$, the map
$$
H^0_\text{\'et}(B,B\times_k A_0) \rightarrow
H^0_\text{\'et}(C,C\times_k A_0)
$$
is a bijection. 
\end{lem}

\begin{proof}
  Let $q$ be a very general $k$-point of $B$.  Let $C$ be a very
  general triangle curve in $B$ such that $q_{1,2}=q$.  Set $r=(q,0)$
  in $B\times_k A_0$.  By Corollary ~\ref{cor-tri1}, for every section
  $s$ of $C\times_k A_0$ over $C$ with $s(q)=r$, there exists a unique
  section $S$ of $B\times_k A_0$ over $B$ with $S|_C = s$.  By Lemma
  ~\ref{lem-trick}, this holds for all sections $s$, whether or not
  $s(q)=r$.
\end{proof}

\begin{lem} \label{lem-const2}
\marpar{lem-const2}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $A_0$ be an Abelian $k$-variety.  For a very
general cubic curve $C$ in $B$, the map
$$
H^0_\text{\'et}(B,B\times_k A_0) \rightarrow
H^0_\text{\'et}(C,C\times_k A_0)
$$
is a bijection. 
\end{lem}

\begin{proof}
  Cubic curves are deformations of triangle curves.  The proof of the
  result for cubic curves uses Lemma \ref{lem-const} and a
  deformation-specialization argument.  There are two parts of the
  argument.  First, sections over cubic curves specialize to sections
  over triangle curves.  Second, sections over triangle curves deform
  uniquely.

\mni
Let $\overline{B}$ be the integral closure of $\PP^b_k$ in the
fraction field of $B$.  Thus $\overline{B}$ is an integral, normal,
projective $k$-scheme containing $B$ as a dense, open subscheme.
Replace $B$ by the smooth locus of $\overline{B}$.  In particular,
$\overline{B}\setminus B$ has codimension $\geq 2$ in $\overline{B}$.
Therefore every general triangle curve in $\overline{B}$, resp. every
general cubic curve in $\overline{B}$, is actually contained in $B$.

\mni
Let $q$ be a very general $k$-point of $B$.  Let $r$ be the point
$(q,0)\in B\times_k A_0$.  Let $C$ be a very general triangle curve in
$B$ containing $q$.  Let $(T,t_0)$ be an irreducible pointed variety
and let $\mc{C}\subset T\times B$ be a flat family of curves in $B$
containing $T\times\{q\}$ and with $\mc{C}_{t_0} = C$.  By the
valuative criterion for stable maps, stable sections $s_t$ of
$\mc{C}_t\times_k A_0$ over $\mc{C}_t$ containing $r$ specialize to
stable sections $s$ of $C\times_k A_0$ over $C$ containing $r$ as $t$
specializes to $t_0$.  Since there are no rational curves in $A_0$,
stable sections are always sections.  This observation is an
elementary special case of \cite[Theorem VI.1.9.3]{K}.

\mni
Next we claim that $s_t$ specializes \emph{uniquely} to a section over
$C$.  This is the same as the claim that the relative Hom scheme over
$T$, $\text{Hom}_T(\mc{C},\mc{C}\times_k A_0,q\mapsto r)$ is
unramified over $T$, cf. \cite[Definition II.1.4]{K}.  To see this,
observe that the Zariski tangent space of the fiber of the relative
Hom scheme over $T$ at a section
$$
(\text{Id},s):\mc{C}_t \rightarrow
\mc{C}_t\times_k A_0
$$
equals
$$
H^0(\mc{C}_t,s^* T_{A_0}\otimes \mc{I})
$$
where $T_{A_0}$ is the tangent bundle of $A_0$ and $\mc{I}$ is the
ideal sheaf of $q$ in $\mc{C}_t$, cf. \cite[Theorem II.1.7]{K}.
Because $A_0$ is an Abelian variety, $T_{A_0} = T_{A_0,0}\otimes_k
\OO_{A_0}$.  Thus the Zariski tangent space of the fiber is
$$
T_{A_0,0}\otimes_k H^0(\mc{C}_t,\mc{I}) = T_{A_0,0}\otimes_k 0 = 0,
$$
i.e., the relative Hom scheme is unramified over $T$.

\mni
Now let $t$ be a very general point of $T$ and let $s_t$ be a section
of $\mc{C}_t\times_k A_0$ over $\mc{C}_t$ sending $q$ to $r$.  By the
last paragraph, $s_t$ specializes uniquely to a section $s$ of
$C\times_k A_0$ over $C$ sending $q$ to $r$.  By Lemma
~\ref{lem-const}, each such section $s$ is the restriction of a unique
section $S$ over all of $B$.  Since $S|_{\mc{C}_t}$ is also a section
over $\mc{C}_t$ specializing to $s$, uniqueness implies $s_t =
S|_{\mc{C}_t}$.  Therefore, for $t$ a very general point of $T$, every
section of $\mc{C}_t\times_k A_0$ over $\mc{C}_t$ sending $q$ to $r$
is the restriction of a unique section of $B\times_k A_0$ over $B$.
Finally, Lemma ~\ref{lem-trick} implies the result for all sections
$s_t$ of $\mc{C}_t\times_k A_0$ over $\mc{C}_t$, whether or not they
map $q$ to $r$.
\end{proof}

\begin{lem} \label{lem-noniso}
\marpar{lem-noniso}
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $Q$ be an Abelian scheme over $B$.  Assume
that $Q$ is strongly non-isotrivial, cf. Definition
~\ref{defn-noniso}.  For a very general triangle curve $C$ in $B$, the
map
$$
H^0_\text{\'et}(B,Q) \rightarrow H^0_\text{\'et}(C,C\times_B Q)
$$
is a bijection.  
\end{lem}

\begin{proof}
  Because $Q$ is strongly non-isotrivial, for a very general pair of
  closed points $p,q\in B$, there is only the zero homomorphism of
  Abelian varieties $Q_p\rightarrow Q_q$.  Since a very general pair
  of closed points is contained in an $f$-linear curve $C'$, the same
  conclusion holds for the Abelian scheme $C'\times_B Q$ over $C'$.
  Thus $C'\times_B Q$ is strongly non-isotrivial.  By Lemma
  ~\ref{lem-Chow2}, $\text{Tr}_{C'/k}(C'\times_B Q)=0$.  Thus, by
  Lemma ~\ref{lem-rigid}, every section of $C'\times_B Q$ over $Q$ is
  rigid and there are only countably many such sections.  By Corollary
  ~\ref{cor-tri2}, for a very general triangle curve $C$ in $B$, every
  section of $C\times_B Q$ over $C$ is contained in a unique
  ``rational section'' of $Q$ over $B$.  By \cite[Theorem
  VI.1.9.3]{K}, these rational sections are all regular.  Therefore
$$
H^0_\text{\'et}(B,Q) \rightarrow H^0_\text{\'et}(C,C\times_B Q)
$$
is a bijection.
\end{proof}

\mni
The most involved step in the proof of Theorem ~\ref{thm-main} is the
analogue of Lemma \ref{lem-noniso} for very general cubic curves $C$
in $B$.  As with Lemma ~\ref{lem-const2}, the argument is a
deformation-specialization argument.  However, both steps are more
involved than in Lemma ~\ref{lem-const2}.  The first step proves that
sections over general cubic curves specialize to sections over
triangle curves.  This no longer follows from Abhyankar's lemma,
\cite[Theorem VI.1.9.3]{K}, as the codimension $1$ fibers of a
compactification of $Q$ are typically uniruled.  Instead we use the
N\'eron extension property.  We recall the definition in the context
which we use it.

\begin{defn} \label{defn-Neron}
\marpar{defn-Neron}
Let $T$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 1$.  A smooth, finite type, separated morphism
$X\rightarrow T$ has the \emph{N\'eron extension property} if for
every triple $(Y\rightarrow T,U,s_U)$ of
\begin{enumerate}
\item[(i)]
a smooth morphism $Y\rightarrow T$,
\item[(ii)]
a dense, open subset $U\subset T$,
\item[(iii)]
and a $T$-morphism $s_U:Y\times_T U \rightarrow X$,
\end{enumerate}
there exists a pair $(V,s_V)$ of
\begin{enumerate}
\item[(i)]
an open subset $V\subset T$ containing $U$ and all codimension $1$
points of $T$
\item[(ii)]
and a $T$-morphism $s_V:Y\times_T V \rightarrow X$ whose restriction
to $Y\times_T U$ equals $s_U$.
\end{enumerate}
If $X\rightarrow T$ has the N\'eron extension property, then it is
called a \emph{N\'eron model}, or a \emph{N\'eron model of its generic
  fiber}.  
\end{defn}

\mni
Let $X_1\rightarrow T$ and $X_2\rightarrow T$ be N\'eron models.  Let
$U \subset T$ be a dense, open subset.  And let $U\times_T X_1 \cong
U\times_T X_2$ be a $T$-isomorphism.  By the N\'eron extension
property, there exists an open subset $V\subset T$ containing $U$ and
all codimension $1$ points and a $T$-isomorphism $V\times_T X_1 \cong
V\times_T X_2$ extending the isomorphism over $U$.  Thus, N\'eron
models are unique in codimension $1$.

\mni
The basic result about existence of N\'eron models is the following.

\begin{lem}\cite[Theorem 3, p.19]{Ner} \label{lem-neron}
\marpar{lem-neron}
Let $W$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 1$.  Let $B$ be a dense open subset of $W$ and let
$A$ be an Abelian scheme over $B$.  There exists an open subset
$\widetilde{B}$ of $W$ containing $B$ and all codimension $1$ points
and a N\'eron model $\widetilde{A}$ over $\widetilde{B}$ whose
restriction over $B$ equals $A$.
\end{lem}

\mni
Let $\overline{B}$ be an integral, normal, projective $k$-scheme
containing $B$ as a dense, open subscheme.  Let $W$ be the smooth
locus of $\overline{B}$.  By Lemma ~\ref{lem-neron}, there exists an
open subset $\widetilde{B}$ of $W$ containing $B$ and all codimension
$1$ points and a N\'eron model $\widetilde{Q}$ over $\widetilde{B}$
whose restriction over $B$ equals $Q$.  By the uniqueness part of
N\'eron models, or by the construction, the N\'eron model
$\widetilde{Q}$ can be chosen to be a smooth group scheme over
$\widetilde{B}$.

\mni
The parameter space for both cubic and triangle curves
is somewhat involved and warrants a bit of notation.

\begin{notat} \label{notat-param}
\marpar{notat-param}
Denote by $P_{1^\text{st}}$ the $\PP^9$-bundle over the Grassmannian
$\text{Grass}(2,\PP^b_k)$ parametrizing pairs $([\Lambda],[E])$ of a
linear $2$-plane $\Lambda \subset \PP^b_k$, i.e., 
$[\Lambda]\in \text{Grass}(2,\PP^b_k)$, together with
a degree $3$ Cartier divisor $E\subset \Lambda$.  Denote by
$\Delta_{1^\text{st}} \subset P_{1^\text{st}}$ the closed subscheme
parametrizing pairs such that $E$ is singular.  
Denote by $\Gamma_{1^\text{st}}
\subset P_{1^\text{st}}$ 
the locally closed subscheme parametrizing pairs such that
$E$ is a union of $3$ non-concurrent lines in $\Lambda$.  Denote by
$P_{2^\text{nd}}\subset P_{1^\text{st}}$ the maximal open subscheme
parametrizing pairs such that
\begin{enumerate}
\item[(i)]
$\overline{B}\times_{\PP^b_k} E$ is contained in $\widetilde{B}$,
\item[(ii)]
and the morphism $\overline{B}\times_{\PP^b_k} E \rightarrow E$ is a finite,
separable morphism of at-worst-nodal curves which is \'etale over a
neighborhood of every node of $E$ and maps every node of 
$\overline{B}\times_{\PP^b_k} E$ to a node of $E$.
\end{enumerate}
Denote $\Gamma_{2^\text{nd}} = \Gamma_{1^\text{st}}\cap
P_{2^\text{nd}}$ and $\Delta_{2^\text{nd}}=\Delta_{1^\text{st}}\cap
P_{2^\text{nd}}$.  
Denote by $P_{3^\text{rd}} \rightarrow P_{2^\text{nd}}$ the
blowing up of $P_{2^\text{nd}}$ along $\Gamma_{2^\text{nd}}$.
Denote the exceptional divisor by by $\Gamma_{3^\text{rd}}$ and denote
the strict transform of $\Delta_{2^\text{nd}}$ by
$\Delta_{3^\text{rd}}$.  

\mni
Denote by $P$ the open complement of $\Delta_{3^\text{rd}}$ in
$P_{3^\text{rd}}$ and denote $\Gamma = \Gamma_{3^\text{rd}}\cap P$.
The scheme $P$ is an integral, smooth parameter space for projective
cubic and triangle curves in $\widetilde{B}$ and the locus $\Gamma$
parametrizing triangle curves is a integral, smooth Cartier divisor in
$P$.  Denote by $\mc{C}\subset \widetilde{B}\times_k P$ the associated
family of projective cubic and triangle curves in $V$ parametrized by
$P$.

\mni
Finally, denote by 
$$
\Sigma \subset \text{Hom}_P(\mc{C},\mc{C}\times_{\widetilde{B}} \widetilde{Q})
$$
the locally closed subscheme of the relative Hom scheme over $P$
parametrizing $\mc{C}$-morphisms.
\end{notat}

\mni
Because $\mc{C}\times_{\widetilde{B}} \widetilde{B}$ is a group scheme
over $\mc{C}$, $\Sigma$ is a group scheme over $P$.  Because the
restriction of $\widetilde{Q}$ over a general curve in $\widetilde{B}$
is strongly non-isotrivial, Lemma ~\ref{lem-rigid} implies that
$\Sigma\rightarrow P$ is ``locally generically finite'', i.e., every
quasi-compact open subset of $\Sigma$ is finite over a dense, open
subset of $P$.

\mni
Let $\Sigma_i$ be an irreducible component of $\Sigma$ and let
$$
F:\mc{C}\times_P \Sigma_i \rightarrow \mc{C}\times_{\widetilde{B}}
\widetilde{Q}
$$
be the restriction of the universal $\mc{C}$-morphism.  
Denote by
$$
y_\Sigma:\mc{C}\times_P \Sigma_i \rightarrow \widetilde{B}
$$
the composition of $F$ with projection to $\widetilde{B}$.
Because $F$ is
a $\mc{C}$-morphism, $F = (\text{Id}_{\mc{C}},G)$ where 
$$
G:\mc{C}\times_P \Sigma_i \rightarrow \widetilde{Q}
$$
is a $\widetilde{B}$-morphism, i.e., $\text{pr}_{\widetilde{B}}\circ G
= y_\Sigma$.  

\begin{lem} \label{lem-deform}
\marpar{lem-deform}
If $\text{char}(k)=0$ and if $\Sigma_i$ dominates $P$, then the image
of $\Sigma_i$ in $P$ intersects $\Gamma$ in a dense subset of $\Gamma$.
\end{lem}


\begin{proof}
  Let $\Sigma_i \hookrightarrow \overline{\Sigma}_i$ be a dense, open
  immersion of $P$-schemes with $\overline{\Sigma}_i$ a normal,
  projective $P$-scheme.  The $\mc{B}$-morphism $G$ gives a rational
  transformation of $\mc{B}$-schemes
$$
G:\mc{C}\times_P \overline{\Sigma}_i \dashrightarrow \widetilde{Q}.
$$
By the universal property of $\Sigma$, $\Sigma_i\subset
\overline{\Sigma}_i$ is the maximal open subscheme over which $G$ is
regular.  Thus to prove the lemma, it is equivalent to prove that
$G$ is regular over an open subscheme of $\overline{\Sigma}_i$ whose
image in $P$ intersects $\Gamma$ in a dense subset.

\mni
Denote by $\widetilde{\Sigma}_i$ the smooth locus of
$\overline{\Sigma}_i$.  

\begin{claim} \label{claim-nonisoA}
\marpar{claim-nonisoA}
There exists an irreducible component $D \subset
\widetilde{\Sigma}_i\times_P \Gamma$ such that $D\rightarrow\Gamma$
is dominant and generically finite.  And $D$ is
a Cartier divisor in $\widetilde{\Sigma}_i$.
\end{claim}

\mni 
Because $\overline{\Sigma}_i\rightarrow P$ is dominant and projective,
it is surjective.  
Thus also $\overline{\Sigma}_i\times_P \Gamma \rightarrow \Gamma$ is
surjective.
Thus some generic point of $\overline{\Sigma}_i\times_P \Gamma$
dominates the generic point of $\Gamma$.
Because $\Gamma$ is a Cartier divisor in $P$,
$\overline{\Sigma}_i\times_P \Gamma$ is a Cartier divisor in
$\overline{\Sigma}_i$.  So each of its generic points is codimension
$1$ in $\overline{\Sigma}_i$.    
Because $\overline{\Sigma}_i$ is normal, 
$\widetilde{\Sigma}_i$ contains every codimension $1$ point, thus it
contains every generic point of $\overline{\Sigma}_i\times_P \Gamma$.
In particular, it contains a generic point dominating the generic
point of $\Gamma$.  Denote the closure of this generic point in
$\widetilde{\Sigma}_i$ by $D$ and give it the reduced, induced scheme
structure.  Then $D$ dominates $\Gamma$.  Because
$\widetilde{\Sigma}_i \rightarrow P$ is generically finite and
$\widetilde{\Sigma}_i$ is integral, the morphism is quasi-finite
at every codimension $1$ point of $\widetilde{\Sigma}_i$.  Thus
$D\rightarrow \Gamma$ is generically finite.
Finally $D$ is a Cartier divisor
in $\widetilde{\Sigma}_i$ since $\widetilde{\Sigma}_i$ is smooth and
$D$ is an integral, codimension $1$, closed subvariety.  This proves
Claim ~\ref{claim-nonisoA}.


%% Because $P$ is smooth, $\Sigma_i$ is generically flat over $P$.  By
%% the argument above, it is also generically finite over $P$.  Also the
%% generic fiber of $\Sigma$ over $P$ is an algebraic group over a
%% characteristic $0$ field, thus smooth.  Therefore $\Sigma_i$ is
%% generically smooth over $P$.

\mni
Denote by
$$
y_\Gamma:
{\mc{C}}\times_P \Gamma \xrightarrow{\text{pr}_{\mc{C}}}
\mc{C} \xrightarrow{\text{pr}_{\widetilde{B}}} \widetilde{B}
$$
the natural morphism.  

\begin{claim} \label{claim-nonisoB}
\marpar{claim-nonisoB}
The morphism ${y}_\Gamma$ is smooth.
\end{claim}

\mni
By chasing diagrams, ${y}_\Gamma$ is the composition of an open
immersion and the base change by $\widetilde{B}\rightarrow \PP^b_k$ of
the same morphism ${y}_\Gamma$ in the special case 
that $\widetilde{B}=\PP^b_k$.  Thus it
suffices to prove Claim ~\ref{claim-nonisoB}
in the special case that
$\widetilde{B}=\PP^b_k$.  In this case ${y}_\Gamma$ is
the fiber 
bundle over $\PP^b_k$ whose fiber over $p\in \PP^b_k$ is the
variety parametrizing all triangles $T$ of non-concurrent lines
containing $p$ as a smooth point, which is clearly smooth.
This proves Claim ~\ref{claim-nonisoB}.

\mni
Denote by
$$
{y}_{\widetilde{\Sigma}}:\mc{C}\times_P
\widetilde{\Sigma}_i \rightarrow \widetilde{B}
$$
the composition
$$
{\mc{C}}\times_P \widetilde{\Sigma}_i
\xrightarrow{\text{pr}_{\mc{C}}} \mc{C}
\xrightarrow{\text{pr}_{\widetilde{B}}} \widetilde{B}.
$$
For every open subset $U \subset \widetilde{\Sigma}$, denote by
$$
{y}_{U}:{\mc{C}}\times_P U \rightarrow \widetilde{B}
$$
the restriction of ${y}_{\widetilde{\Sigma}}$ over $U$.

\begin{claim} \label{claim-nonisoC}
\marpar{claim-nonisoC}
There exists an open subset $U\subset \widetilde{\Sigma}$ intersecting
$D$ such that ${y}_{U}$ is smooth.
\end{claim}

\mni
The smooth locus of ${y}_{\widetilde{\Sigma}}$ is an open subscheme of
$\mc{C}\times_P \widetilde{\Sigma}$.  Its complement is closed.
Because $\mc{C}\rightarrow P$ is proper, the image of the closed
complement in $\widetilde{\Sigma}$ is closed.  The open complement 
$U\subset \widetilde{\Sigma}$ is the maximal open subscheme such that
${y}_{U}$ is smooth.  The claim is precisely that $U$ intersects $D$.

\mni
Since $\text{char}(k)=0$, the morphism $D\rightarrow \Gamma$ is smooth
on a dense open subset $U_D\subset D$.  Thus the product
$$
{\mc{C}}\times_P U_D \rightarrow {\mc{C}}\times_P \Gamma
$$
is smooth.  
Denote by 
$$
{y}_{U_D}:{\mc{C}}\times_P U_D \rightarrow
\widetilde{B}
$$
the composition of this product with
$y_\Gamma$.  By Claim ~\ref{claim-nonisoB}, $y_\Gamma$ is smooth.
Thus the composition ${y}_{U_D}$ is smooth.
Chasing diagrams, ${y}_{U_D}$ equals the restriction of
${y}_{\widetilde{\Sigma}}$ to ${\mc{C}}\times_P U_D$.  Since
${\mc{C}}\times_P D$ is a Cartier divisor in ${\mc{C}}\times_P
\widetilde{\Sigma}_i$, $y_{\widetilde{\Sigma}}$ is smooth at every
point of ${\mc{C}}\times_P U_D$. Since the singular locus of
$y_{\widetilde{\Sigma}}$ does not intersect the inverse image of
$U_D$, $U_D$ is contained in $U$.  Therefore $U$ intersects $D$.  This
proves Claim ~\ref{claim-nonisoC}.

\begin{claim} \label{claim-nonisoD}
\marpar{claim-nonisoD}
There exists an open subset $V\subset \widetilde{B}$ containing $B$
and all codimension $1$ points such that $G$ is regular on the open subset
$y_U^{-1}(V) \subset {\mc{C}}\times_P U$.
\end{claim}

\mni
Since $y_U$ is smooth, 
$$
{y}_U:{y}_U^{-1}(B) \rightarrow B
$$
is also smooth.  Since $Q\rightarrow B$ is an Abelian scheme, the
rational transformation of smooth $B$-schemes
$$
G:y_U^{-1}(B) \dashrightarrow Q
$$
is a regular morphism by Weil's extension theorem,
cf. \cite[\S 4.4]{Ner}, \cite[Theorem VI.1.9.3]{K}.  
Because $\widetilde{Q} \rightarrow \widetilde{B}$ is a N\'eron model,
by the Neron extension
property, there exists an open subset $V$ of $\widetilde{B}$
containing $B$ and all codimension $1$ points such that
$$
G:y_U^{-1}(V) \dashrightarrow \widetilde{Q}
$$
is a regular morphism.  This proves Claim ~\ref{claim-nonisoD}.  

\mni
Let $P_V\subset P$ denote the open subscheme
parametrizing curves in $\widetilde{B}$
entirely contained in $V$.  
Because $V$ contains
all codimension $1$ points of $\widetilde{B}$, a general triangle
curve is contained in $P_V$.  Since $\Gamma$ is irreducible, $P_V\cap
\Gamma$ is dense.  Since $D\rightarrow \Gamma$ is dominant, $D\times_P
P_V$ is also dense in $D$.  Since $U\cap D$ is also dense in $D$, the
two open subsets of $D$ intersect.
In other words, $U\times_P P_V\subset \widetilde{\Sigma}_i$ is an open
subset intersecting $D$.  Since $D$ is integral, this subset is dense
in $D$.  Since $D\rightarrow \Gamma$ is dominant, the image of
$U\times_P P_V$ in $P$ intersects $\Gamma$ in a dense subset.
By definition of $P_V$, 
$$
{\mc{C}}\times_P (U\times_P P_V) \subset y_U^{-1}(V).
$$
Therefore $G$ is regular over $U\times_P P_V$ by Claim
~\ref{claim-nonisoD}.  Thus $U\times_P P_V\subset \overline{\Sigma}_i$
is an open subset satisfying the conditions at the beginning of the
proof.  Therefore the image of $\Sigma_i$ in $P$ intersects $\Gamma$
in a dense subset.
\end{proof}    

\begin{lem} \label{lem-noniso2}
\marpar{lem-noniso2}
Assume $\text{char}(k)=0$.  
Let $B$ be an integral, smooth, quasi-projective $k$-scheme of
dimension $b\geq 2$.  Let $Q$ be an Abelian scheme over $B$.  Assume
that $Q$ is strongly non-isotrivial, cf. Definition ~\ref{defn-noniso}.
For a very
general cubic curve $C$ in $B$, the map
$$
H^0_\text{\'et}(B,Q) \rightarrow H^0_\text{\'et}(C,C\times_B Q)
$$
is a bijection.  
\end{lem}

\begin{proof}
The group $H^0_\text{\'et}(B,Q)$ is finitely generated by the
Lang-N\'eron theorem.  
Applying the N\'eron extension property to each
of the finitely many generators, there exists a single open subset of
$\widetilde{B}$ containing $B$ and all codimension $1$ points such
that every
section of $Q$ over $B$ extends to a section of $\widetilde{Q}$ 
over this open
subset.  Replacing $\widetilde{B}$ by this open subset, 
we may assume that every
section of $Q$ over $B$ extends to a section of $\widetilde{Q}$ over
$\widetilde{B}$.  
Then restriction to curves in $\widetilde{B}$ determines a
homomorphism of group schemes over $P$
$$
\text{restr}:H^0_\text{\'et}(B,Q)\times P \rightarrow \Sigma,
$$
where the first group scheme is the finite, flat group scheme over $P$
whose geometric fibers are all canonically identified with
$H^0_\text{\'et}(B,Q)$.  The lemma is equivalent to the statement that
the induced homomorphism of geometric generic fibers over $P$ is a
bijection.  

\mni
Denote by $\eta\in P$ the generic point and 
denote by $\gamma \in P$ the generic point of $\Gamma$.  
Denote
$R= \OO_{P,\eta_\Gamma}$.  Because $P$ is smooth and $\Gamma\subset
P$ is a Cartier divisor, this is a DVR.  Consider the base change of
$\text{restr}$ over $\SP R$.  
The geometric generic fiber of $\Sigma\times_P \SP R$ over $\SP R$ is
$$
\Sigma_{\overline{\eta}} = \Sigma\otimes_{\OO_P}
\overline{\kappa(\eta)},
$$
and the geometric closed fiber is
$$
\Sigma_{\overline{\gamma}} = \Sigma\otimes_{\OO_P}
\overline{\kappa(\gamma)}.
$$
By Lemma ~\ref{lem-rigid},
$\Sigma\times_P \SP R$ is a group scheme over
$\SP R$ which is ``locally quasi-finite'', i.e., every quasi-compact
component $\Sigma_i\times_P \SP R$
is quasi-finite over $\SP R$.
By Zariski's Main Theorem, the morphism to $\SP R$ is the composition
of a dense open immersion and a finite morphism.  By Lemma
~\ref{lem-noniso2}, points of $\Sigma_{\overline{\eta}}$ specialize to
points of $\Sigma_{\overline{\gamma}}$.  
Thus $\Sigma_i\times_P \SP R \rightarrow \SP R$ is finite.  
Since the characteristic is zero,
all group schemes are reduced.  Thus
$\Sigma\times_P \SP R$ is unramified over $\SP R$. 
In particular, the specialization homomorphism
$$
\text{spec}:\Sigma_{\overline{\eta}}\rightarrow
\Sigma_{\overline{\gamma}}
$$
is injective.  But this is compatible with the restriction
homomorphism, i.e.,
$$
\begin{CD}
H^0_\text{\'et}(B,Q) @> = >> H^0_\text{\'et}(B,Q) \\
@V \text{restr}_{\overline{\eta}} VV @VV
\text{restr}_{\overline{\gamma}} V \\
\Sigma_{\overline{\eta}} @> \text{spec} >> \Sigma_{\overline{\gamma}}
\end{CD}
$$
is a commutative diagram of group homomorphisms.  By Lemma
~\ref{lem-noniso}, the right vertical homomorphism is a bijection.
Since $\text{spec}$ is injective, also the right vertical homomorphism
is a bijection, proving the lemma.
\end{proof}

\begin{proof}[Proof of Theorem ~\ref{thm-main}]
\textbf{(ii)}
First we prove (ii), i.e., we prove that $\mc{P}(B,f,A)$ holds,
cf. Definition ~\ref{defn-P}.
By Corollary ~\ref{cor-Chow2}, there exists a
dense, open subscheme $U\subset B$, a finite, \'etale, Galois morphism
$U'\rightarrow U$, and an isogeny of Abelian schemes over $U'$,
$$
u=u_0\oplus u_Q:(U'\times_k A_0) \times_{U'} Q \rightarrow U'\times_B
A
$$
where $A_0$ is an Abelian $k$-variety and $Q$ is a strongly
non-isotrivial Abelian scheme over $U'$.  By the elementary reductions
Lemma ~\ref{lem-red0} and Lemma~\ref{lem-red1}, it suffices to prove
$\mc{P}(U',f',U'\times_B A)$ holds.  By the isogeny reduction, Lemma
~\ref{lem-Chow3}, it suffices to prove the two cases
$\mc{P}(U',f',U'\times_k A_0)$
and $\mc{P}(U',f',Q)$ hold.  The first case follows from Lemma
~\ref{lem-const} for triangle curves, resp. Lemma ~\ref{lem-const2} for
cubic curves.  The second case follows from Lemma ~\ref{lem-noniso}
for triangle curves, resp. Lemma ~\ref{lem-noniso2} for cubic curves
in characteristic $0$.  This proves (ii).

\mni
\textbf{(i)}
Next we prove (i), by reducing it to (ii).  Let $T$ be an $A$-torsor
over $B$.  Because this is a torsor for the \'etale topology, there
exists a dense open subset $U\subset B$ and a finite, \'etale, Galois
morphism $U'\rightarrow U$ with Galois group $G$ such that $U'\times_B
T$ is a trivial $U'\times_B A$-torsor over $U'$.  By (i), for a very
general triangle curve $C$ in $B$, 
$$
H^0_\text{\'et}(U',U'\times_B A) \rightarrow
H^0_\text{\'et}(U'\times_B C, U'\times_B C\times_B A)
$$
is a bijection.  This is an isomorphism of Galois modules.  Thus the
induced morphism of Galois invariants is an isomorphism.  By \'etale
descent, the morphism of Galois invariants is the restriction map
$$
\text{Hom}_{U-\text{sch}}(U,U\times_B T) \rightarrow
\text{Hom}_{U\times_B C-\text{sch}}(U\times_B
C, U\times_B C \times_B T).
$$
In particular, if $C\times_B T$ has a section over $C$, then
$U\times_B T$ has a section over $U$.  This section extends over all
of $B$ by \cite[Theorem VI.1.9.3]{K}.  Thus, if $C\times_B T$ is a
trivial torsor, then $T$ is a trivial torsor.  

\mni
Also, since
sections of $T$ over very general cubic curves specialize to sections
of $T$ over very general triangle curves, if $C\times_B T$ has a
section over $C$ for a very general cubic curve, then is also has a
section over a very general triangle curve.  Therefore, by the
previous paragraph, $T$ is a trivial torsor.  

\mni
Finally, consider the statement with planar surfaces in place of
curves.  Let $C$ be a very general triangle curve in $B$ satisfying
(ii).  Since $C=B\times_{\PP^b_k}E$, where $E$ is a plane
cubic curve in a linear $2$-plane $\Pi \subset \PP^b_k$, $C$ is
contained in the planar surface $S=B\times_{\PP^b_k} \Pi$.  If $C$ is
very general, then $S$ is very general.  Moreover, since (ii)
holds for $B$ for one triangle curve in $S$, it holds for every very general
triangle curve in $S$.  But for a very general triangle curve in $S$,
(ii) also applies with $S$ in place of $B$.  Therefore both
$$
H^0_\text{\'et}(B,A)\rightarrow H^0_\text{\'et}(C,C\times_B A)
$$
and
$$
H^0_\text{\'et}(S,S\times_B A) \rightarrow H^0_\text{\'et}(C,C\times_B
A)
$$
are bijections.  Since the first map factors through the second map by
restriction
$$
H^0_\text{\'et}(B,A) \rightarrow H^0_\text{\'et}(S,S\times_B A),
$$
this restriction is also a bijection.  And (i) follows for planar
surfaces in a similar way.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Examples and the proof of Proposition \ref{prop-neg}
%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Examples and the proof of Proposition \ref{prop-neg}} \label{sec-ex}
\marpar{sec-ex}

\mni
In this section we give some examples demonstrating limits to further
generalization of Theorem ~\ref{thm-main}.  

\mni
\textbf{Example (i).}
First of all, the
restriction map
$$
H^1_\text{\'et}(B,A) \rightarrow H^1_\text{\'et}(C,C\times_B A)
$$
can fail to be surjective for all triangle curves, resp. all cubic
curves.  For an example, let $B=\PP^2$, let $f$ be the identity map,
and let $A = B\times_k A_0$, where $A_0$ is a simple Abelian
$k$-variety of dimension $g \geq 2$.  
Let $l$ be an integer not divisible by
$\text{char}(k)$.  Associated to the multiplication map
$$
\begin{CD}
0 @>>> B\times_k A_0[l] @>>> B\times_k A_0 @> \text{mult}_l >>
B\times_k A_0 @>>> 0
\end{CD}
$$
there is a long exact sequence of cohomology groups part of which is
$$
\begin{CD}
0 @>>> H^0_\text{\'et}(B,B\times_k A_0)\otimes \ZZ/l\ZZ @>>>
H^0_\text{\'et}(B,B\times_k A_0[l]) @>>> H^0_\text{\'et}(B,B\times_k
A_0)[l] @>>> 0.
\end{CD}
$$
Because $\text{Alb}(P)=0$, 
$$
H^0_\text{\'et}(B,B\times_k A_0) = A_0(k).
$$
Because $A_0$ is
divisible, the exact sequence above gives
$$
H^1_\text{\'et}(B,B\times_k A_0)[l] \cong H^1_\text{\'et}(B,B\times_k
A_0[l]) \cong H^1_\text{\'et}(B,\ZZ/l\ZZ)^{\oplus 2g}.
$$

\mni
A similar argument applies to plane cubic curves and triangle curves
in $B$.  Of course the Albanese variety of a plane curve is not zero.
But because $A_0$ is a simple Abelian variety of dimension $g\geq 2$,
every morphism from $C$ to $A_0$ is a constant morphism.  Thus
$$
H^1_\text{\'et}(C,C\times_k A_0)[l] \cong H^1_\text{\'et}(C,C\times_k
A_0[l]) \cong H^1_\text{\'et}(C,\ZZ/l\ZZ)^{\oplus 2g}.
$$
Thus the restriction homomorphism on $l$-torsion is equivalent to
$$
H^1_\text{\'et}(\PP^2_k,\ZZ/l\ZZ)^{\oplus 2g} \rightarrow
H^1_\text{\'et}(C,\ZZ/l\ZZ)^{\oplus 2g}.
$$
Now
$$
H^1_\text{\'et}(\PP^2_k,\ZZ/l\ZZ) = 0
$$
whereas
$$
H^1_\text{\'et}(C,\ZZ/l\ZZ) = (\ZZ/l\ZZ)^{\oplus 2}
$$
for a smooth plane cubic and
$$
H^1_\text{\'et}(C,\ZZ/l\ZZ) = \ZZ/l\ZZ
$$
for a triangle curve.  Thus the restriction map is not surjective.

\mni
\textbf{Example (ii).}
This example is similar.  Let $B=\AAA^2_k$ and let $f:\AAA^2_k
\hookrightarrow \PP^2_k$ be the usual inclusion.  Let $A_0$ be an
Abelian $k$-variety having a $p$-torsion $k$-point, 
$$
a\in A_0(k)[p],
$$
e.g., $A_0$ is an ordinary elliptic curve over $k$.  Let
$A_0\rightarrow A_1$ be the \'etale isogeny of Abelian varieties whose
kernel is generate by $a$.  There is an exact sequence
$$
\begin{CD}
0 @>>> \ZZ/p\ZZ @> a >> A_0 @>>> A_1 @>>> 0.
\end{CD}
$$
This gives rise to a long exact sequence.  For the same reason as
in Example (i), the induced map
$$
H^1_\text{\'et}(\AAA^2_k,\ZZ/p\ZZ) \rightarrow
H^1_\text{\'et}(\AAA^2_k,
\AAA^2_k\times_k A_0)[p]
$$
is an injection.  Thus to produce an $A_0$-torsor over $\AAA^2_k$ of
order $p$ whose restriction to $C$ is trivial, it suffices to produce
a $\ZZ/p\ZZ$-torsor $T$ over $\AAA^2_k$ whose restriction to $C$ is
trivial.  Let $f\in k[x,y]$ be a polynomial function on $\AAA^2_k$
vanishing on $C$ and whose degree $d$ is not divisible by $p$.  

\mni
Let
$T\rightarrow \AAA^2_k$ 
be the Artin-Schreier cover determined by the ring homomorphism
$$
k[x,y] \rightarrow k[x,y,t]/\langle t^p -t - f \rangle.
$$
Because $d$ is not divisible by $p$, there is no element $g\in k[x,y]$
such that $g^p-g - f =0$.  Since $k[x,y]$ is normal and $t^p-t-f$ is a
monic polynomial in $t$, also there is no
element $g\in k(x,y)$ such that $g^p-g-f=0$.  Thus $T$ is integral and
so $T\rightarrow \AAA^2_k$ is a nontrivial torsor.  On the other hand,
since $f$ is zero on $C$,
$$
C\times_{\AAA^2_k} T \cong C \times \SP k[t]/\langle t^p-t \rangle,
$$
which is the trivial $\ZZ/p\ZZ$-torsor over $C$.  Thus
$$
H^1_\text{\'et}(\AAA^2_k,\AAA^2_k\times_k A_0)[p] \rightarrow
H^1_\text{\'et}(C,C\times_k A_0)[p]
$$
is not injective.

\mni
\textbf{Example (iii).}
Let $B=\PP^2_k$ and let $f$ be the identity map.  
Let $A_0$ be an ordinary elliptic curve over $k$ and let
$A=\PP^2_k\times_k A_0$.  Then
$$
H^0_\text{\'et}(\PP^2_k,\PP^2_k\times_k A_0) = A_0(k).
$$
But every open subset of the set of plane cubic curves in $\PP^2_k$
contains curves $C$ isogenous to $A_0$.  For these, the map
$$
A_0(k) \rightarrow H^0_\text{\'et}(C,C\times_k A_0)
$$
is not surjective.  Thus, it is necessary to use very general cubic
curves in Theorem ~\ref{thm-main} -- general cubic curves do not suffice.


%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% %% Section: Morphisms to Abelian varieties
%% %% 
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% \section{Morphisms to Abelian varieties} \label{sec-mor}
%% \marpar{sec-mor}

%% \mni
%% In this section we collect some well-known results about morphisms to
%% Abelian varieties.  The first results are applications of standard deformation
%% theory to morphisms to Abelian varieties.  
%% Let $S$ be a scheme.  For a group scheme $\mc{A}$ over $S$,
%% or for a torsor for a group scheme,
%% there exists
%% a locally free $\OO_S$-module $\mc{T}$ and an isomorphism
%% $$
%% \textit{Hom}_{\OO_S}(\mc{T},\OO_{\mc{A}}) = \Omega_{\mc{A}/S},
%% $$
%% cf. \cite[p. 102, Proposition 2]{Ner}.  
%% Let 
%% $$
%% i:X\hookrightarrow X'
%% $$
%% be a closed immersion of $S$-schemes whose ideal sheaf $\mc{I}$ is
%% square-zero.  Let $j:Z'\rightarrow X'$ be a closed immersion with
%% ideal sheaf $\mc{J}$ (not necessarily square-zero).  Let 
%% $$
%% f:X \rightarrow \mc{A}, \ \ g':Z'\rightarrow \mc{A}
%% $$
%% be $S$-morphisms whose restrictions to the closed subscheme $X\cap Z'
%% = X\times_{X'} Z'$ agree, i.e.,
%% $$
%% f|_{X\cap Z'} = g|_{X\cap Z'}.
%% $$

%% \begin{prop}\cite[Proposition III.2.2.4]{Illusie1} \label{prop-def}
%% \marpar{prop-def}
%% If $\mc{A}$ is smooth over $S$,
%% there is a canonically defined element
%% $$
%% \omega\in H^1(X,\mc{T}\otimes_{\OO_S}(\mc{I}\cap \mc{J}))
%% $$
%% whose vanishing is equivalent to the existence of an $S$-morphism
%% $$
%% f':X'\rightarrow \mc{A}, \ \ f'|_X = f, \ \ f'|_{Z'} = g'.
%% $$
%% If $\omega = 0$, the set of such $S$-morphisms $f'$ is canonically a
%% torsor for the group
%% $$
%% H^0(X,\mc{T}\otimes_{\OO_S}(\mc{I}\cap \mc{J})).
%% $$
%% \end{prop}

%% \begin{proof}
%% First of all, it is straightforward to eliminate $Z'$.  
%% Define $X*Z'$ to be the closed subscheme of $X'$ with ideal sheaf
%% $\mc{I}\cap \mc{J}$.  A pair $(f,g')$ satisfying the condition above
%% is equivalent to an $S$-morphism $X*Z'\rightarrow \mc{A}$.  Thus,
%% replacing $X$ by $X*Z'$, the proposition reduces to the case when $Z'$
%% is the empty set.  Thus, assume $Z'$ is empty.

%% \mni
%% Now, according to \cite[Proposition III.2.2.4]{Illusie1}, there is a
%% canonically defined element in the hypercohomology group
%% $$
%% \omega \in \mathbb{E}\text{xt}^1_{\OO_X}(f^*L_{\mc{A}/S},\mc{I})
%% $$
%% whose vanishing is equivalent to the existence of an $S$-morphism
%% $$
%% f':X'\rightarrow \mc{A}, \ \ f'|_X = f, \ \ f'|_{Z'} = g'.
%% $$
%% If $\omega = 0$, the set of such $S$-morphisms $f'$ is canonically a
%% torsor for the hypercohomology group
%% $$
%% \mathbb{E}\text{xt}^0(f^*L_{\mc{A}/S},\mc{I}).
%% $$

%% \mni
%% Here $L_{\mc{A}/S}$ is the cotangent complex of $\mc{A}$ over $S$.
%% Because $\mc{A}$ is smooth over $S$, there is a canonical
%% quasi-isomorphism
%% $$
%% L_{\mc{A}/S} \cong \Omega_{\mc{A}/S}[0] =
%% \textit{Hom}_{\OO_S}(\mc{T},\OO_{\mc{A}})[0].
%% $$
%% Plugging this in for $L_{\mc{A}/S}$ and using adjunction, there are
%% canonical isomorphisms
%% $$
%% \mathbb{E}\text{xt}^i_{\OO_X}(f^*L_{\mc{A}/S},\mc{F}) \cong
%% H^i(X,\mc{T}\otimes_{\OO_S} \mc{F})
%% $$
%% for every integer $i$ and for every $\OO_X$-module $\mc{F}$.  
%% \end{proof}

%% \mni
%% One immediate application is the following.

%% \begin{cor} \label{cor-def1}
%% \marpar{cor-def1}
%% Let $k$ be a field, let $A$ be a smooth group scheme over $k$, let $Y$ be
%% a $k$-scheme, and let $D$ be a Cartier divisor in $Y$ with associated
%% invertible sheaf $\mc{L} = \OO_Y(D)$. 
%% Let $f:D\rightarrow A$ be a
%% $k$-morphism.  Let $r\geq 1$ be an integer.  If $h^1(D,\mc{L}^{-i}|_D)=0$ for
%% $i=1,\dots,r-1$, then there exists a $k$-morphism 
%% $f_r:rD \rightarrow A$ whose restriction to $D$ equals $f$.  If also
%% $h^0(D,\mc{L}^{-i}|_D) = 0$ for $i=1,\dots,r-1$, then the morphism
%% $f_r$ is unique.
%% \end{cor}

%% \begin{proof}
%% In this case $S=\SP k$ and $\mc{T}$ is simply a $k$-vector space.  The
%% result is proved by induction on $r$.  The base case $r=1$ is
%% tautological.  Thus assume $r>1$ and the result is known for $r-1$.
%% Denote $(r-1)D$ by $X$ and denote $rD$ by $X'$.  The ideal sheaf
%% $\mc{I}$
%% of
%% $X$ in $X'$ is the pushforward from $D$ of $\mc{L}^{-i}|_D$.  Thus,
%% according to Proposition ~\ref{prop-def}, the obstruction to lifting
%% $f_{r-1}$ to $f_r$ is an element in the cohomology group
%% $$
%% H^1(D,\mc{T}\otimes_k \mc{L}^{-(r-1)}|_D) = \mc{T}\otimes_k
%% H^1(D,\mc{L}^{-(r-1)}|_D).
%% $$
%% And the set of such lifts is a torsor under 
%% $$
%% \mc{T}\otimes_k H^0(D,\mc{L}^{-(r-1)}|_D).
%% $$
%% In particular, if $h^0(D,\mc{L}^(-(r-1))|_D) = 0$, there is a unique
%% lift.  
%% \end{proof}

%% \mni
%% Other applications are to the Hom scheme.  Let $S$ be a scheme and let
%% $\mc{A}$ be a smooth group scheme over $S$, or a torsor for a smooth group
%% scheme over $S$.  
%% Let
%% $$
%% \pi:\mc{C} \rightarrow S
%% $$
%% be a projective, flat $S$-scheme.  The relative Hom functor
%% over $S$ parametrizing morphisms from fibers of $\mc{C}$ to fibers of
%% $\mc{A}$ is representable by a scheme
%% $$
%% \text{Hom}_S(\mc{C},\mc{A})
%% $$
%% whose connected components are quasi-projective over $S$,
%% cf. \cite[Theorem I.1.10]{K}.  Because $\mc{A}$ is a group scheme over
%% $S$, the Hom scheme is also a group scheme over $S$.  

%% \mni
%% Let $Z$ be a closed subscheme of $\mc{C}$ 
%% which is flat over $S$.  Denote the ideal sheaf of $Z$ by $J_Z$.  
%% There is a restriction morphism of Hom schemes,
%% $$
%% \text{Hom}_S(\mc{C},\mc{A}) \rightarrow \text{Hom}_S(Z,\mc{A})
%% $$
%% This is a homomorphism of group schemes over $S$.  Thus, for every
%% $S$-morphism $g:Z\rightarrow \mc{A}$, the fiber of the restriction
%% morphism 
%% $$
%% \text{Hom}_S(\mc{C},\mc{A},g)
%% $$
%% is a torsor for the fiber over $0$,
%% $$
%% \text{Hom}_S(\mc{C},\mc{A},0|_Z).
%% $$

%% \begin{cor} \label{cor-def2}
%% \marpar{cor-def2}
%% \begin{enumerate}
%% \item[(i)]
%% If $\pi_*J_Z =0$ compatibly with arbitrary base change of $S$, then
%% $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$ is unramified over $S$.  
%% \item[(ii)]
%% If
%% $R^1\pi_*J_Z = 0$ compatibly with arbitrary base change on $S$, then
%% $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$ is smooth over $S$.
%% \end{enumerate}
%% Notice, in particular, that the hypotheses are independent of the
%% particular Abelian scheme $\mc{A}$.  
%% The hypothesis in (i) holds, for instance, if $\pi_*\OO_{C}=\OO_S$
%% compatibly with arbitrary base change of $S$ and $Z$ surjects to $S$.
%% \end{cor}

%% \begin{proof}
%% Because $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$ is locally finitely
%% presented over $S$, it is unramified over $S$, resp. smooth over $S$,
%% if and only if it is formally unramified over $S$, resp. formally
%% smooth over $S$.  By hypothesis, formation of $\pi_*J_Z$, resp. of
%% $R^1\pi_*J_Z$, is compatible with arbitrary base change.  Thus,
%% without loss of generality, assume $S$ is $\SP R'$ for a local Artin
%% ring $R'$.  Let $\mf{m}\subset R'$ be the maximal ideal, and let $I\subset
%% R'$ be an ideal such that $\mf{m}I = (0)$.  

%% \mni
%% To prove that
%% $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$ is formally unramified in every
%% case, it suffices to prove in this case that there is at most one lift
%% of an $S$-morphism $f:\mc{C}\otimes_{R'}(R'/I) \rightarrow \mc{A}$ to
%% an $S$-morphism $f':\mc{C} \rightarrow \mc{A}$.  By Proposition
%% ~\ref{prop-def}, the set of lifts is a torsor under the group
%% $$
%% \mc{T}\otimes_{R'} I\otimes_{R'}
%% H^0(\mc{C}\otimes_{R'}(R'/\mf{m}),J_Z\otimes_{R'}(R'\mf{m})) = 0.
%% $$
%% Thus the Hom scheme is unramified so long as
%% $$
%% h^0(\mc{C}\otimes_{\OO_S}\kappa, J_Z\otimes_{\OO_S} \kappa) = 0
%% $$
%% for every $\OO_S$-algebra $\kappa$ which is a field, i.e., $\kappa =
%% R'/\mf{m}$.  This is equivalent to the hypothesis that $\pi_*J_Z =
%% (0)$ compatibly with arbitrary base change on $S$.  

%% \mni
%% A similar argument shows the Hom scheme is smooth so long as
%% $$
%% h^1(\mc{C}\otimes_{\OO_S}\kappa, J_Z\otimes_{\OO_S}\kappa) = 0
%% $$
%% for every $\OO_S$-algebra $\kappa$ which is a field.  This is implied
%% by the hypothesis that $R^1\pi_* J_Z = (0)$ compatibly with arbitrary
%% base change on $S$.
%% \end{proof}

%% \mni
%% There is another classical result for these Hom schemes which is
%% really our starting point for this article.

%% \begin{thm} \cite[\S VI.1]{K}
%% \begin{enumerate}
%% \item[(i)]
%% If $\mc{C}$ is smooth over $S$, then its fibers have \emph{uniruled
%%   modifications} in the sense of \cite[Definition VI.1.6]{K} (in fact
%% even \emph{ruled modifications}).
%% \item[(ii)]
%% Assume that for every DVR $R$ and for every morphism $\SP R \rightarrow
%% S$ mapping the generic point of $\SP R$ to the image in $S$
%% of a generic point of 
%% $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$,
%% the
%% fiber product $\mc{C}\times_S \SP R$ is normal and has \emph{uniruled
%%   modifications over the closed fiber}. To be precise, 
%% for every normal scheme
%% $D$ and every projective, birational morphism $D\rightarrow \mc{C}\times_S
%% \SP R$ which is an isomorphism over the generic fiber, the exceptional
%% set is uniruled.  Then every connected component of the 
%% Hom scheme $\text{Hom}_S(\mc{C},\mc{A},0|_Z)$ is projective over $S$.
%% \end{enumerate}
%% \end{thm}

%% \begin{proof}
%% The first part is \cite[Lemma VI.1.6.2.1]{K}.  For the second part,
%% first observe by limit theorems it suffices to consider the case when
%% $S$ is Noetherian.  Then the valuative criterion of properness implies
%% the components of the Hom scheme are proper over $S$, and hence
%% projective over $S$, if the valuative criterion holds for every
%% morphism from a DVR as above.  Finally, the condition on uniruled
%% modifications implies the valuative criterion by \cite[Theorem
%% VI.1.9.3]{K}.  
%% \end{proof}

%% \mni
%% The hypothesis (ii) is typically 
%% verified if we assume the general fibers
%% of $\mc{C}$ over $S$ are smooth, and the singular fibers are of some
%% special type, e.g., ordinary double points.  In this case, the fibers
%% of $\mc{C}\times_S \SP R$ are $A_n$-singularities.  An explicit
%% resolution of singularities shows that $\mc{C}\times_S \SP R$ has
%% uniruled modifications.  

%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% %% Section: Constant families
%% %% 
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% \section{Constant families} \label{sec-const}
%% \marpar{sec-const}

%% \mni
%% To demonstrate the method, in this section we prove the result for 
%% constant families of Abelian varieties.  As in the previous section,
%% $k$ is an algebraically closed field (not necessarily of
%% characteristic $0$).  Let $B$ be a normal, quasi-projective variety.
%% Let $f:B\rightarrow \PP^N_k$ be a quasi-finite morphism.  Denote by
%% $B_\text{smooth}$ the maximal smooth, open subscheme of $B$.  
%% Let $A$ be an Abelian variety over $k$.  
%% By
%% Weil's extension theorem, cf. \cite[p. 109]{Ner}, every $k$-rational
%% transformation from $B$ to $A$ is regular on $B_\text{smooth}$.  So
%% the set of $k$-rational transformations from $B$ to $A$ equals the
%% Hom-set,
%% $$
%% \text{Hom}_{k-\text{sch}}(B,A).
%% $$
%% For every subvariety $V$ of $B_\text{smooth}$,
%% there is a well-defined restriction map
%% $$
%% \text{Hom}_{k-\text{sch}}(B,A) \rightarrow
%% \text{Hom}_{k-\text{sch}}(V,A).
%% $$
%% Incidentally, for every subvariety $V$ of $B$ such that $V\cap
%% B_\text{smooth}$ is dense in $V$, it follows that there is a
%% well-defined restriction map
%% $$
%% \text{Rat. transf.}_k(B,A) \rightarrow \text{Rat. transf.}_k(V,A).
%% $$

%% \begin{prop} \label{prop-const}
%% \marpar{prop-const}
%% Let $B$ be a normal, quasi-projective variety.  Let $f:B\rightarrow
%% \PP^N_k$ be a quasi-finite, generically unramified morphism.  
%% Let $A$ be an Abelian variety over $k$.  Let $\mc{C}\rightarrow S$ 
%% be a flat,
%% one-parameter family of closed curves in $B_\text{smooth}$ whose
%% general fiber is smooth, and such that $\mc{C}_0$ is the image of a
%% very general triangle curve for some $k$-point $0$ of $S$. 
%% For a very general point $t$ of $S$, the restriction morphism
%% $$
%% \text{Hom}_{k-\text{sch}}(B_\text{smooth},A) \rightarrow
%% \text{Hom}_{k-\text{sch}}(\mc{C}_t,A)
%% $$
%% is a bijection.      
%% \end{prop}

%% \mni
%% We first prove a couple of lemmas.

%% \begin{lem} \label{lem-const1}
%% \marpar{lem-const1}
%% Let $B$ be a normal, quasi-projective variety.  Let $f:B\rightarrow
%% \PP^N_k$ be a quasi-finite, generically unramified morphism.  Let $A$
%% be an Abelian variety over $k$.  Let $q$ be any closed point of the
%% smooth locus $B_\text{smooth}$
%% of $B$, and let $C$ be a very general triangle curve
%% containing $q$.  The restriction map
%% $$
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0)) \rightarrow 
%% \text{Hom}_{k-\text{sch}}((C,q),(A,0))
%% $$
%% is a bijection.  
%% \end{lem}

%% \begin{proof}
%% As in the proof of Lemma ~\ref{lem-tri}, we may reduce to the case
%% where $B$ is projective.  Thus assume $B$ is projective.  
%% Let $q$ be a closed point of $B_\text{smooth}$.  Denote the product
%% $B\times_k A$ by $X$ and denote by $\pi:X\rightarrow B$ the projection
%% onto $B$.  Denote by $p$ the point $(q,0)$ of $X$.  

%% \mni
%% Since $B$ is normal,
%% the complement of $B_\text{smooth}$ in $B$ has codimension $\geq 2$.  
%% Thus a very general
%% triangle curve $C$ with $q_{1,2}=q$ is contained in $B_\text{smooth}$.
%% By Corollary ~\ref{cor-tri1}(iii), every section of $X_C\rightarrow C$
%% mapping $q$ to $p$ is contained in the image of a rational section of
%% $X_B\rightarrow B$.  
%% Of course sections and rational sections 
%% of $X_B\rightarrow B$, resp. of $X_C\rightarrow C$, are precisely the
%% same as morphism and rational transformations from $B$ to $A$,
%% resp. from $C$ to $A$.  As discussed above, every rational
%% transformation from $B$ to $A$ is regular on $B_\text{smooth}$.  Thus,
%% the graph of every morphism from $C$ to $A$ mapping $q$ to $0$ is
%% contained in the graph of a morphism from $B_\text{smooth}$ to $A$, which
%% necessarily maps $q$ to $0$.  This proves that
%% $$
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0)) \rightarrow
%% \text{Hom}_{k-\text{sch}}((C,q),(A,0)) 
%% $$
%% is surjective.  

%% \mni
%% By the rigidity lemma,
%% $\text{Hom}_{k-\text{sch}}((B,q),(A,0))$ is a countable union of
%% $k$-points, i.e., every morphism is rigid.  The graphs of any two
%% distinct morphisms intersects in $X$ over a proper subvariety of $B$.
%% Varying the morphisms gives countably many proper subvarieties of $B$.
%% Choosing the vertices $r$ and $s$ of $C$ to lie outside these
%% countably many proper subvarieties of $B$, any section of
%% $X_C\rightarrow C$ is contained in at most one section of
%% $X_{B_\text{smooth}} \rightarrow B_\text{smooth}$.  Therefore the
%% restriction map
%% $$
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0)) \rightarrow
%% \text{Hom}_{k-\text{sch}}((C,q),(A,0))
%% $$
%% is a bijection.  
%% \end{proof}

%% \mni
%% Let $S$ be a scheme, and let
%% $\mc{A}\rightarrow S$ be an Abelian scheme.  
%% Let $\mc{C}\rightarrow S$ be a
%% projective, flat morphism,
%% let $Z\subset \mc{C}$ be a closed subscheme which is flat over $S$,
%% and let $g:Z\rightarrow \mc{A}$ be an $S$-morphism.
%% There is a relative Hom-scheme over $S$
%% $$
%% \textit{Hom}_S(\mc{C},\mc{A},g)
%% $$
%% parametrizing families of morphism from fibers of $\mc{C}$ over $S$ to
%% fibers of $\mc{A}$ over $S$ which 
%% restrict to $g$ on $Z$, cf. \cite[Definition II.1.4]{K}.
%% Moreover, the connected components of
%% $\textit{Hom}_S(\mc{C},\mc{A},g)$ are quasi-projective over $S$.

%% \begin{lem} \label{lem-const2}
%% \marpar{lem-const2}
%% \begin{enumerate}
%% \item[(i)]
%% If the geometric fibers of $\mc{C}$ over $S$ are integral, and if
%% $Z\rightarrow S$ is surjective, then
%% the morphism
%% $$
%% \textit{Hom}_S(\mc{C},\mc{A},g) \rightarrow S
%% $$
%% is unramified.  
%% \item[(ii)]
%% If the geometric fibers of $t$ are smooth
%% then every
%% connected component of $\textit{Hom}_S(\mc{C},\mc{A},g)$ is proper,
%% and thus projective, 
%% over $S$.  This also holds if $\mc{C}$ is a family of reduced, at
%% worst nodal curves whose generic fibers are smooth.
%% \end{enumerate}
%% \end{lem}

%% \begin{proof}
%% By standard deformation theory, the Zariski tangent space of the fiber
%% of $\textit{Hom}_S(\mc{C},\mc{A},g)$ over $S$ at a point
%% $F_s:\mc{C}_s\rightarrow \mc{A}_s$ equals
%% $$
%% H^0(\mc{C}_s,F_s^* T_{\mc{A}_s}\otimes \mc{I}_{Z_s}),
%% $$
%% where $T_{\mc{A}_s}$ is the tangent sheaf of $\mc{A}_s$ over
%% $\kappa(s)$, and where $\mc{I}_{Z_s}$ is the ideal sheaf of $Z_s$ in
%% $\mc{C}_s$.  Because $\mc{A}_s$ is an Abelian variety, $T_{\mc{A}_s}
%% \cong \OO_{\mc{A}_s}^{\oplus d}$ where $d=\text{dim}(\mc{A}_s)$.  Thus
%% the Zariski tangent space is
%% $$
%% H^0(\mc{C}_s,\mc{I}_{Z_s})^{\oplus d}.
%% $$
%% Because $\mc{C}_s$ is geometrically integral over $\kappa(s)$, and
%% because $\mc{Z}_s$ is nonempty,
%% $$
%% h^0(\mc{C}_s,\mc{I}_{Z_s}) = 0.
%% $$
%% Thus $\textit{Hom}_S(\mc{C},\mc{A},g)$ is unramified over $S$.

%% \mni
%% By limit theorems and 
%% the valuative criterion of properness for Noetherian schemes, to prove
%% the connected components of 
%% $\textit{Hom}_S(\mc{C},\mc{A},g)$ are proper over $S$, it suffices to
%% prove the following.  
%% If $S = \SP R$ for a DVR $R$ with fraction field
%% $K$, the restriction map
%% $$
%% \textit{Hom}_{\SP R}(\mc{C}_R,\mc{A}_R,g_R) \rightarrow
%% \textit{Hom}_{\SP K}(\mc{C}_K,\mc{A}_K,g_K)
%% $$
%% is surjective.  
%% Moreover, by the conditions on the fibers of the original family, we
%% may assume either all fibers of $\mc{C}$ over $\SP R$ are smooth or
%% all fibers are reduced, at worst nodal curves with smooth generic
%% fiber.  

%% \mni
%% Let $F_K:\mc{C}_K \rightarrow \mc{A}$ be an
%% $S$-morphism restricting to $g_K$.  If the fibers of $\mc{C}$ are
%% smooth over $S$, then $\mc{C}$ is regular, and thus has \emph{uniruled
%%   modifications} in the sense of \cite[Definiton VI.1.6]{K} by
%% Abhyankar's theorem, cf. \cite[Lemma VI.1.6.2.1]{K}.  
%% If the fibers are reduced,
%% at worst nodal curves whose generic fiber is smooth, then $\mc{C}$ is
%% normal and
%% has
%% finitely many $A_n$-singularities.  Thus, by resolution of
%% $A_n$-singularities, there exists a proper, birational morphism
%% $\widetilde{\mc{C}} \rightarrow \mc{C}$  whose exceptional set is a
%% union of finitely many trees of rational curves.  Thus again $\mc{C}$
%% has uniruled modifications.  By \cite[Theorem VI.1.9]{K}, either $F_K$
%% extends to a regular morphism on all of $\mc{C}$, or else
%% the exceptional locus of the rational morphism maps onto 
%% a uniruled subvariety of the closed
%% fiber $\mc{A}_0$ of $\mc{A}$.  Since the closed fiber is an Abelian
%% variety, it contains no uniruled subvarieties, thus $F_K$ extends to a
%% regular morphism $F_R$ on all of $\mc{C}$.  Because $\mc{C}$ is separated,
%% the extension is unique and is an $S$-morphism.  Because $Z$ is
%% separated and $Z_K$ is dense in $Z$, also $F_R|_Z = g_R$.  
%% \end{proof}

%% \begin{proof}[Proof of Proposition ~\ref{prop-const}]
%% Let $S$ be a one-dimensional, regular $k$-scheme, let $0\in S$ be a $k$-point,
%% and 
%% let $\mc{C}$ be a closed subscheme of $S\times B_\text{smooth}$ which
%% is flat over $S$, whose general fiber is smooth, and such that
%% $\mc{C}_0$ is a very general triangle curve $C_0$ as in Lemma
%% ~\ref{lem-const1}.  Also, after making a finite base change of $S$ if
%% necessary, we may assume there exists an $S$-morphism $\sigma:S\rightarrow
%% \mc{C}$ such that $\sigma(0) = q$.  Denote $\sigma(S)$ by $Z$ and
%% denote by $g:Z\rightarrow S\times_k A$ the unique $S$-morphism whose
%% projection onto $A$ is the constant morphism with value $0$.  Denote
%% by 
%% $$
%% \textit{Hom}_S(\mc{C},S\times_k A,g)
%% $$
%% the relative Hom-scheme as in Lemma ~\ref{lem-const2}. 
%% For each 
%% $$
%% F\in \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0))
%% $$
%% define
%% $$
%% F_\sigma:S\times_k B \rightarrow S\times_k A, \ F_\sigma(s,b) = (s,
%% F(b)- F(\sigma(s))).
%% $$
%% The restriction of $F_\sigma$ to $\mc{C}$ 
%% is an $S$-morphism restricting on $Z$ to $g$.  The association
%% $F\mapsto F_\sigma$ defines an $S$-morphism
%% $$
%% S\times_k
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0)) 
%% \rightarrow \textit{Hom}_S(\mc{C},S\times_k A,g).
%% $$

%% \mni
%% By Lemma ~\ref{lem-const2}(i), $\textit{Hom}_S(\mc{C},S\times_k A,g)$
%% is unramified over $S$.  Therefore, for every $F$, the image of
%% $S\times{F}$ is an irreducible component of
%% $\textit{Hom}_S(\mc{C},S\times_k A,g)$.  By Lemma ~\ref{lem-const1},
%% every element in
%% the fiber of $\textit{Hom}_S(\mc{C},S\times_k A,g)$ over $0$
%% is contained in
%% one of these irreducible components.  By Lemma ~\ref{lem-const2}(ii),
%% $\textit{Hom}_S(\mc{C},S\times_k A,g)$ is projective over $S$.  Thus
%% every irreducible component which dominates $S$ actually surjects to
%% $S$, and thus intersects the fiber over $0$.  Therefore the
%% irreducible components which dominate $S$ are precisely the images
%% $S\times{F}$ as $F$ varies among elements in
%% $\text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0))$.  For a very
%% general point $t$ of $S$, the fiber of
%% $\textit{Hom}_S(\mc{C},S\times_k A,g)$ over $t$ intersects only
%% irreducible components which dominate $S$.  Thus, the restriction
%% map
%% $$
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},\sigma(t)),(A,0))
%% \rightarrow \text{Hom}_{k-\text{sch}}((\mc{C}_t,\sigma(t)),(A,0))
%% $$
%% is surjective.  For the same reason as in the case $t=0$, this map is
%% injective for $t$ very general.  Therefore the restriction map is a
%% bijection.

%% \mni
%% There is a natural exact sequence of Abelian groups
%% $$
%% \begin{CD}
%% 0 @>>> \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0)) @>>>
%% \text{Hom}_{k-\text{sch}}(B,A) @>\text{ev}_q >> A(k)
%% \end{CD}
%% $$
%% Because of constant morphisms, the last arrow is surjective, and there
%% is even a canonical splitting
%% $$
%% \text{Hom}_{k-\text{sch}}(B_\text{smooth},A) =
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q),(A,0))\times A(k).
%% $$
%% Similarly, there is a canonical splitting
%% $$
%% \text{Hom}_{k-\text{sch}}(\mc{C}_t,A) =
%% \text{Hom}_{k-\text{sch}}((\mc{C}_t,\sigma(t)),(A,0))\times A(k).
%% $$
%% And there is a
%% commutative diagram 
%% $$
%% \begin{CD}
%% \text{Hom}_{k-\text{sch}}(B_\text{smooth},A) @> \text{res} >>
%% \text{Hom}_{k-\text{sch}}(\mc{C}_t,A) \\
%% @V\cong VV @VV \cong V \\
%% \text{Hom}_{k-text{sch}}((B_\text{smooth},q),(A,0))\times A(k) @>>>
%% \text{Hom}_{k-\text{sch}}((\mc{C}_t,\sigma(t)),(A,0)) \times A(k).
%% \end{CD}
%% $$
%% Since 
%% $$
%% \text{Hom}_{k-\text{sch}}((B_\text{smooth},q), (A,0)) \rightarrow
%% \text{Hom}_{k-\text{sch}}((\mc{C}_t,\sigma(t)),(A,0))
%% $$
%% is a bijection, the bottom horizontal map in the diagram is a
%% bijection.  Since the vertical maps are also bijections, the top
%% horizontal map is also a
%% bijection.  Therefore, for $t$ very general, the restriction map
%% $$
%% \text{Hom}_{k-\text{sch}}(B_\text{smooth},A) \rightarrow
%% \text{Hom}_{k-\text{sch}}(\mc{C}_t,A)
%% $$
%% is a bijection.  
%% \end{proof}



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