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\begin{document}

\title[Sections over curves]{Rational connectivity and 
sections of families over curves}

\

\author[T. Graber]{Tom Graber}
\address{Mathematics Department, University of California, Berkeley,
970 Evans Hall \#3840, Berkeley CA 94720-3840, USA}
\email{graber@math.berkeley.edu}
\thanks{T.G. was partially supported by an NSF postdoctoral fellowship.}

\author[J. Harris]{Joe Harris}
\address{Mathematics Department, Harvard University,
1 Oxford st., Cambridge MA 02138, USA}
\email{harris@math.harvard.edu}

\author[B. Mazur]{Barry Mazur}
\address{Mathematics Department, Harvard University,
1 Oxford st., Cambridge MA 02138, USA}
\email{mazur@math.harvard.edu}

\author[J. Starr]{Jason Starr}
\address{Mathematics Department, M.I.T., Cambridge MA 02139, USA}
\email{jstarr@math.mit.edu}
\thanks{J.S. was partially supported by NSF grants DMS-0201423 and
  DMS-0353692.}

\date{\today}

\maketitle

\section{Introduction}

\subsection{Statement of results}

We will work over the field of complex numbers, although our results
hold over any uncountable algebraically closed field of characteristic
zero.  A {\em variety} will be a reduced scheme of finite type, but not
necessarily irreducible.  
Recall that a property is said to hold at a general point of 
a variety $V$ if it holds for all points in a dense open subset 
of $V$.  A property is said to hold at a very general point of
$V$ if it holds at all points in a countable intersection of
dense open subsets of $V$.

A variety $V$ is said to be {\em rationally connected} if two general
points of $V$ can be joined by a rational curve.
In \cite{GHS}, it is proved that a one-parameter family of rationally
connected varieties always has a rational section: explicitly, we have
the following theorem.

\ps

\begin{thm}\label{oldmainth} Let $\pi : X \to C$ be a proper morphism 
of complex varieties, with $C$ a smooth connected curve. 
If the general fiber of
$\pi$ is rationally connected, then $\pi$ has a section.
\end{thm}

The goal of this paper is to state and prove a converse to this
statement (cf. Theorem~\ref{mainth} below).

\

We should first of all discuss what we mean by this, inasmuch as the
literal converse of Theorem~\ref{oldmainth} is clearly false. To this
end, let's focus on the question: under what circumstances does a
family $\pi : X \to B$ of varieties have the property that its
restriction to a general curve $C \subset B$ has a section?

\

This is certainly the case if the family $\pi : X \to B$ has a global
rational section. It is also the case by Theorem~\ref{oldmainth} if
the general fiber of $\pi$ is rationally connected, and by extension
it is the case if $X$ contains a subvariety $Z \subset X$ dominating
$B$ and whose fiber over a general point of $B$ is rationally
connected. (We can think of the case where the family $\pi : X \to B$
has a global rational section as a special case of this, a single
point being a rationally connected variety!) In this paper, we will
prove that in fact these are the only circumstances under which it may
occur.  To make this claim precise, we start by making the 
following definition.

\begin{defn}
Let $\pi : X \to B$ be an arbitrary morphism of complex varieties. By
a \textbf{pseudosection} of $\pi$ we will mean a subvariety $Z \subset
X$ such that the restriction $\pi|_Z : Z \to B$ is dominant with
rationally connected general fiber.
\end{defn}

Our main result is the following.

\begin{thm}\label{mainth}
Let $B$ be any irreducible variety.  
For every positive integer $d$ there exists a
bounded family $\cH_d$ of maps $h:C \rightarrow B$ from smooth
irreducible curves to $B$ such that for any proper morphism $\pi : X
\to B$ of relative dimension $d$ or less, if $h:C \to B$ is a map
parametrized by a very general point of $\cH_d$, the pullback
$$
\pi_C : X_C = X \times_B C \to C
$$
has a section if and only if $\pi$ has a pseudosection.

If $B$ is normal and quasi-projective, we can take $\cH_d$ to
be the family of smooth one-dimensional
linear sections of $B$ under a sufficiently
positive projective embedding.
\end{thm}

What we are saying here, in other words, is that if we have any 
family of varieties $\pi : X \to B$ satisfying the condition that
every one-parameter subfamily has a section, it does so by virtue of
the fact that $X$ contains a family of rationally connected varieties.

\

As a corollary of this, we will at the end of the paper settle a
question left hanging in [GHS]: whether or not the statement of
Theorem~\ref{oldmainth} holds for the larger class of
$\cO$-\textit{acyclic} varieties---that is, varieties $X$ with $H^i(X,
\cO_X) = 0$ for all $i > 0$. In fact, it was suggested by Serre in a
letter to Grothendieck ([GS], p.  152) that this might be the case
(though Serre immediately adds that it is ``sans doute trop
optimiste"). In Section~\ref{enriq}, we show this does not hold:
specifically, by applying Theorem~\ref{mainth} to the universal family
over a parameter space of Enriques surfaces with a particular
polarization, we will deduce the following result.

\begin{cor}\label{enriqcor}
There exists a one-parameter family $X \to C$ of Enriques surfaces
with no rational section.
\end{cor}

While this method produces the {\em existence} of a family of Enriques
surface over a complex curve with no section, it does not provide an
answer to the following natural questions.
\begin{itemize}
\item What are the possible base curves for such families?  (For example
is there a pencil of Enriques surfaces with no section?)
\item Can such a family be defined over a number field?
\item Are there local obstructions to the existence of sections?
\end{itemize}

A recent result of Guillaume Lafon \cite{L} clarifies the situation
greatly.  He constructs Enriques surfaces over $\Q(t)$ that
do not have rational points over $\C((t))$.  

We remark, however, that the Enriques surfaces we produce have points
everywhere locally, so the existence of local obstructions is not
the only reason that Enriques surfaces over function fields can
fail to have rational points.

{\bf Acknowledgments:}  The authors are grateful to the referee.
His thorough remarks improved the mathematical content and clarity of this 
paper.

\section{Stable maps and stable sections}

Our proof of Theorem~\ref{mainth} involves an induction on the
relative dimension of $f:X\rightarrow B$ where the base case
(fiber dimension zero) is proved by a version of the Lefschetz
hyperplane theorem.  In the course of the proof we will need to use
specializations of irreducible curves in $B$.  There are several
possible compactifications of the Chow variety of irreducible curves
in $B$, but the one we will use is the \emph{Kontsevich space of
stable maps}.  The reader who is unfamiliar with stable maps is
referred to the article \cite{FP}.  A stable map to $X$ is a 
morphism $h:C \rightarrow X$ for
which $C$ is a connected, projective curve which has at-worst-nodes as
singularities and such that the morphism $h$ has a finite automorphism
group.

There is an equivalence relation on stable maps which is the obvious
one, and the Kontsevich moduli space of stable maps is the
corresponding coarse moduli space of equivalence classes of stable maps.

\

If $X$ is a quasi-projective variety, we let 
$\M_g(X,\beta)$ denote the Kontsevich space of stable maps
$h:C\rightarrow X$ such that $C$ has arithmetic genus $g$ and such
that the push-forward fundamental class $h_*[C]$ equals $\beta \in
H_2(X,\Z)$.  This is a quasi-projective
coarse moduli scheme which is projective if $X$ is projective 
(cf. \cite{FP}).  We will often not need the
decorations, so let $\M(X) := \cup_{g,\beta} \M_g(X,\beta)$ denote the
space of all stable maps to $X$.  


Given a morphism of quasi-projective 
schemes $\pi : X \to B$, there is an
induced map on Kontsevich spaces, $\M (\pi) : \M(X) \to \M(B)$.
(Technically we must restrict to those stable maps with $g>1$ or with
$\pi_*\beta \neq 0$, but this will always be the case for us when we
apply $\M (\pi)$.)  
This map
takes the moduli point $h : C\to X$ to the point $\pi \circ h : \tilde
C \to B$, where $\tilde C$ is the stabilization of $C$ relative to the
morphism $\pi \circ h$.  It is obtained by contracting those
components of $C$ of genus 0 which are contracted by $\pi \circ h$ and
meet the rest of $C$ in fewer than 3 points.  If $\pi$
is projective, then $\M(\pi)$ is projective on each connected component
of $\M(X)$.


\begin{defn}If $h:C \to X$ and $h':C' \to X$ are
stable maps, we say that $h'$ is a \textbf{submap} of $h$ if 
there is an element $i\in \M(C)$ such that $[h']=\M(h)([i])$.
\end{defn}

This terminology is suggested by the special case where 
$h'$ factors as a composition of $h$ with a closed embedding
$i: C' \to C$.  We use the more general notion for convenience.

The following definition makes precise what it means for
one stable map to be a submap of a specialization of another stable
map.

\begin{defn}

Given a quasi-projective variety $X$, and
 two families of stable maps to $X$, $D$ and $E$, 
we say
that $D$ \textbf{dominates} $E$ if a general map parametrized by $E$ can be
realized as a submap of a specialization of maps parametrized by $D$.
Precisely, for a general point of 
$E$ parametrizing a stable map $h:C\rightarrow X$, we can find a family
of stable maps over $\spec (\C[[t]])$ such that $h$ is a submap of the
map over the special fiber, and the map over the generic fiber is
pulled back by a morphism from $\spec \left(\C((t)) \right)$ to $D$.
\end{defn}

Although the definition here requires only that a general
map parametrized by $E$ be obtainable as a submap of a 
specialization, it
is equivalent to require that {\em every} map parametrized
by $E$ arises in this way.  To see this, consider the family
of stable maps parametrized by $D$.

$$\xymatrix{
\cC_D \ar[r]^{h_D} \ar[d]_{\pi_D}& X\\
D & }
$$

Now define $\M^0(\cC_D)$ to be the open and closed subset of
$\M(\cC_D)$ parametrizing stable maps to fibers of $\pi_D$.
The condition that $E$ is dominated by $D$ is equivalent to
the statment that a general point of $E$ maps to a point in $\M(X)$
which is in the closure of $\M(h_D)(V)$ for some irreducible component
$V$ of $\M^0(\cC_D)$.  However, if a general point of $E$ maps
to a point in such a closure, then so does every point of $E$.



\


We need a criterion for when a family of stable maps to a projective
variety is dominated by a family of embedded complete intersection
curves.  We will repeatedly make use of the following criterion.

\begin{lm}\label{smoothinglm}
Suppose $E$ is a family of stable maps to a normal, irreducible,
projective variety
$B$ such that for a general map $h: C \to B$ parametrized by $E$,
there is an open subset $U\subset B$ contained in the smooth locus of
$B$ and such that:
\begin{enumerate}
\item $h^{-1}(U)$ is dense in $C$,
\item $h^{-1}(U)$ contains all the nodes of $C$, and
\item the restricted map $h^{-1}(U) \rightarrow U$ is a closed embedding.

\end{enumerate}
Then $E$ is dominated by the family of one dimensional linear sections
of $B$ under a sufficiently positive projective embedding.


\end{lm}
  
\bpf 
Let $h:C\to B$ be a general element of our family.  We embed $B$ in a
projective space and find an integer $a$ such that the ideal of the
reduced image curve $h(C)$ is generated by polynomials of degree $a$.
The sufficiently positive embedding required is the $a$th Veronese
re-embedding.  Now we know that we can realize $h(C)$ as an
intersection of hyperplanes.  Since $h(C)$ is a local
complete intersection in a neighborhood of the image of 
each of the nodes of $C$, 
if we choose $\dim(B)-1$ generic
hyperplane sections of $B$ which contain $h(C)$, their intersection
will agree with $h(C)$ in a neighborhood of the image each of these nodes.
Moreover, by Bertini's Theorem, this intersection will be smooth
away from $h(C)$.  We conclude that the intersection
of $\dim(B)-1$ generic hyperplane sections of $B$ containing
$h(C)$
will be a generically reduced curve $C'$ which contains $h(C)$ as a
subcurve and such that there exists an open set $U'\subset U$ containing
the images of all of the nodes of $C$ such that 
$h^{-1}(U') \rightarrow U' \cap C'$ is an isomorphism.  

\

Now we choose any one parameter family of smooth complete
intersections in $B$ whose flat limit is $C'$, i.e., a morphism from $\spec
(\C[[t]])$ to the Hilbert scheme of complete intersections in $B$
whose general fiber maps to a smooth complete intersection and whose
special fiber maps to $C'$.  We think of the general fiber as a stable
map and perform stable reduction to the corresponding map $\spec
(\C[[t]]) \rightarrow \M(B)$.  Denote the special fiber of the stable
reduction by $\overline{h}:\overline{C} \rightarrow B$.
Since $U'\cap C'$ is already at-worst-nodal, there is an open
subset $V\subset \overline{C}$ such that $\overline{h}:V \rightarrow U'
\cap C'$ is an isomorphism.  In other words, we have a factorization
$i:h^{-1}(U') \rightarrow \overline{C}$ of $h:h^{-1}(U') \rightarrow
B$.  Since every point of $C$ in $C-h^{-1}(U')$ is smooth, we can apply
the valuative criterion of properness to extend this factorization to
a morphism $i:C \rightarrow \overline{C}$.  So $h:C \rightarrow B$ is
a submap of $\overline{h}:\overline{C} \rightarrow B$, which shows
that $E$ is dominated by the family of smooth curves in $B$ which are
complete intersections of $d-1$ hyperplanes. 

\epf

We remark that the lemma above is not the most general result, but to
prove a stronger version would lead us too far astray.  We leave
it to the interested reader to prove that in the above lemma it
suffices to assume that for the general map $h:C\rightarrow B$
parametrized by $E$, the preimage of the smooth locus,
$h^{-1}(B_{\text{smooth}})$, is a dense open set which contains every
node of $C$.

\

Although our main concern is to understand sections of a map $\pi : X
\to B$ over smooth curves in $B$, the specialization methods we use
force us to consider the more general notion of stable maps.
Similarly, we need to replace the notion of section over a curve with
an object which specializes as we specialize the base curve.  

\

\begin{defn} Given a morphism $\pi: X \to B$, and
a stable map $h:C \to B$, we define a \textbf{stable section} of $\pi$
over $[h]$ to be a stable map $\tilde{h}:\tilde{C} \to X$ such that
$\M(\pi)([\tilde h])=[h].$
\end{defn}

Notice that for a stable section $\tilde h$, the class
$\pi_*(\tilde{h})_*[\tilde{C}]$ is just $h_*[C]$
and $g(C') = g(C)$.  Since $h$ is a stable map, either $g(C) > 1$ or
$h_*[C]$ is nonzero, i.e., the map $\M(\pi)$ really is defined in a
neighborhood of $[\tilde{h}]$.

\

If $C$ is a smooth connected projective 
curve in $B$, then a stable section of $\pi$ over
$C$ is simply a section of $\pi$ over $C$ with some trees of
$\pi$-contracted rational curves in $X$ attached.  The notion is more
interesting when $C$ has nodes.  In this case, the existence of a
stable section over $C$ does not guarantee the existence of any
sections over $C$ whatsoever.  For example, a typical stable section
$\tilde{h}$ over a curve $C=C_1 \cup C_2$, where $t_1\in C_1$ is glued
to $t_2\in C_2$, would consist of sections $\tilde{h}_1$ and
$\tilde{h_2}$ of $\pi$ over $C_1$ and $C_2$ separately together with a
tree of $\pi$-contracted rational curves in $X$ joining
$\tilde{h}_1(t_1)$ to $\tilde{h}_2(t_2)$ (and some trees of
$\pi$-contracted rational curves attached elsewhere).  Such rational
curves are exactly the sort which are contracted under the
stabilization process associated with $\M(\pi)$.

\

The point of this definition is that given a family of curves in the
base $B$ specializing to some stable map $h:C\rightarrow B$ (possibly
reducible), and given an honest section over the generic curve in this
family, then we cannot conclude the existence of a section of $\pi$
over $h$, but we do conclude the existence of a stable section of
$\pi$ over $h$.  In other words, the existence of a stable section is
preserved under specialization.  This follows immediately from the
properness of the irreducible components of $\M(X)$.  Another
elementary fact is the following lemma.

\begin{lm} If $h':C'\to B$ is a submap of $h:C\to B$, then the existence
of a stable section of $\pi:X \to B$ 
over $h$ implies the existence of a stable section of $\pi$
over $h'$.  
\end{lm}

\bpf
We have the following commutative diagram:
$$
\xymatrix{& \tilde C \ar[rd]^{\tilde h} \ar[d]_q & \\
\hat C \ar[r]^i \ar[dr]_{q'} & C \ar[dr]^h & X \ar[d]^\pi \\
& C' \ar[r]^{h'} & B}
$$
where the morphisms $q$ and $q'$ are isomorphisms 
except for possibly contracting
some trees of rational curves.  To define a stable section of $h'$ one
can simply complete the upper left hand corner of this diagram with a curve
$C''$ giving a stable section of $q$ over $i$.  
To construct such a $C''$, note that 
over each irreducible component of $\hat C$ the pullback of
$\tilde C$ admits a section.  To complete this to a stable section, we simply
need to find, for each node of $\hat C$, a tree of rational curves
in $\tilde C$ connecting the image point of one branch of the node
to the image point of the other.  Since every fiber of $q$ is either
a point or a tree of rational curves, this is trivial.
  
\epf

Taken together, we get the following fundamental lemma.

\begin{lm}\label{specializationlm} 
If $D$ and $E$ are families of stable maps to $B$, with $D$ dominating
$E$ and if a general map parametrized by $D$ admits a stable section,
then so does every map parametrized by $E$.
\end{lm}

As an application, we can strengthen the easy direction of our
main theorem.

\begin{prop}  If $\pi : X \to B$ is a morphism of projective
varieties with $B$ smooth and if $\pi$ admits a pseudosection, then
for any smooth curve $C$, and for any morphism $h: C \to B$, the
pullback family $X_C \to C$ admits a section.
\end{prop}

\bpf 
Since any map from a smooth curve factors through the normalization of
its image, it suffices to prove this statement for maps birational
onto their image (note that since $C$ and $B$ are smooth, such maps
satisfy the hypotheses of Lemma~\ref{smoothinglm}.)  
Let $Z$ be a pseudosection of $\pi$.  We already
know by Theorem~\ref{oldmainth} that the proposition is true for any
smooth curve such that the general fiber of $Z$ over the curve is
rationally connected.  In particular it holds for a generic complete
intersection curve in $B$ under any projective embedding.  
By Lemma~\ref{smoothinglm}, our map $h$
can be realized as a submap of a limit of such curves.  Then
Lemma~\ref{specializationlm} implies that $\pi$ admits a stable
section over $[h]$.  Since $C$ is smooth, this implies that $X_C$
admits a section over $C$.  \epf

\section{Proof of main theorem}

In our proof of Theorem~\ref{mainth}, we begin by assuming that $B$ is
normal and that both $X$ and $B$ are projective.  After handling this
``special'' case, we give the (easy) argument which reduces the
general case to the special case.


\

We will prove the theorem by induction on the relative dimension of
$X$ over $B$.  We start with the case of relative dimension zero.

\begin{prop}\label{dim0}
Let $B\subset \P^n$ be an irreducible normal variety and $\pi : X \to B$ a
generically finite proper morphism.  Then $\pi$ admits a rational
section if and only if $\pi$ admits a section when restricted to a
general one-dimensional linear section of $B$.
\end{prop}

\bpf
It suffices to prove this when $X$ is irreducible and $\pi$ is dominant.  
We are also free to assume that $X$ is normal and $\pi$ is proper, 
since the statement depends only on the birational geometry of $X$. 
Let $b$ be the dimension of $B$, and let $G$ denote the family of
codimension $b-1$ linear subspaces of $\P^n$.  A standard
application of Bertini's Theorem shows that there
is a dense open subset $U_B$ of $G$, such that for any linear space $L$ 
parametrized by $U_B$, the intersection $L\cap B$ is a smooth, one-dimensional,
irreducible subvariety.  Applying a stronger characteristic
zero version of Bertini's Theorem 
such as \cite[III.10.9, Ex III.11.3]{H}  to the morphism
from $X$ to $\P^n$ (obtained by composing $\pi$ with the inclusion), 
we can find a dense open subset $U_X \subset G$ such that
for any linear space $L$ parametrized by a point of $U_X$, the pullback to
$X$ of $L$ is a smooth, irreducible, one-dimensional subvariety.  Choosing
a subspace $L$ parametrized by a point in $U_X \cap U_B$ we find that
the restricted morphism
$$ \pi^{-1}(L\cap B) \to (L\cap B) $$ is a finite morphism
of smooth proper curves, so it
admits a section if and only if this morphism has degree 1.  For a general
$L$, the degree of this morphism agrees with the degree of $\pi$ and
the result follows.

\epf

We remark that it is also possible to prove this lemma by applying a
suitable version of the Lefschetz hyperplane theorem 
for fundamental groups such as \cite[Thm. 3]{ham} or 
\cite[Thm. 1.2, Part II]{gm}.

\

In handling the case of positive relative dimension, one of the main
ingredients needed is the following bend-and-break lemma for sections.

\begin{lm}\label{bendbreaklm}
Let $\pi : X \to C$ be a proper morphism with $C$ a smooth connected curve.  
Let $p\in X$ be an arbitrary point.  If there is a positive dimensional family
of sections of $\pi$ passing through $p$, then there is a rational
curve in $X$ passing through $p$ which is contracted by $\pi$.
\end{lm}

\bpf
This lemma is a version of Mori's Bend-and-Break Lemma from \cite{Mo}
which in turn relies on the Rigidity Lemma \cite[p. 43]{Mu}.

Let $q=\pi(p)$.  Suppose we have a one-parameter family of sections
passing through $p$.  This gives us a rational map 
$f:D\times C \dashrightarrow X$
over $C$ whose restriction to a general fiber $\{d\}\times C$ is a
section passing through $p$.  Suppose, by way of contradiction, that
$f$ is a regular morphism in a neighborhood of $D \times \{q\}$.
Since $f$ contracts $D \times \{q\}$, by the Rigidity Lemma it also
contracts $D\times \{c\}$ for all $c\in C$, i.e., our family is
constant which is a contradiction.  Hence,
$f$ is not regular near $D\times \{q\}$.  So there is at least one
point of indeterminacy in $D \times \{q\}$.  We may form the minimal
blow-up of $D\times C$ necessary to resolve the indeterminacy locus of
$f$.  The exceptional divisor of this blow-up is a tree of rational
curves which intersects the proper transform of $D \times \{q\}$ and
which is mapped to a tree of $\pi$-contracted rational curves in $X$.
Therefore some rational curve in the exceptional divisor maps to a
$\pi$-contracted rational curve which meets $p$.  

\epf


We will apply this lemma in two ways.  The first application is to get
a uniform bound on the dimensions of spaces of sections.  First we
need a definition.

\begin{defn} If $\pi : X \to B$ is a proper morphism, we define the
  \textbf{rational curve locus}, $V(\pi)$, to be the union of all
$\pi$-contracted rational curves in $X$.  
\end{defn}

Let us pause to describe what sort of object $V(\pi)$ is.  If $\pi$
is projective and we
fix a relatively
ample divisor $H$ on $X$, then for each integer $d$, the locus of
all $\pi$-contracted rational curves of $H$-degree 
less than or equal to $d$ is a Zariski closed subset of $X$ by
\cite[Sec. 5.6]{D}.  Hence, we see that $V(\pi)$
is a countable union of closed subvarieties. 


\

The next lemma applies Lemma \ref{bendbreaklm} to give a uniform bound
on the dimensions of spaces of sections.  Let $\pi:X \to C$ be a projective
morphism of relative dimension $d$.  Let $\Sigma$ be an irreducible variety
parametrizing a family of sections of $\pi$.

\begin{lm}\label{dimensionbound}
If there exists a section parametrized by $\Sigma$ whose image
is not contained in $V(\pi)$, then $\dim (\Sigma)\leq d$.
\end{lm}

\bpf
Choose a very general point $c$ of $C$ and set $X_c=\pi^{-1}(c)$. 
Let $ev_c : \Sigma \to X_c$ be
the map which evaluates a section at $c$.  Our hypotheses ensure that
$\dim (X_c) = d$ and that $ev_c(\Sigma)\not\subset V(\pi)$.
Lemma~\ref{bendbreaklm} then implies that $ev_c$ is generically finite
onto its image, yielding the desired bound.  

\epf

\

Let $B \hookrightarrow \PP^N$ be a closed immersion.  Let $D
\subset \text{Grass}(N-b+1,\PP^N)$ be the dense open subset
parametrizing linear spaces $\Lambda$ such that $\Lambda \cap B$ is
smooth, irreducible, and one-dimensional.
Let $\rho_D:C_D \rightarrow D$ denote the
universal family of intersections $\Lambda \cap B$ 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}
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 in $D$.
\end{enumerate}

The \textbf{space of triangles}, 
is the locally closed subvariety $T(B) \subset C_D^{(2)} \times
C_D^{(2)} \times C_D^{(2)}$ 
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 $T(B)$ 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}

Of course, $T(B)$ depends not only on $B$, but also on a choice of
projective embedding of $B$.   
There is a morphism $\text{ev}:T(B) \rightarrow B \times B \times B$
which maps 
a triple 
$$
( (C_1,q_{1,3},q_{1,2}),
(C_2,q_{2,1},q_{2,3}), (C_3,q_{3,2},q_{3,1}) )
$$ 
to
$(q_{1,2},q_{1,3},q_{2,3})$.  

\

Let $q\in B$ be a point.  
Define $D_q \subset D$ to be the closed subset where $q \in h(C)$.
Define $T(B)_q \subset T(B)$ to be
the closed set where $q \in h(C_1 \cup C_2 \cup C_3)$.  
For each $i=1,2,3$ define
$T(B)_{q,i} \subset T(B)_q$ to be the closed subset where $q \in
h(C_i)$.  

\begin{lm} \label{TDirred}
\begin{enumerate}
\item
The morphism $\text{ev}$ is flat with irreducible
fibers.
\item
The variety $T(B)$ is irreducible.
\item
For each $i=1,2,3$, the variety $T(B)_{q,i}$ is irreducible.  
\item
The triangle associated to a general point of $T(B)$ satisfies the
hypotheses of Lemma \ref{smoothinglm}.
\end{enumerate}
\end{lm}

\begin{proof}
Let $\Delta \subset B \times B\times B$ denote the union of the three
big diagonals, i.e., the set of triples $(q_{1,2},q_{1,3}, q_{2,3})$
such that some $q_{i,j} = q_{i',j'}$.
Given $(q_{1,2}, q_{1,3}, q_{2,3}) \in B \times B \times B - \Delta$,
the fiber of {\it ev}
over this point is an open subset of the product of Grassmannians
which parametrizes triples $(\Lambda_1,\Lambda_2,\Lambda_3) \in D
\times D \times D$ such that $\text{span}(q_{1,2},q_{1,3}) \subset
\Lambda_1$, $\text{span}(q_{2,1}, q_{2,3}) \subset \Lambda_2$ and
$\text{span}(q_{3,1}, q_{3,2}) \subset \Lambda_3$.  So $\text{ev}:T(B)
\rightarrow B \times B \times B - \Delta$ is an open subset of a fiber
product of three $\text{Grass}(N-b-1,\PP^{N-2})$-bundles.  Therefore
it is flat and has irreducible fibers.

\

Statement $(2)$ follows from $(1)$ and the fact that $B \times B
\times B$ is irreducible.  Statement $(3)$ follows by an argument similar
to that for
$(1)$.  For definiteness, suppose $i=1$.  There is a projection
$\text{pr}_1: T(B)_{q,1} \rightarrow D$
$$
( (C_1,q_{1,3},q_{1,2}),
(C_2,q_{2,1},q_{2,3}), (C_3,q_{3,2},q_{3,1}) ) \mapsto C_1.
$$
The image is a subset of the
subvariety $D_q$.  This subvariety is isomorphic to
an open subset of $\text{Grass}(N-b,\PP^{N-1})$ and so is irreducible
(possibly empty).  Define $C_{D,q}^{(2)}$ to be the preimage of $D_q$
in $C_D^{(2)}$.  This is also irreducible since 
$C_D^{(2)} \rightarrow D$ is flat with irreducible
fibers.  There is a projection $T(B)_{q,1} \rightarrow
C_{D,q}^{(2)}$.  
For each
$(C_1,q_{1,3},q_{1,2}) \in C_{D,q}^{(2)}$ the fiber in $T(B)_{q,1}$ is
an open subset of a fiber product of two
$\text{Grass}(N-b-1,\PP^{N-2})$-bundles over $B$: the map to $B$
corresponds to the choice of $q_{2,3}$ and the Grassmannian bundles
correspond to the choice of $\Lambda_2$ and $\Lambda_3$ containing
$\text{span}(q_{2,1}, q_{2,3})$ and $\text{span}(q_{3,1},q_{3,2})$
respectively.  Thus $T(B)_{q,1} \rightarrow C_{D,q}^{(2)}$ is flat
with irreducible fibers.  Therefore $T(B)_{q,1}$ is
irreducible.    

\

Finally $(4)$ is trivial to verify.
\end{proof}





\

Our second application of Lemma~\ref{bendbreaklm} is in the
proof of the following lemma, which is the main step in the proof of 
Theorem~\ref{mainth}.

\begin{lm}\label{trianglelm}
Let $\pi:X \to B$ be a morphism of projective varieties with $B$ normal
and irreducible, 
and let $p \in X$ be any point such that $p \notin V(\pi)$.  Suppose also
that $p$ is not contained in the closure of the image of any rational
section of $\pi$. 
Then a very general triangle passing through $q=\pi(p)$ admits
no stable sections passing through $p$.
\end{lm}

\bpf
It suffices to exhibit a single triangle with this property.  
Choose an irreducible subfamily $H\subset D$ of curves
passing through $q$, such that for general $b \in B$ a finite (but
positive) number of members of $H$ pass through $b$.  Our strategy 
is to show that if we construct a triangle by choosing two
very general members of $H$ (which will necessarily intersect at $q$)
as $C_1$ and $C_2$,
and a very general member of $D$ which intersects $C_1$ and $C_2$
as $C_3$, then the result will be a triangle satisfying
the desired property.

We construct a
subset $\Omega \subset X$ which is a countable union of subvarieties
of $X$ in the following way.  For every finite type family of sections
of $\pi$ over curves in $H$ which take the value $p$ at $q$, we
have a map from the base of this family to $\M(X)$.  Form the closure
of the image of this map, and define $\tilde\Omega$ to be the
countable union of all such closed subvarieties of $\M(X)$ arising from
the countably many Chow varieties of sections as above.  Notice that
$\tilde\Omega$ is not necessarily quasi-compact, but it is a closed
subset of $\M(X)$ (which is also not quasi-compact).

\

We can restrict the universal curve of $\M(X)$ over $\tilde\Omega$,
and there is a map from the total space of this universal curve to
$X$.  We define $\Omega$ to be the image of this map, so $\Omega$ is a
countable union of closed subvarieties of $X$.  Let $\tilde\Omega_0$
be any irreducible component of $\tilde\Omega$ and let $\Omega_0
\subset X$ be the (closed) image of the universal curve over
$\tilde\Omega_0$.  Notice that 
$p\in\Omega_0$, since it is in the image of each section parametrized
by a general point of $\tilde\Omega_0$.

\

Consider the restricted morphism $\M(\pi):\tilde\Omega_0 \rightarrow
\M(B)$.  The general point of $\tilde\Omega_0$ parametrizes a section over a
member of $H$, so the image of $\tilde\Omega_0$ under $\M(\pi)$ is contained
in the closure $\overline{H}$ of $H$ in $\M(B)$.  By Lemma~\ref{bendbreaklm},
the morphism 
$\M(\pi):\tilde\Omega_0 \rightarrow \overline{H}$ is generically finite,
since $p\notin V(\pi)$.  
Therefore the map from the universal curve over $\tilde\Omega_0$ to the
universal curve over $\overline{H}$ is generically finite.  By
construction, the evaluation morphism from the universal curve over
$\overline{H}$ to $B$ is generically finite.  So finally we conclude
the restricted morphism $\pi:\Omega_0 \rightarrow B$ is generically
finite, i.e., for a general point $b\in B$ there are only finitely many
preimages of $b$ in $\Omega_0$.  


\


\begin{claim}There are no rational sections of $\pi$ whose image is
contained in $\Omega$.
\end{claim}
We will prove this by showing that the closure of any rational 
section $\rho$ of $\Omega$
must contain $p$, in violation of the hypotheses on $\pi$.
First
observe that $\rho$ factors through one of the subsets
$\Omega_0\subset \Omega$.  Now for a general point $b$ in $B$,
$\rho:B \dashrightarrow \Omega_0$ is actually regular in a neighborhood of
$b$ and the image $\rho(b)$ lies on some honest section $\tilde h:C
\rightarrow X$ over a curve $C$ in $H$ which contains $q$ and $b$ and
such that $\tilde h(q)=p$.  Since $\pi:\Omega_0 \rightarrow B$ is
unramified at $\rho(b)=\tilde h(b)$ (by genericity of the choice of $b$), 
we have that $\tilde h:C
\rightarrow X$ and $\rho|_C: C \dashrightarrow X$ are equal as rational
maps.  We conclude that
$\tilde h:C \rightarrow X$ factors through the closure of the image of
$\rho$, in particular $p=\tilde h(q)$ lies on the closure of the image
of $\rho$.  

\

Applying Lemma \ref{dim0} to each of the countably many components
of $\Omega$, we now conclude that for a very general curve $C_3$ in
$D$, there is no section of 
$$\pi: \pi^{-1}(C_3) \to C_3$$ 
whose
image lies in $\Omega$.

\

We know that for each irreducible component $\Omega_0$ of
$\Omega$, $p$ is in $\Omega_0 - V(\pi)$.  Thus $\Omega_0 \cap V(\pi)$
is a proper closed subset of $\Omega_0$, and by the finiteness
of $\pi:\Omega_0 \to B$, we conclude that for
a very general $b$ in $B$,  
$\Omega \cap V(\pi)\cap\pi^{-1}(b) = \emptyset$.  Choose a very
general curve $C_3$ in $D$ as above, and choose a very general
point $r$ on $C_3$.  Then $\Omega\cap \pi^{-1}(r)$ is a
countable set disjoint from $V(\pi)$.
Given any point in this set, there are at most countably many sections
of $\pi$ over $C_3$ which take this value at $r$ by Lemma \ref{dim0}.  
Hence there are at most
countably many sections of $\pi$ over $C_3$ whose value at
$r$ is contained in $\Omega$.  The image of any such section 
cannot be contained in
$\Omega$ and thus meets $\Omega$ in at most countably many points
with countably many images in $C_3$.  Choosing another
point $s\in C_3$ not to lie in any
of these countably many countable sets, we conclude that for any
section $\tilde{h}_3$ of $\pi$ over $C_3$ such that $\tilde{h}_3(r)$
is in $\Omega$, we have that $\tilde{h}_3(s)$ is \emph{not}
in $\Omega$.

\

Now we take our triangle to be $C=C_1 \cup C_2 \cup C_3$ where $C_1$
and $C_2$ are members of $H$ which join $q$ to $r$ and $q$ to $s$
respectively.  By way of contradiction, suppose there is a stable
section $\tilde{h}$ of $\pi$ over $C$ whose image contains $p$.  As we
have discussed, such a stable section consists of honest sections
$\tilde{h}_1$, $\tilde{h}_2$ and $\tilde{h}_3$ over $C_1$, $C_2$, and
$C_3$ respectively, perhaps with some trees of $\pi$-contracted rational
curves attached which connect $\tilde{h}_1(r)$ to $\tilde{h}_3(r)$,
which connect $\tilde{h}_2(s)$ to $\tilde{h}_3(s)$ and which connect
$\tilde{h}_1(q)$ and $\tilde{h}_2(q)$ to $p$. 
Since $p$ is not contained in $V(\pi)$, there
is no tree of $\pi$-contracted rational curves which meets $p$.
Therefore $\tilde{h}_1(q) = \tilde{h}_2(q) = p$. 
By the definition of $\Omega$, the images
$\tilde{h}_1(C_1)$ and $\tilde{h}_2(C_2)$ are therefore 
necessarily contained in
$\Omega$.  Since $r$ and $s$ are very general on $B$, both $\Omega
\cap V(\pi) \cap \pi^{-1}(r)$ and $\Omega \cap V(\pi) \pi^{-1}(s)$ are 
empty, hence 
no $\pi$-contracted rational curves over $r$ or $s$ which meet
$\Omega$.  In particular, there is no tree of $\pi$-contracted 
rational curves
which meets either $\tilde{h}_1(r)$ or $\tilde{h}_2(s)$.  So we must
have $\tilde{h}_1(r) = \tilde{h}_3(r)$ and $\tilde{h}_2(s) =
\tilde{h}_3(s)$.    The upshot is that,
after pruning any extraneous trees of $\pi$-contracted rational
curves, we have that $\tilde{h}$ is an honest section of $\pi$ over
the reducible curve $C$.

\
 
But now we have our contradiction: we have seen that for any section
$\tilde{h}_3$ of $\pi$ over $C_3$ such that $\tilde{h}_3(r)$ is
contained in $\Omega$, then $\tilde{h}_3(s)$ is not contained in
$\Omega$.  On the other hand we have by the last paragraph that
$\tilde{h}_3(r) = \tilde{h}_1(r)$ is contained in $\Omega$ and also
$\tilde{h}_3(s) = \tilde{h}_2(s)$ is contained in $\Omega$.  Therefore
we conclude there is no stable section $\tilde{h}$ of $\pi$ over $C$.

\epf

Of course, Lemma \ref{trianglelm} 
tells us nothing in case the fibers of $\pi$
are uniruled.  Thanks to a construction of Campana and 
Koll\'ar-Miyaoka-Mori and
using Theorem~\ref{oldmainth}, we can always reduce to the case that
the fibers of $\pi$ are nonuniruled.

\begin{defn} Given a morphism $\pi : X \to B$, 
the \textbf{relative MRC fibration} is a dominant
rational map $\phi : X \dashrightarrow W$ that fits
into a commutative diagram
$$\xymatrix{
X \ar@{-->}[r]^\phi \ar[d]_\pi& W\ar[ld]^{\pi'}\\
B & }
$$
such
that a general fiber of $\phi$ is rationally connected and a general
fiber of $\pi'$ is not uniruled.
\end{defn}

Note that although the general fiber of a rational map is not
well-defined, it is well-defined up to birational
equivalence.  Hence the concept of the rational connectivity
of the general fiber of a rational map makes sense.
Since the existing references only establish the existence of the
absolute MRC fibration (the case where $B=\Spec (K)$ for some
field $K$), we point out that the simplest way
to construct this relative version is to simply use the
absolute MRC fibration for the fiber over the generic point of $B$, and
choose for $W$ any $B$-model for the resulting $K$-scheme.  A very
readable account of
the construction in the case of $B=\Spec (K)$ can be found in 
\cite{D}.  While
$K$ is assumed to be algebraically closed there, this hypothesis is
never actually used in the construction.

Finally, we warn the reader that the definition of the MRC fibration
is usually more complicated.  The equivalence of our definition
with the usual one is an easy consequence of Theorem \ref{oldmainth}.

\

Before applying Lemma~\ref{trianglelm} to the proof of 
our main theorem, we note a
corollary which is interesting in its own right.

\begin{cor} 
If $B$ is a normal, connected, quasi-projective variety, 
and $\cH$ is any family
of smooth curves in $B$ which dominates the family of triangles
$T(B)$,  then for any projective morphism $\pi : X \to B$, the
following two conditions are equivalent.

\begin{enumerate}
\item A general point of $X$ lies in a pseudosection.
\item For a general curve $C$ parametrized by $\cH$, a general
point of $X_C = \pi^{-1}(C)$ lies on a section of $\pi:X_C \to C$.
\end{enumerate}
Note that the existence of such a family $\cH$ is ensured by
Lemma~\ref{smoothinglm}.
\end{cor}

\bpf
Direction $(1) \Rightarrow (2)$ follows from a stronger version of
Theorem~\ref{oldmainth}:  if $\pi : Z \to C$ is a proper morphism from
an irreducible variety to a
smooth curve, and if a general fiber of $\pi$ is rationally connected,
then a general point of $Z$ is contained in a section of $\pi$,
cf. ~\cite[2.13]{KMM} and also ~\cite[Thm. IV.6.10]{K}. 

\

Direction $(2) \Rightarrow (1)$
is more interesting.  Suppose first that a general fiber of $\pi$ is not
uniruled and that $(2)$ holds.  
By the properness of the spaces
of stable sections of bounded degree, for every curve $C$ parametrized by
$\cH$, every point of $X_C$ lies on a stable section.  Therefore
for every triangle $h:C \rightarrow B$ in $T(B)$, every point of $X_C$
lies on a stable section.  Also a very general point of $X$ is not
contained in $V(\pi)$.  So, by Lemma~\ref{trianglelm}, a very general
point of $X$ is contained in a rational section.  It follows from a
straightforward uncountability argument that a general point of $X$ is
contained in a rational section, so $(1)$ holds.

\

Next suppose that the
general fiber of $\pi$ is uniruled and that $(2)$ holds.
Let  
$$\xymatrix{
X \ar@{-->}[r]^\phi \ar[d]_\pi& W\ar[ld]^{\pi'}\\
B & }
$$
be a relative
MRC fibration.  Let $f:X' \rightarrow X$ be a proper birational
morphism such that $f\circ \phi$ extends to a regular morphism
$\phi':X' \rightarrow W$.  
Let $p\in X$ be a point over which $f$ is an isomorphism.  Let $C$ be
a curve in $\cH$ which contains $\pi(p)$.  Let $s:C \rightarrow
X_C$ be a section which contains $p$.  Then the rational
map $f^{-1} \circ s:C \dashrightarrow X'_C$ is defined.  Since
$\pi$ is proper and $C$ is smooth, this rational map
extends to a section of $X'_C$.  
So $(2)$ holds for $X'
\rightarrow B$.  It suffices to prove that $(1)$ holds for $X'$, 
since the image of a pseudosection in $X'$ is a pseudosection
in $X$.  Thus, we will assume from now on that $\phi$ is a regular
morphism.


\

For any section $s:C \rightarrow X_C$, the
composition $\phi \circ s:C \rightarrow W_C = C \times_B W$ is a section of
$W_C$.  Since a general point of $X_C$ is contained in a section $s$,
a general point of $W_C$ is contained in a section $\phi \circ
s$, i.e., $(2)$ holds for $W$.
Since a general fiber of $\pi'$ is not uniruled, a general point
of $W$ is contained in a rational section $Z \subset W$.  The preimage
$\phi^{-1}(W)$ of a rational section through a general point of $W$ is
a pseudosection of $\pi$ passing through a general point of $X$.  So
$(1)$ holds for $X$.
\epf

Our proof of Theorem~\ref{mainth} proceeds similarly.  First we will
prove the result for maps whose general fiber is not uniruled, and
then we will handle the general case by appealing to the relative MRC
fibration.  In addition we will use an induction on the relative
dimension of $X$ over $B$.  We have already considered the case of
fiber dimension zero in Proposition~\ref{dim0}, thus suppose that $d >
0$.  By way of induction, assume that we have already constructed a
family $\cH_{d-1}$ of smooth curves in $B$ which cover $B$ and such
that for any morphism $\pi : X \to B$ of relative dimension less than
$d$, $\pi$ admits a section when restricted over a very general curve
in $\cH_{d-1}$ if and only if $\pi$ admits a pseudosection.  We
construct $\cH_d$ as follows.  First we construct a family of
reducible nodal curves by letting $\tilde \cH_d$ be the family of maps
$f:C\to B$ of the form $C= C_0 \cup C_1 \cup \cdots \cup C_{d+1}$,
where $[f_{C_0}]$ is a member of $\cH_{d-1}$ and the other $C_i$,
$i=1,\dots, d+1$ are triangles which
each meet $C_0$ in a single node which is embedded in the smooth locus
of $B$.  Now take $\cH_d$ to be any family of smooth curves that
dominates $\tilde \cH_d$.  By Lemma~\ref{smoothinglm}, we can take
$\cH_d$ to be the family of linear sections of $B$ under a
sufficiently positive projective embedding.

\

We need to check that $\cH_d$ satisfies the desired property.  Namely,
suppose $\pi: X \to B$ is a projective morphism of relative dimension
less than or equal to $d$ which does not admit a pseudosection.  Then
we need to show that over a very general member of $\cH_d$, $\pi$ does
not admit a section.  By Lemma~\ref{specializationlm}, it suffices to
check that over a very general member of $\tilde\cH_d$, $\pi$ does not
admit a stable section.

\

First we will consider the case where the general fiber of $\pi$ is
not uniruled.  In order to later handle the uniruled case, it will be
useful for us to prove a statement that seems stronger than necessary.
In particular, we will prove a statement that applies to morphisms
$\pi$ which do admit pseudosections.
As above, we let $V(\pi)$ be the rational curve locus which is the
union of all $\pi$-contracted rational curves in $X$. This is a
subset of $X$ which is a countable union of subvarieties.  We let
$Y(\pi)$ be the union of $V(\pi)$ and the closures of the
images of all rational sections of $\pi$.
This is also a countable union of subvarieties of $X$.  Note that
Lemma~\ref{trianglelm} says exactly that for any point $p$ in
$X-Y(\pi)$, a very general triangle through $\pi(p)$ admits
no stable sections containing $p$.

\

\begin{lm}\label{technicallm}
Any stable section of $\pi$ over a very general member of
$\tilde\cH_d$ has values over $C_0$ contained in $Y(\pi)$.
\end{lm}

\

Before proving the lemma, we remark that (given the inductive
hypothesis) it immediately implies our theorem in the case where the
fibers are not uniruled.  If $\pi : X \to B$ is a morphism whose
general fiber is not uniruled and which does not admit a pseudosection
then $Y(\pi) = V(\pi)$ is a countable union of {\em proper}
subvarieties of $X$, i.e., it is a countable union of subvarieties
$Y(\pi)_0$ of $X$ such that the fiber dimension of $Y(\pi)_0
\rightarrow B$ is strictly less than $d$. By the induction assumption,
for a very general $C_0$ in $\cH_{d-1}$, there can be no honest
section of $\pi$ over $C_0$ 
contained in any of the subvarieties $Y(\pi)_0$.
Thus, by the lemma, there can be no stable section of $\pi$ over $C =
C_0 \cup C_1 \cup \dots \cup C_{d+1}$.

\

\bpf
We now prove the lemma.  We imagine assembling our very general member
of $\tilde \cH_d$ one component at a time.  Pick a very general $C_0
\in \cH_{d-1}$ and let $\Sigma$ denote the parameter space of all
sections of $\pi$ over $C_0$ which are not contained in $Y(\pi)$.
This is the complement of a countable union of subvarieties 
in a countable union of subvarieties of $\M(X)$.   Denote the irreducible
components of $\Sigma$ by $\Sigma^\alpha$, and
by Lemma~\ref{dimensionbound} we conclude that each 
$\Sigma^\alpha$ has dimension less than or equal to $d$.

\

Our strategy now is simple.  The condition that a section over $C_0$
extends to a stable section over $C_0 \cup C_i$ should impose a condition by
Lemma~\ref{trianglelm}, and so after imposing $d+1$ conditions there
should be no sections left.  To prove this, we consider the chain
$$\Sigma_{d+1} \subset \Sigma_d \subset \cdots \subset \Sigma$$ 
where $\Sigma_i$ is defined to be the subset of $\Sigma$ parametrizing
sections of $\pi$ over $C_0$ which are not contained in $Y(\pi)$ and
which can extended to stable sections 
over $C_0 \cup C_1 \cup \cdots \cup C_i$.  That is,
if we let $q_1, \ldots, q_{d+1}$ be the very general points at which
we attach the triangles, $\Sigma_i$ parametrizes those sections of
$\pi$ over $C_0$ whose value at $q_j$ agrees with the value of some
stable section of $\pi$ over $C_j$ for all $j\leq i$.  This is 
a countable union of closed subsets of $\Sigma$.  We will prove
by induction on $i$ that $\dim(\Sigma_i) \leq d-i$ for each $i$, in
particular $\Sigma_{d+1}$ is empty.

\

We have already seen that every component of $\Sigma=\Sigma_0$ has
dimension at most $d$, so this establishes the base case $i=0$.  By
way of induction, assume that every component of $\Sigma_k$ has
dimension at most $d-k$.  Now we want to show the result for $k+1$.
For any one of the countably many irreducible components
$\Sigma_k^\alpha$ of $\Sigma_k$, for a general point $q_{k+1}$ of $C_0$,
a very general
section of $\pi$ over $C_0$
parametrized by $\Sigma_k^\alpha$ maps
$q_{k+1}$ to a point not in $Y(\pi)$.
Choosing
a very general point $q_{k+1}$, we can arrange that for every
irreducible component $\Sigma^\alpha_k$ of $\Sigma_k$, there is a
section $\tilde{h}^\alpha_0$ in $\Sigma^\alpha_k$ has the property that
$p^\alpha=\tilde{h}^\alpha_0(q_{k+1})$ is not in $Y(\pi)$.

\

Now for each $\alpha$, for a very general triangle $C_{k+1}$ through
$q_{k+1}$, we conclude by Lemma~\ref{trianglelm} that there is no
stable section over $C_{k+1}$ which passes through $p^\alpha$.  So if
we choose a very general triangle $C_{k+1}$, we can arrange that for
every $\alpha$, there is no stable section over $C_{k+1}$ which passes
through any of the points $p^\alpha$.  So none of the sections
$\tilde{h}^\alpha_0$ extend to a stable section over $C_0\cup C_{k+1}$.
Thus, for each $\alpha$, $\Sigma_{k+1}\cap \Sigma_k^\alpha$ 
has dimension strictly less than
$\text{dim}(\Sigma_k^\alpha) \leq d-k$.  Since we have
$$\Sigma_{k+1} = \bigcup_\alpha \left( \Sigma_{k+1} \cap \Sigma_k^\alpha
\right)$$ 
we conclude that every irreducible component of $\Sigma_{k+1}$ has
dimension at most $d-k-1$, as desired.
In particular, we conclude that $\Sigma_{d+1}=\emptyset$,
i.e., over $C_0$ every section of $\pi$ which can be extended to stable
sections over $C$ is contained in $Y(\pi)$.  \epf

\

As discussed above, Lemma~\ref{technicallm} proves the induction step
in case the fibers of $\pi$ are not uniruled.  So to finish the
inductive proof of Theorem~\ref{mainth}, we are left to consider the
case where the fibers of $\pi$ are uniruled.  We argue by
contradiction.

\

By way of contradiction, assume that we have a morphism $\pi : X \to
B$ with no pseudosection, but which admits a section when restricted
to a very general element of $\cH_d$.  
Let $\phi : X \to W$ be the relative MRC
fibration.  We may resolve the indeterminacy locus of $\phi$ by
blowing up:
$$\xymatrix{
X' \ar[d]_f \ar[dr]^{\phi'}& \\
X \ar@{-->}[r]^{\phi \quad} \ar[d]_\pi& W\ar[dl]^{\pi'}\\
B &}
$$
Let $Z\subset X$ denote the fundamental locus of the morphism $f$
(i.e., the image under $f$ of the exceptional divisor of $f$).

\

Notice that the relative dimension of $\pi|_Z:Z\rightarrow B$ is
strictly less than $d$.  Suppose that $\pi|_Z:Z\rightarrow B$ admits a
section when restricted over a very general curve $C$ in $\cH_d$.  By
Lemma~\ref{specializationlm}, we conclude that $\pi|_Z:Z\rightarrow B$
admits a stable section when restricted over a stable map in
$\tilde\cH_d$.  In particular, since every curve in $\cH_{d-1}$ occurs
as the $C_0$-submap of a stable map in $\tilde\cH_d$, we conclude that
$Z\rightarrow B$ admits a section when restricted over a very general
curve $C_0$ in $\cH_{d-1}$.  By the induction hypothesis, this implies
that there is a pseudosection of $\pi|_Z:Z \rightarrow B$.  But, in
particular, this implies there is a pseudosection of $\pi:X
\rightarrow B$ which contradicts our assumption.  So we conclude that
for a very general curve $C$ in $\cH_d$, $\pi|_Z:Z \rightarrow B$
admits no section when restricted over $C$.  On the other hand, our
assumption is that $\pi:X \rightarrow B$ does admit a section over
$C$, so there exists a section over $C$ which is not contained in
$Z$.  This is the same as a rational section of $\pi\circ f:X'
\rightarrow B$ over $C$.  Since $C$ is smooth, 
this rational section of $\pi\circ f$
extends to a regular section of $\pi\circ f$ over $C$.

\

Thus we find that $\pi \circ f$ admits a section over a very
general curve
$C$ in $\cH_d$.  Now if $\pi \circ f$ admits a pseudosection, so
does $\pi$ by simply taking the image of the pseudosection under
$f$.  Therefore we conclude that $\pi\circ f: X'
\rightarrow B$ admits no pseudosection, but it does admit a section
when restricted over a very general curve in $\cH_d$.  Therefore, as
far as deriving a contradiction is concerned, we can replace
$X$ by $X'$.  So from now on we  
assume that $\phi:X \rightarrow W$ is a regular morphism.

\

Let $W' \subset W$ denote the closure of the 
locus of points over which the fiber of
$\phi$ is not rationally connected.  Any rational section of $\pi'$
not contained in $W'$ gives rise to a pseudosection of $\pi$, which
does not exist by hypothesis.  Therefore all rational sections of
$\pi'$ are contained in $W'$.  Applying Lemma~\ref{technicallm}, we
find that over a general member $C$ of $\tilde\cH_d$, any stable
section of $\pi'$ maps $C_0$ into the subset $W'\cup V(\pi')$.  Thus,
any stable section of $\pi$ over $C$ maps $C_0$ into the subset
$\phi^{-1}\left(W'\cup V(\pi')\right)$.

\

On the other hand, $\phi^{-1}\left(W'\cup V(\pi')\right)$ is a
countable union of proper subvarieties of $X$, each of which has
relative dimension at most $d-1$ over $B$.  So by the induction
hypothesis, every section of $\pi:X \rightarrow B$ over the very general
curve $C_0$ in $\cH_{d-1}$ has image which is not contained in
$\phi^{-1}\left(W'\cup V(\pi')\right)$.  
We conclude that over a very general member $C$ of
$\tilde\cH_{d}$, $\pi$ admits no stable section.  By
Lemma~\ref{specializationlm}, $\pi$ admits no section over a very
general member of $\cH_d$, and this is a contradiction of our
assumptions.

\

This establishes the inductive step needed and completes the proof of
Theorem \ref{mainth}.

\epf

\section{The general case}\label{general}

In the last section we proved the main theorem in case $B$ is normal
and quasi-projective and $\pi:X \rightarrow B$ is projective.  In this
section we will show how to reduce the general case to this case.  We
proceed by induction on the dimension.  

\

Suppose that $B$ is an algebraic variety of finite type.  By Chow's
lemma we can find a projective, birational morphism $B_1 \rightarrow
B$ such that $B_1$ is quasi-projective.  By Noether normalization,
the normalization $B_2 \rightarrow B_1$ of $B_1$ is a finite morphism.
Thus $f:B_2 \rightarrow B$ is a projective, birational morphism such
that $B_2$ is quasi-projective and normal.  Let $\cH_d$ be the family
of curves $C$ in $B_2$ constructed in the last section.  The
restriction of $f$ to a general curve in this family 
is a nonconstant morphism,
and hence a stable map.  Therefore, replacing $\cH_d$ by a Zariski
dense open subset, we may consider $\cH_d$ to be a family of stable
maps $h:C\rightarrow B$ with smooth domain $C$.  The claim is that
Theorem~\ref{mainth} holds for $B$ and $\cH_d$.  We will prove this by
induction, but before proceeding to the induction argument we
introduce a little more notation.

\

Suppose that $\pi:X \rightarrow B$ is a proper morphism of relative
dimension at most $d$ which admits no pseudosection.  We need to prove
that for a very general map $h:C\rightarrow B$ in $\cH_d$, $\pi$
admits no section over $h$.  The base change $\pi_2:X\times_B B_2
\rightarrow B_2$ is a proper morphism of relative dimension at most
$d$ which admits no pseudosection, since the image under $\pi_1:X
\times_B B_2 \rightarrow X$ of a pseudosection of $\pi_2$ is a
pseudosection of $\pi$.  By again applying Chow's lemma, we can find a
projective, birational morphism $\phi:X_2 \rightarrow X\times_BB_2$ such that
$\pi_2\circ\phi:X_2 \rightarrow B_2$ is projective.  Any pseudosection
of $\pi_2\circ \phi$ maps under $\phi$ to a pseudosection of $\pi_2$.
Therefore $\pi_2\circ\phi$ admits no pseudosection.

\

Now $\pi_2\circ\phi:X_2 \rightarrow B_2$ satisfies the hypotheses of
the last section.  By the proof of the main theorem in that section,
for a very general curve $C$ in $\cH_d$, $\pi_2\circ\phi$ admits no
section over $C$.  Let $Z\subset X$ denote the fundamental locus of
the birational, projective morphism $X_2 \rightarrow X$, i.e., the
locus over which this morphism is not an isomorphism.  

\

If $d=0$, we are essentially done.  The locus $Z\subset X$ is a proper
subvariety, and since $\pi$ is generically finite, also $\pi(Z)\subset
B$ is a proper subvariety.  If we choose a very general map
$h:C\rightarrow B$ in $\cH_0$, then the image $h(C)$ does not lie in
$\pi(Z)$.  But then any section of $\pi:X \rightarrow B$ over $h$
determines a rational section of $\pi_2\circ\phi: X_2 \rightarrow B_2$
over $C$.  Since $C$ is smooth 
this rational section extends to a regular section.  This
contradicts the result of the last section.  So we conclude that for a
very general map $h:C\rightarrow B$ in $\cH_0$, there is no section of
$\pi:X \rightarrow B$ over this map.

\

Now we proceed by induction via an argument very similar to that
in the end of the last section.  We have established the base case $d=0$,
so we suppose that $d>1$.  By way of induction, we suppose the theorem
has been proved for $d-1$.  Consider $\pi|_Z:Z \rightarrow B$.  This
morphism has fiber dimension at most $d-1$.  By our induction
assumption, we conclude that $\pi|_Z:Z \rightarrow B$ has no section
when restricted over a very general map $h_0:C_0 \rightarrow B$ in
$\cH_{d-1}$.  By Lemma~\ref{specializationlm}, we conclude that
$\pi|_Z:Z\rightarrow B$ has no section when restricted over a very
general map $h:C \rightarrow B$ in $\cH_d$ (since $\cH_d$ dominates
$\cH_{d-1}$).  So if we choose a very general map $h:C\rightarrow B$
in $\cH_d$, then for any section $\tilde{h}:C \rightarrow X$ of $\pi$
over $h$, we have that $\tilde{h}(C)$ is not contained in $Z$.  So the
regular section $\tilde{h}$ determines a rational section of
$\pi_2:X_2 \rightarrow B_2$ over $C$.  Since $C$ is smooth, this
rational section extends to a
regular section.  This contradicts the result of the last section.  So
we conclude that for a very general map $h:C\rightarrow B$ in $\cH_d$,
there is no section of $\pi:X \rightarrow B$ over $h$.

\section{Application: families of Enriques surfaces}\label{enriq}

In this section we will show how to apply Theorem~\ref{mainth} to a
family of Enriques surfaces to deduce Corollary~\ref{enriqcor}, that
is, to find a one-parameter family of Enriques surfaces without a
section.

\subsection{A family of quartic Enriques surfaces}

The family we will be starting with is the universal family over a
parameter space for quartic Enriques surfaces: that is, a family of
polarized Enriques surfaces $S$ with a polarization $M \in \pic(S)$ of
self-intersection 4 that includes a general such surface.  Now, for
the purposes of applying Theorem~\ref{mainth} and deducing
Corollary~\ref{enriqcor}, we can just write down the family as in
Definition~\ref{princenriq} below -- we do not need to know that it is
actually the generic quartic Enriques surface, and the reader who
does not particularly care can jump directly to
Definition~\ref{princenriq} -- but since we are going to be working
closely with the family it seems worthwhile to take a few paragraphs
and establish its origin.

\

To begin with, since Enriques surfaces 
have fundamental group $\Z/2\Z$ and have as universal covering space a K3
surface, a quartic Enriques surface $S$ is the quotient of an octic K3
surface $(T,L)$ -- that is, a K3 surface $T$ with a polarization $L \in
\pic(T)$ of self-intersection $c_1(L)^2 = 8$ -- by an involution
$\tau$ of $T$ preserving $L$.  For a general octic K3 $(T,L)$, the
linear system of sections of $L$ is base-point-free and defines an
embedding of $T$ into $\P^5$, and the image surface is the
intersection of three quadric hypersurfaces in $\P^5$ with defining
equations $Q_1$, $Q_2$, and $Q_3$.

\

Next, since $\tau^*L \cong L$, the action of $\tau$ can be lifted to
an action on $H^0(T,L)$, and hence to an involution of $\P^5$ carrying
$T$ to itself. Moreover, if we let $M$ be the line bundle
on the quotient Enriques surface $S$ obtained by descent, then by Riemann-Roch
$$
h^0(S,M) \; = \; \frac{c_1(M)^2}{2} + \chi(\cO_S) \; = \; \frac{4}{2}
+ 1 \; = \; 3
$$
the action of $\tau$ on $H^0(T,L)$ must have eigenvalues $1$ and $-1$,
each with multiplicity 3. We thus have a canonical direct-sum
decomposition
$$
H^0(T,L) \; = \; \Gamma \oplus \Psi
$$
with $\dim \Gamma = \dim \Psi = 3$.

\

Applying the same principle, we see that the action of $\tau$ on
$H^0(T,L^2)$ has eigenvalue 1 with multiplicity
$$
h^0(S,M^2) \; = \; \frac{c_1(M^2)^2}{2} + \chi(\cO_S) \; = \; 
\frac{16}{2} + 1 \; = \; 9
$$
and correspondingly eigenvalue $-1$ with multiplicity $h^0(T,L^2)-9 =
18-9 = 9$. On the other hand, given that $H^0(T,L) = \Gamma \oplus
\Psi$ as above, we can write
$$
\sym^2H^0(T,L) \; = \; \sym^2 \Gamma \oplus (\Gamma \otimes \Psi) 
\oplus \sym^2\Psi
$$
with the action of $\tau$ on $\sym^2H^0(T,L)$ having (+1)-eigenspace
$\sym^2 \Gamma\oplus\sym^2\Psi$ of dimension 12 and $(-1)$-eigenspace
$\Gamma \otimes \Psi$ of dimension 9. It follows that the kernel of
the restriction map
$$
\sym^2H^0(T,L) \; \longrightarrow \; H^0(T,L^2)
$$
-- that is, the vector space of quadrics in $\P^5$ vanishing on $T$
 -- must be contained in the direct sum $\sym^2
\Gamma\oplus\sym^2\Psi$. In other words, we can choose homogeneous
coordinates
$$
[Z,W] \; = \; [Z_0,Z_1,Z_2,W_0,W_1,W_2]
$$
on $\P^5$ so that the action of $\tau$ is given by
$$
\tau \; : \; [Z_0,Z_1,Z_2,W_0,W_1,W_2] \; \mapsto \; 
[Z_0,Z_1,Z_2,-W_0,-W_1,-W_2]
$$
and the defining equations of the double cover $T$ of a general
quartic Enriques surface $S$ may be written in the form
$$
Q_\alpha(Z,W) \; = \; Q'_\alpha(Z) + Q''_\alpha(W).
$$

\

We are now prepared to write down the families of K3 and Enriques
surfaces we will be studying in the sequel. To start with, let $\Gamma$
and $\Psi$ be 3-dimensional vector spaces and denote by $\P^5$ the
projective space of 1-dimensional subspaces of $\Gamma\oplus\Psi$ And
let
$$
\P^{11} \; = \;\P(\sym^2 \Gamma^\vee\oplus\sym^2\Psi^\vee)
$$
be the projective space of 1-dimensional subspaces of the
(12-dimensional) vector space of quadrics on $\P^5$ of the form above.
Finally, we let $[Z,W] = [Z_0,Z_1,Z_2,W_0,W_1,W_2]$ be homogeneous
coordinates on $\P^5$ with $\Gamma$ the zero locus of $W_0$, $W_1$ and
$W_2$, and $\Psi$ likewise the zero locus of $Z_0$, $Z_1$ and $Z_2$;
and we let $\tau$ be the involution $[Z_0,Z_1,Z_2,W_0,W_1,W_2]
\mapsto[Z_0,Z_1,Z_2,-W_0,-W_1,-W_2]$ of $\P^5$.

\begin{defn}\label{princK3}
By the \textbf{principal family of K3 surfaces} we will mean the
family $\pi : Y \to B$ with $B = \P^{11} \times \P^{11} \times
\P^{11}$ and $Y$ the subvariety of $B \times \PP^5$ defined by
$$
Y \; = \; \left\{\;(Q_1,Q_2,Q_3, p)\in B\times \PP^5 \mid
Q_1(p)=Q_2(p)=Q_3(p)=0   \; \right\},
$$
with $\pi : Y \to B$ the projection on the first factor.
\end{defn}

Note that the action of $\tau$ on the second factor of $B \times \P^5$
carries $Y$ into itself, so that we can make the following definition.

\begin{defn}\label{princenriq}
By the \textit{principal family of Enriques surfaces} we will mean the
family $\pi : X \to B$ with $B$ again as above and $X$ the quotient of
the variety $Y$ above by the involution $\tau$ of $\P^5$.
\end{defn}

It may be a misnomer to call these families of K3 and Enriques
surfaces, since they are only generically that: there are degenerate
fibers, and even fibers of dimension greater than 2.  But it is
convenient to use the term, and we hope the reader will forgive this.

\subsection{Proof of Corollary~\ref{enriqcor}}

In order to apply Theorem~\ref{mainth} to the principal family of
Enriques surfaces and deduce Corollary~\ref{enriqcor}, we simply have
to show that $X \to B$ admits no pseudosections. We will do this by
analyzing the corresponding family $Y \to B$ of K3 surfaces, since
their equations are in simpler form. We start with a straightforward
result.

\begin{lm}
Let $Y \to B$ be the principal family of K3 surfaces of
Definition~\ref{princK3}. The total space $Y$ is smooth, and its Chow
ring, $A^*(Y)$, is generated by restrictions of pullbacks of hyperplane classes
under the inclusion
$$
Y \; \hookrightarrow \; \P^{11} \times\P^{11} \times\P^{11} \times \P^5.
$$
\end{lm}

\bpf
To start, introduce the variety
$$
W \; = \;  \{ (Q,p) : p \in Q \} \; \subset \; \P^{11} \times \P^5.
$$
Via the projection $\eta : W \to \P^5$ on the second factor, $W$ is a
$\P^{10}$-bundle over $\P^5$; it is therefore smooth, and its Chow ring
is generated over the Chow ring of $\P^5$ by any class whose
restriction to the fibers of $\eta$ is the hyperplane class on
$\P^{10}$---for example, the restriction of the pullback of the
hyperplane class from $\P^{11}$, via the inclusion $W \hookrightarrow
\P^{11} \times \P^5$. Since the total space $Y$ of our principal
family of K3 surfaces is (via projection to $\P^5$) simply the triple
fiber product
$$
Y \; = \;  W \times_{\P^5} W \times_{\P^5} W
$$
the lemma follows.

\epf

As an immediate corollary of this lemma, we have the following
description of cycles $Z \subset X$ of relative dimension 0 over $B$.

\begin{prop}\label{lef}
Let $X \to B$ be the principal family of Enriques surfaces as in
Definition~\ref{princenriq}.  If $Z \subset X$ is any cycle of
codimension 2, the degree of the projection $\pi|_Z : Z \to B$ is
divisible by 4.
\end{prop}

\bpf 
Let $\eta: Y \to X$ be the quotient map.  Let $T$ be the class of a
general fiber of $Y$ over $B$.  By the preceding Lemma, the class of
any cycle in $Y$ is a polynomial (with integer coefficients) in the
restrictions to $Y$ of the pullbacks of the hyperplane classes to
$\P^{11} \times\P^{11} \times\P^{11} \times \P^5$. But the first three
of these classes restrict to 0 on a general fiber, so the class of
$\eta^{-1} Z\cdot T$ must be a multiple of the restriction to $T$ of
the hyperplane class on $\P^5$ This has degree divisible by 8.  As
$\eta$ has degree 2, the proposition follows.  

\epf \

As an immediate consequence of Proposition~\ref{lef}, we see that
\textit{the principal family $X \to B$ of Enriques surfaces has no
rational sections}: the image of such a section would give a
codimension 2 cycle of $X$ with degree one over $B$.  

\

In order to show that $X \to B$ admits no pseudosections, it remains
to prove that $X$ cannot contain a subvariety $Z \subset X$ whose
general fiber over $B$ is an irreducible rational curve. To do this,
suppose that $Z$ is such a subvariety.  Let $\tilde Z$ be a resolution
of singularities of $Z$.  We then have a commutative diagram
$$\xymatrix{
\tilde Z \ar[r]^f \ar[rd]^\mu & X\ar[d]^\pi\\
& B}$$

\

Consider the class $f_*(c_1(\omega_{\tilde Z/B}))$ in $A^2(X)$.  Since the
general fiber of $\tilde Z$ over $B$ is a smooth rational curve, this
class has degree $-2$ when restricted to a general fiber of $\pi$.  This
contradicts the fact that all elements of $A^2(X)$ have degree over
$B$ divisible by 4.

\

We have thus established the following result.

\begin{lm}
The principal family $X \to B$ of Enriques surfaces admits no
pseudosections.
\end{lm}

\ni Applying Theorem~\ref{mainth} we may deduce
Corollary~\ref{enriqcor}.

\section{Application: torsors for Abelian varieties}\label{Torsors}

It follows from Theorem~\ref{mainth} that any family $\pi:X\rightarrow
B$ of smooth, connected, projective curves of positive genus over some
smooth variety $B$ has a section over $B$ if and only if the
restriction of this family over every curve $C\subset B$ has a
section: since the fibers contain no rational curves, every
pseudosection is a rational section, and every rational section is
everywhere defined.  Similarly, we have the following corollary.

\begin{cor}\label{torsorcor}
Let $B$ be a smooth 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 $T_C \to C$ is a trivial torsor for $A_C \to C$.
\end{cor}

Since torsors for an Abelian scheme are classified by \'etale
cohomology with coefficients in the group scheme, we can rephrase
Corollary~\ref{torsorcor} by saying that the map
$$ H^1_{\text{\'et}}(B,A) \rightarrow \prod_{C\subset B}
H^1_{\text{\'et}}(C,A_C)$$ 
is injective.  Note that the Lefschetz hyperplane theorem for Picard
groups tells us that the same is true if we replace the Abelian scheme
$A\rightarrow B$ by the commutative group scheme $\mathbb{G}_m \times
B \rightarrow B$.  It is an interesting question, for which other
(possibly noncommutative) group schemes over $B$ does this hold.

\section{Further questions and conjectures}

In this section we will consider some questions and conjectures raised
by Theorem~\ref{mainth}.

\subsection{Arithmetic question}

For arithmetic questions related to rationally connected varieties, we
refer the reader to \cite{Abel} and \cite{Aim}.  
Let us just mention the 
following
question which is an analogue of our main theorem in the case
of fiber dimension 1.
Let $K$ be a number field, let $B$ be a smooth scheme defined over
$K$, and let $\pi:X \to B$ be a proper, smooth morphism of schemes
whose geometric fibers are connected curves of positive genus.
Suppose that for every number field extension $L/K$, the induced
mapping on rational points $\pi:X(L) \to B(L)$ is surjective -- we
refer to this property by saying $\pi$ is \emph{arithmetically
surjective}.  Does it then follow that $\pi:X \to B$ has a section?
We may also ask the same question when the geometric fibers of $\pi$
are Abelian varieties.

\subsection{Possible extensions}

We would like to take a moment here to discuss possible extensions of
Theorem~\ref{mainth}. To begin with, we interpreted the theorem as
stating that a family of varieties $\pi : X \to B$ such that every
one-parameter subfamily has a section has this property ``by virtue
of" the fact that $X$ contains a family of rationally connected
varieties. But the statement of the theorem asserts only the existence
of a pseudosection in $\pi : X \to B$; it does not assert any direct
connection between the sections of $X_C \to C$ over very general
curves $C$ and the pseudosection.  Accordingly, we could ask the
following.

\begin{ques} 

Does there exist a family $\cH'_d $ of curves on $B$, whose general
member is smooth and irreducible, with the property that for any
proper morphism $\pi : X \to B$ of relative dimension $d$, for very
general $[C] \in \cH'_d$ every section of the pullback
$$
 \pi_C : X_C = X \times_B C \to C
$$
lies in a pseudosection of $\pi : X \to B$?  

\end{ques}

One special case of this question is when $\pi:A \rightarrow B$ is an
Abelian scheme over a smooth variety $B$.  In this case we are asking
whether we can find a family of curves $C\subset B$ such that for a
very general member of this family, the map
$$ H^0_{\text{\'et}}(B,A) \to H^0_{\text{\'et}}(C,A_C) $$
is surjective.  So, in this case, the question above is an
$H^0$-analogue of the $H^1$-interpretation of Theorem~\ref{mainth}.

\subsection{Dependence on $d$}

A simpler question is whether we can eliminate the dependence of the
family $\cH_d$ of curves on $d$. The answer to this is ``no."
Proving this is not so easy, and for full details the reader is referred
to \cite{jason}.  Here we will just sketch an argument, since it
may shed some light on how fast the size of the curves in $\cH_d$ have
to grow with $d$.

\

Briefly, for any $e$ we will write down families of hypersurfaces in
$\P^n$ parametrized by $B = \P^2$ with the property that their
restriction to any curve $C \subset \P^2$ of degree $e$ or less has a
section, but which admits no pseudosections at all. To do
this, let $m$ and $n$ be any integers; let $\P^n$ be projective
$n$-space with homogeneous coordinates $[X_0,\dots,X_n]$ and let
$\P^N$ be the projective space parametrizing hypersurfaces of degree
$m$ in $\P^n$. Let $\cX \subset \P^N \times \P^n \to \P^N$ be the
universal hypersurface of degree $m$ in $\P^n$; that is, the zero
locus of the polynomial
 $$
 F(a,X) \; = \; \sum a_IX^I
 $$
which is linear in the coordinates $a_I$ on $\P^N$ and of degree $m$
in the $X_i$. Finally, let $\P^2 \hookrightarrow \P^N$ be a general
map of the form
 $$
 \P^2 \; \xrightarrow{\nu_e} \;  \P^{\binom{e+2}{2}-1} \; \to \; \P^N
 $$
where $\nu_e$ is a Veronese map of degree $e$ and the second map is a
general linear inclusion; and let
 $$
 \pi \; : \; X = \P^2 \times_{\P^N}  \cX \; \longrightarrow \; \P^2
 $$
be the pullback of the universal hypersurface to $\P^2$ via this inclusion.

Now assume that
 $$
 \binom{e+2}{2} \; = \; n+1
 $$
and that $m$ is large.  Consider the following two assertions:

\

\ni $\bullet$ \thinspace the restriction of the family $\pi : X \to
\P^2$ to any curve $C \subset \P^2$ of degree $e$ or less has a
section; but

\

\ni $\bullet$ \thinspace the family $\pi : X \to \P^2$ itself has no
pseudosection.

\

The first of these assertions is straightforward to prove: under the
inclusion $\P^2 \hookrightarrow \P^N$, the span of a curve $C \subset
\P^2$ of degree $e$ or less has dimension $\binom{e+2}{2} - 2 = n-1$ or
less. Thus the hypersurfaces appearing as fibers of the restriction
$X_C \to C$ of the family $\pi : X \to \P^2$ to $C$ are all linear
combinations of $n$ hypersurfaces $G_1,\dots,G_n \subset \P^n$, and
any point of intersection of these hypersurfaces gives a section of
$X_C \to C$.

\

As for the second assertion, we will not prove it here, but we give a
``plausibility argument'' which suggests it is true.  To begin with, a
general fiber of $\pi : X \to \P^2$ is a general hypersurface of
degree $m$ in $\P^n$; by a result of Clemens \cite{Cl}, for $m$
large this will contain no rational curves. Thus to prove the second
assertion we need only show that $\pi : X \to \P^2$ has no rational
sections.

\

Since rational sections over $\P^2$ are tricky to parametrize we will
restrict to a general curve $C \subset \P^2$ of degree $e+1$, and
present evidence that the restriction $X_C \to C$ has no section.  To
do this, we start by counting the dimension of the family of sections
of the product $C \times \P^n$ there are of a given degree $k$---that
is, graphs of maps $C \to \P^n$ of degree $k$---and then estimating
the number of conditions it imposes on such a section to require it
lies on the hypersurface $X_C \subset C \times \P^n$. For the first, a
map $C \to \P^n$ of degree $k$ is given by a line bundle $L$ of degree
$k$ on $C$, together with $n+1$ sections of $L$ up to scalars. The
line bundles of degree $k$ on $C$ are parametrized by the Jacobian of
$C$, which has dimension
 $$
 g \; = \; \binom{e}{2}.
 $$
If $k$ is large, moreover, each such line bundle will have $k-g+1$
global sections, so the dimension of the family of maps $C \to \P^n$
of degree $k$ is
 $$
 g + (n+1)(k-g+1) - 1 \; = \; (n+1)(k+1) - ng - 1.
 $$

\

Now we count how many conditions it is for the graph of such a map
to lie in $X_C$. This is straightforward: when we pull the polynomial
$F(a,X)$ defining the universal hypersurface back to $C$, the
coefficients pull back to sections of $\cO_C(e)$ and the coordinates
$X_i$ to sections of $L$, so that the pullback of $F$ is a section of
the bundle
 $$
 M \; = \; L^{\otimes m} \otimes \cO_C(e).
 $$
The number of conditions for this section to vanish identically should
thus be 
\begin{align*} h^0(M) \; &= \; \deg(M) - g + 1 \\ &= \; km +
e(e+1) - g + 1 \end{align*} 
and the expected dimension of the family
of sections of $X_C \to C$ of degree $k$ is accordingly
 $$
   (n+1-m)k - (n-1)(g-1) - e(e+1).
 $$
In particular, for $m$ large this is negative, suggesting that there 
should be no sections.


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\end{document}
