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\begin{document}

\centerline{\bf  {\large Families of Rationally Connected Varieties}}

\

\

\centerline{\today}

\

\noindent {\bf Tom Graber}

\noindent Mathematics Department, Harvard University,

\noindent 1 Oxford st., Cambridge MA 02138, USA


\noindent graber{\char'100}math.harvard.edu

\

\noindent {\bf Joe Harris}

\noindent Mathematics Department, Harvard University,

\noindent 1 Oxford st., Cambridge MA 02138, USA


\noindent harris{\char'100}math.harvard.edu

\

\noindent {\bf Jason Starr}

\noindent Mathematics Department, M.I.T.,

\noindent Cambridge MA 02139, USA


\noindent jstarr{\char'100}math.mit.edu

\

\tableofcontents


\section{Introduction}

\subsection{Statement of results}

We will work throughout over the complex numbers, so that the results
here apply over any algebraically closed field of characteristic 0.

\ps

Recall that a smooth projective variety $X$ is said to be {\em
rationally connected} if two general points $p, q \in X$ are contained
in the image of a map $g:C\rightarrow X$ of a connected, complete,
at-worst-nodal curve $C$ of arithmetic genus $0$ to $X$.  This turns
out to be equivalent to the a priori stronger condition that for any
finite subset $\Gamma \subset X$, there is a map $g:\PP^1\rightarrow
X$ whose image contains $\Gamma$ and such that $g^*T_X$ is an ample
bundle.  If $\text{dim}(X) \geq 3$, one can arrange that $g$ is an
embedding with ample normal bundle.
%can be joined
%by a chain of rational curves. In case $\dim(X) \geq 3$ this turns out
%to be equivalent to the a priori stronger condition that for any
%finite subset $\Gamma \subset X$ there is a smooth rational curve $C
%\subset X$ containing $\Gamma$ and having ample normal bundle.

\ps

Rationally connected varieties form an important class of
varieties. In dimensions 1 and 2 rational connectivity coincides with
rationality, but the two notions diverge in higher dimensions and in
virtually every respect the class of rationally connected varieties is
better behaved. For example, the condition of rational connectivity is
both open and closed in smooth proper families; there are geometric
criteria for rational connectivity (e.g. any smooth projective variety
with negative canonical bundle is rationally connected, so we know in
particular that a smooth hypersurface $X \subset \P^n$ of degree $d$
will be rationally connected if and only if $d \leq n$), and there
are, at least conjecturally, numerical criteria for rational
connectivity (see Conjecture~\ref{mumford} below).  In this paper we
will prove a conjecture of Koll\'ar, Miyaoka and Mori that represents
one more basic property of rational connectivity (also one not shared
by rationality): that if $X \to Y$ is a morphism with rationally
connected image and fibers, then the domain $X$ is rationally
connected as well. This will be a corollary of our main theorem:

\ps

\begin{thm}\label{mainth}
Let $f : X \to B$ be a morphism from a smooth projective variety to a
smooth projective curve over $\C$. If the general fiber of $f$ is
rationally connected, then $f$ has a section.
\end{thm}

Since this is really a statement about the birational equivalence
class of the morphism $f$, we can restate it in the equivalent form

\begin{thm}
If $K$ is the function field of a curve over $\C$, any rationally
connected variety $X$ defined over $K$ has a $K$-rational point.
\end{thm}

In this form, the theorem directly generalizes Tsen's theorem, which
is exactly this statement for $X$ a smooth hypersurface of degree $d
\leq n$ in projective space $\P^n$ (or more generally a smooth
complete intersection in projective space with negative canonical
bundle, c.f. ~\cite[theorem IV.6.5]{K}). It would be interesting to
know if in fact rationally 
connected varieties over other $C_1$ fields necessarily have rational
points.

\ps

As we indicated, one basic corollary of our main theorem is

\begin{cor}\label{totalspace}
Let $f : X \to Y$ be any dominant morphism of complex varieties. If
$Y$ and 
the general fiber of $f$ are rationally connected, then $X$ is
rationally connected.
\end{cor}

\ni {\em Proof}. Since we work over $\mathbb{C}$, we can assume that
$X$ and $Y$ are smooth projective varieties. Let $p$ and $q$ be
general points of $X$. We can find a map $g:\PP^1\rightarrow Y$ whose
image contains $f(p)$ and $f(q)$;  let $X' = \PP^1 \times_Y X$ be the
pullback of $f'$ by $g$.  
By Theorem~\ref{mainth}, there is a section $D$
of $X'$ over $\PP^1$. We can then connect $p$ to $q$ by a chain of
rational curves in $X'$ in three stages: connect $p$ to the point $D
\cap X_p$ of intersection of $D$ with the fiber $X_p$ of $f$ through
$p$ by a rational curve; connect $D \cap X_p$ to $D \cap X_q$ by $D$,
and connect $D \cap X_q$ to $q$ by a rational curve in $X_q$. \qed

There is a further corollary of Theorem~\ref{mainth} based on a
construction of Campana and Koll\'ar--Miyaoka--Mori: the {\it maximal
rationally connected fibration} associated to a variety $X$ (see [Ca],
[K] or [KMM]). Briefly, the maximal rationally connected fibration
associates to a variety $X$ a (birational isomorphism class of)
variety $Z$ and a rational map $\phi : X \to Z$ with the properties
that
\begin{itemize}
\item the fibers $X_z$ of $\phi$ are rationally connected; and conversely 
\item almost all the rational curves in $X$ lie in fibers of $\phi$:
for a very general point $z \in Z$ any rational curve in $X$ meeting
$X_z$ lies in $X_z$.
\end{itemize}
\ni The variety $Z$ and morphism $\phi$ are unique up to birational
isomorphism, and are called the \textit{mrc quotient} and \textit{mrc
fibration} of $X$, respectively.  They measure the failure of $X$ to
be rationally connected: if $X$ is rationally connected, $Z$ is a
point, while if $X$ is not uniruled we have $Z=X$. As observed in
Koll\'ar ([K], IV.5.6.3), we have the following Corollary:

\begin{cor}\label{quot}
Let $X$ be any variety and $\phi : X \to Z$ its maximal rationally
connected fibration. Then $Z$ is not uniruled.
\end{cor}

\ni {\em Proof}. Suppose that $Z$ were uniruled, so that for a
general point $z \in Z$ we could find a map $g:\PP^1\rightarrow Z$
whose image contains $z$.  
By Corollary~\ref{totalspace}, the pullback $X'= \PP^1\times_Z X$ of
$\phi$ by $g$
will be rationally connected, which means that every
point of the fiber $X_z$ will lie on a rational curve not contained in
$X_z$, contradicting the second defining property of mrc fibrations.
\qed

There are conjectured numerical criteria for a variety $X$ to be
either uniruled or rationally connected. They are

\begin{conj}\label{uniruled}
Let $X$ be a smooth projective variety. Then $X$ is uniruled if and
only if $H^0(X,K_X^m)=0$ for all $m > 0$.
\end{conj}

\ni and

\begin{conj}\label{mumford}
Let $X$ be a smooth projective variety. Then $X$ is rationally
connected if and only if $H^0(X,(\Omega^1_X)^{\otimes m})=0$ for all
$m > 0$.
\end{conj}

For each of these conjectures, the ``only if" part is known, and
straightforward to prove; the ``if" part represents a very difficult
open problem (see for example [K], IV.1.12 and IV.3.8.1). As another
consequence of our main theorem, we have an implication:

\begin{cor}
Conjecture~\ref{uniruled} implies Conjecture~\ref{mumford}
\end{cor}

\ni {\em Proof}. 
This is proved in [K, IV.5.7], assuming theorem~\ref{mainth}; 
for completeness we remind the reader of the proof.   
Let $X$ be any smooth projective variety that is not
rationally connected; assuming the statement of
Conjecture~\ref{uniruled}, we want to show that
$H^0(X,(\Omega^1_X)^{\otimes m})\neq 0$ for some $m > 0$. Let $\phi :
X \to Z$ be the mrc fibration of $X$. By hypothesis $Z$ has dimension
$n >0$, and by Corollary~\ref{quot} $Z$ is not uniruled. If we assume
Conjecture~\ref{uniruled}, then, we must have a non-zero section
$\sigma \in H^0(Z,K_Z^m)$ for some $m > 0$. But the line bundle
$K_Z^m$ is a summand of the tensor power $(\Omega^1_Z)^{\otimes nm}$,
so we can view $\sigma$ as a global section of that sheaf; pulling it
back via $\phi$, we get a nonzero global section of
$(\Omega^1_X)^{\otimes nm}$ \qed

\ps

\ni {\bf Acknowledgments}. We would like to thank Johan de Jong,
 J\'anos Koll\'ar and Barry Mazur for many conversations, which were
 of tremendous help to us.

\ps  

\section{Preliminary definitions and constructions}

We will be dealing with morphisms $\pi : X \to B$ satisfying a number
of hypotheses, which we collect here for future reference. In
particular, for the bulk of this paper we will deal with the case $B
\cong \P^1$; we will show in section~\ref{barb} below both that the
statement for $B \cong \P^1$ implies the full Theorem~\ref{mainth}
and, as well, how to modify the argument that follows to apply to
general $B$.

\begin{hyp}\label{hyp}
$\pi : X \to B$ is a nonconstant morphism of smooth connected
projective varieties over $\C$, with $B \cong \P^1$. For general $b
\in B$, the fiber $X_b = \pi^{-1}(b)$ is rationally connected of
dimension at least 2.
\end{hyp}

It is necessary to work with a class of maps of curves $g:C\rightarrow
X$ more general than embeddings.  We choose to work with Kontsevich
stable maps:

\begin{defn}~\label{stabmap}
A pair $(f:C\rightarrow X, (p_1,\dots,p_n)$ of a connected, complete,
at-worst-nodal 
curve $C$ to $X$ and an ordered set of closed points, called
\emph{marked points}, of $C$ is a
\emph{Kontsevich stable map} (or simply 
\emph{stable map}) if 
\begin{enumerate}
\item the marked points $p_i$ are all distinct,
\item the marked  points $p_i$ are contained in the smooth locus of $C$, and
\item for every irreducible component $C_j$ of $C$ with arithmetic
  genus 0 on which $f$ is constant, there are at least three
  \emph{special points} (nodes or 
  marked points), and for every irreducible component $C_j$ of $C$
  with arithmetic genus 1 on which $f$ is constant, there is at least
  one special point.
\end{enumerate}
\end{defn}

Given a numerical equivalence class $[D]\in N^1(X)$, the intersection
number $f_*[C].[D]$ is defined in the usual way.  Thus $f_*[C]$
defines a class in $N_1(X)$.  For a fixed nonnegative integer $g$, a
fixed nonnegative integer $n$, and a class $\beta\in N_1(X)$, the
\emph{Kontsevich moduli stack} $\Kgnb{g,n}{X,\beta}$ is defined to
be the Deligne-Mumford stack which parametrizes flat families of Kontsevich
stable maps $(f:C\rightarrow X,(p_1,\dots,p_n))$ such that $C$ has
arithmetic genus $g$ and the class $f_*[C]$ equals $\beta$.  
We denote by $\kgnb{g,n}{X,\beta}$ the coarse moduli space of
$\Kgnb{g,n}{X,\beta}$.  
For
definitions and results about Kontsevich stable maps, the reader is
referred to (~\cite{FP}, ~\cite{BM}).  

\ps

Now suppose we have a class $\beta \in N_1(X)$ having intersection
number $d$ with a fiber of the map $\pi$.  By ~\cite[theorem 3.6]{BM} 
we have then a natural
morphism
$$
\varphi : \mg(X,\beta) \to \mg(B,d)
$$
defined by composing a map $f : C \to X$ with $\pi$ and collapsing
components of $C$ as necessary to make the composition $\pi f$ stable.

\begin{defn}
Let $\pi : X \to B$ be a morphism satisfying \ref{hyp}, and let $f : C
\to X$ be a stable map from a nodal curve $C$ of genus $g$ to $X$ with
class $f_*[C] = \beta$. We say that $f$ is {\it flexible} relative to
$\pi$ if the map $\varphi : \mg(X,\beta) \to \mg(B,d)$ is dominant at
the point $[f] \in \mg(X,\beta)$; that is, if any neighborhood of
$[f]$ in $\mg(X,\beta)$ dominates a neighborhood of $[\pi f]$ in
$\mg(B,d)$.
\end{defn}

Now, it's a classical fact that the variety $\mg(B,d)$ has a unique
irreducible component whose general member corresponds to a flat map
$f : C \to B$ (see for example \cite{C} and \cite{H}, and ~\cite[prop
1.5]{F} for a modern treatment). Since the map
$\varphi : \mg(X,\beta) \to \mg(B,d)$ is proper, it follows that if
$\pi : X \to B$ admits a flexible curve then $\varphi$ will be
surjective.  Moreover, $\mg(B,d)$ contains points $[f]$ corresponding
to maps $f : C \to B$ with the property that every irreducible
component of $C$ on which $f$ is nonconstant maps isomorphically via
$f$ to $B$. (For example, we could simply start with $d$ disjoint
copies $C_1,\dots,C_d$ of $B$ (with $f$ mapping each isomorphically to
$B$) and identify $d+g-1$ pairs of points on the $C_i$, each pair
lying over the same point of $B$.

\begin{prop}
If $\pi : X \to B$ is a morphism satisfying \ref{hyp} and $f : C \to
X$ a flexible stable map, then $\pi$ has a section.
\end{prop}

Our goal in what follows, accordingly, will be to construct a flexible
curve $f : C \to X$ for an arbitrary $\pi : X \to B$ satisfying
\ref{hyp}.

\subsection{The first construction}

To manufacture our flexible curve, we apply two basic constructions,
which we describe here. (These constructions, especially the first,
are pretty standard: see for example section II.7 of [K].)  We start
with a basic lemma:

\begin{lm}\label{bundle}
Let $C$ be a smooth curve and $E$ any vector bundle on $C$; let $n$ be
any positive integer. Let $p_1,\dots,p_N \in C$ be general points and
$\xi_i \subset E_{p_i}$ a general one-dimensional subspace of the
fiber of $E$ at $p_i$; let $E'$ be the sheaf of rational sections of
$E$ having at most a simple pole at $p_i$ in the direction $\xi_i$ and
regular elsewhere. For $N$ sufficiently large we will have
$$
H^1(C, E'(-q_1-\dots-q_n)) = 0
$$ 
for any $n$ points $q_1,\dots,q_n \in C$.
\end{lm}

\ni {\it Proof}. To start with, we will prove simply that
$H^1(C,E')=0$. Since this is an open condition, it will suffice to
exhibit a particular choice of points $p_i$ and subspaces $\xi_i$ that
works.  Denoting the rank of $E$ by $r$, we take $N = mr$ divisible by
$r$ and choose $m$ points $t_1,\dots,t_m \in C$. We then specialize to
the case
\begin{align*}
p_1 = \dots = p_r &= t_1; & \xi_1,\dots,\xi_r &\mbox{ spanning }
E_{t_1} \\ 
p_{r+1} = \dots = p_{2r} &= t_2; & \xi_{r+1},\dots,\xi_{2r}
&\mbox{ spanning } E_{t_2}
\end{align*}
and so on. In this case we have $E' = E(t_1 + \dots + t_m)$, which we
know has vanishing higher cohomology for sufficiently large $m$.

Given this, the statement of the lemma follows: to begin with, choose
any $g+n$ points $r_1,\dots,r_{g+n} \in C$. Applying the argument thus
far to the bundle $E(-r_1-\dots -r_{g+n})$, we find that for $N$
sufficiently large we will have $H^1(C, E'(-r_1-\dots-r_{g+n})) =
0$. But now for any points $q_1,\dots,q_n \in C$ we have
$$
q_1 + \dots + q_n = r_1+\dots + r_{g+n} - D
$$
for some effective divisor $D$ on $C$. It follows then that
\begin{align*}
h^1(C, E'(-q_1-\dots-q_n)) &= h^1(C, E'(-r_1-\dots -r_{g+n})(D)) \\
&\leq h^1(C, E'(-r_1-\dots -r_{g+n})) \\
&=0
\end{align*}
\qed

%The relevance of this to our present circumstances will perhaps be
%made clear by the following:
%
%\begin{lm}\label{normal}
%Let $X$ be a smooth projective variety, $C$ and $C'\subset X$ two
%nodal curves meeting at points $p_1,\dots,p_\delta$; suppose $C$ and
%$C'$ are smooth with distinct tangent lines at each point $p_i$. Let
%$D = C \cup C'$ be the union of $C$ and $C'$; and let $N_{C/X}$ and
%$N_{D/X}$ be the normal sheaves of $C$ and $D$ in $X$. We have then an
%inclusion of sheaves
%$$
%0 \to N_{C/X} \to N_{D/X}|_C
%$$
%identifying the sheaf of sections of $N_{D/X}|_C$ with the sheaf of
%rational sections of $N_{C/X}$ having at most a simple pole at $p_i$
%in the normal direction determined by $T_{p_i}C'$. Moreover, if
%$\tilde D \subset \spec \C[\e]/(\e^2) \times X$ is a first-order
%deformation of $D$ in $X$ corresponding to a global section $\sigma
%\in H^0(N_{D/X})$, then $\tilde D$ smooths the node of $D$ at $p_i$ if
%and only if the restriction $\sigma|_U$ of $\sigma$ to a neighborhood
%$U$ of $p$ in $C$ is not in the image of $N_{C/X}$.
%\end{lm}
%
%\ps

It is well-known that given an embedded curve $D\subset X$ which is
at-worst-nodal, and given an irreducible component $C\subset D$ which
intersects $\overline{D-C}$ in the nodes $p_1,\dots,p_\delta$, the
restriction $N_{D/X}|_C$ (as a sheaf) consists of rational sections of
$N_{C/X}$ with at worst simple poles at each $p_i$ whose normal
direction is determined by the other branch of $D$ through $p_i$.  
We will need a version of this result when the embedded curves
are replaced by stable maps to $X$.  

\ps

For an embedded
curve LCI curve $C\subset X$,  the space of first-order deformations and the
obstruction group are given by $H^0(C,N_{C/X})$ and $H^1(C,N_{C/X})$
respectively.  Suppose given an unmarked stable map $f:C\rightarrow X$.
By ~\cite{BF} and ~\cite{B}, the space of
first-order deformations of the stable map
and the obstruction group are given by the hypercohomology groups
\begin{equation}
\text{Def}(f) = {\mathbb H}^1(C,{\mathbb
  R}\textit{Hom}_{\OO_C}(\Omega^\cdot_f,\OO_C)), 
\text{Obs}(f) ={\mathbb H}^2(C,{\mathbb
  R}\textit{Hom}_{\OO_C}(\Omega^\cdot_f,\OO_C)),
\end{equation}
where $\Omega^\cdot_f$ is the complex
\begin{equation}
\begin{CD}
-1 & & 0 \\
f^*\Omega_X @> df^\dagger >> \Omega_C.
\end{CD}
\end{equation}
We will only need to use the deformation theory of stable maps when
the stable map satisfies the additional hypothesis that the unramified
locus $U\subset C$ contains all nodes of $C$ (in particular there are
no irreducible components on which $f$ is constant).  In this case
${\mathbb R}\textit{Hom}_{\OO_C}(\Omega^\cdot_f,\OO_C)$ is
quasi-isomorphic to a complex $\cN_f[-1]$ where $\cN_f$ is a coherent
sheaf, the \emph{normal sheaf} of $f$, whose restriction to $U$ is
isomorphic to the dual of the kernel of $f^*\Omega_X\rightarrow
\Omega_C$.  

\ps

Suppose $(f:C\rightarrow X, (p_1,\dots,p_\delta))$ and
$(f':C'\rightarrow X,(p'_1,\dots,p'_\delta))$ are stable maps such
that 
\begin{enumerate}
\item the unramified locus of $f$ (resp. $f'$) contains all nodes of
  $f$ as well as all marked points,
\item for every $i=1,\dots, \delta$, $f(p_i)=f'(p'_i)$, and
\item for every $i=1,\dots,\delta$, the tangent lines $df(T_{C,p_i}),
  df'(T_{C',p'_i}) \subset T_{X,f(p_i)}$ are distinct.  
\end{enumerate}
By ~\cite[theorem 3.6]{BM}, there is a unique stable map
$F:D\rightarrow X$ where $D$ is the connected sum of $C$ and $C'$ with
$p_i$ identified to $p'_i$, and such that $F|_{C}=f$, $F|_{C'}=f'$.
Condition (3) above insures that the points $p_i$ are in the
unramified locus of $F$. 

\begin{lm}\label{normal2}
Let $X$ be a smooth projective variety, $(f:C\rightarrow
X,(p_1,\dots,p_\delta))$, $(f':C\rightarrow X,(p'_1,\dots,p'_\delta))$
be as above.  Let $(F:D\rightarrow X)$ be the stable map obtained by
gluing each $p_i$ to $p'_i$.  The normal sheaf of the map $\cN_F$ is
locally free near each of the nodes $p_i$ of $D$ and we have an
inclusion of sheaves
$$
0 \to \cN_f \to \cN_F|_C
$$
identifying $\cN_F|_C$ with the sheaf of rational sections of $\cN_f$
having at most a simple pole at each $p_i$ in the normal direction
determined by $df'(T_{p'_i}C')$.  Moreover, if
$(\tilde{F}:\tilde{D}\rightarrow X)$ is a first-order deformation of
$F$ corresponding to a global section $\sigma\in H^0(D,\cN_F)$, then
$\tilde D$ smooths the node of $D$ at $p_i$ if and only if the
restriction $\sigma|_U$ of $\sigma$ to a neighborhood $U$ of $p_i$ in
$C$ is not in the image of $\cN_f$.  
\end{lm}

\ps

\begin{hyp}~\label{hyp2}
Suppose $\pi:X\to B$ is a morphism satisfying our basic
hypotheses~\ref{hyp}.  An unmarked stable map $f:C\rightarrow X$
satisfyies our \emph{second hypotheses} if $C$ is smooth and
$\mu:=\pi\circ f$ is surjective.
\end{hyp}

Suppose $f:C\rightarrow X$ satisfies our second hypotheses and $C$ has
genus $g$.
For a general point $p \in C$, let $X_p = \pi^{-1}(\pi(f(p)))$
be the fiber of $\pi$ through $f(p)$. By hypothesis, $X_p$ is a smooth,
rationally connected variety, so that we can find a stable map $(f':C'
\subset X_p)$ from a smooth rational curve and a point $p'\rightarrow
C'$ (at which $f'$ is unramified) such that $f(p)=f'(p')$, such that
the image of $df'(T_{p'}C')$ in $T_{f(p)}X/df(T_pC) = (\cN_{C'/X})_p$ is
arbitrary, and such that $(f')^*TX_p$ is an ample bundle on $C'$.

\ps

Choose a large number of general points $p_1,\dots,p_\delta \in C$,
and for each $i$ let $f'_i:C'_i \subset X_{p_i}$ be a stable map as in
the last paragraph.  Let $F:D\rightarrow X$ denote the stable map
obtained by gluing each $C'_i$ to $C$ by identifying $p_i$ and $p'_i$. 
Combining the preceding two lemmas, we see that for $\delta$
sufficiently large, the normal sheaf $\cN_F$ 
will be generated by its global sections; in
particular, by Lemma~\ref{normal2} there will be a smooth deformation
$\tilde{f}:\tilde{C}\rightarrow X$ of $F$ (i.e. a general stable map
which is a 
member of a $1$-parameter family of stable maps specializing to $F$). 
Moreover, for any given $n$ we can choose the
number $\delta$ large enough to ensure that $H^1(C,
\cN_F|_C(-r_1-\dots-r_{g+n})) = 0$ for some $g + n$ points
$r_1,\dots,r_{g+n} \in C$; it follows that $H^1(\tilde C,
\cN_{\tilde{f}}(-r_1-\dots-r_{g+n})) = 0$ for some $r_1,\dots,r_{g+n}
\in \tilde C$ and hence that
$$
H^1(\tilde C, \cN_{\tilde{f}}(-q_1-\dots-q_n)) = 0
$$
for any $n$ points on $\tilde C$.

\ps 

The process of taking a stable map $f:C\rightarrow X$ which satisfies
our second hypotheses, 
attaching rational curves
in fibers and smoothing as above to get a new stable map $\tilde{f}:\tilde{C}
\rightarrow X$ with smooth domain is our first 
construction. It has the properties that:
\begin{enumerate}
\item $\tilde{f}:\tilde{C}\rightarrow X$ also satisfies our second
  hypotheses, 
\item the genus $g$ of the new curve $\tilde C$ is the same as the
genus of the curve $C$ we started with;
\item the degree $d$ of $\tilde{\mu}=pi\circ\tilde{f}:\tilde
  C\rightarrow B$ is the same as the degree of $\mu=\pi\circ
  f:C\rightarrow B$;
\item the branch divisor of $\tilde{\mu}$
is a small deformation of the branch divisor
of $\mu$; and again,
\item for any $n$ points $q_1,\dots,q_n \in \tilde C$ we have
$H^1(\tilde C, \cN_{\tilde{f}}(-q_1-\dots-q_n)) = 0$.
\end{enumerate}

\ps

Here is one application of this construction. Suppose we have a stable
map $f:C \subset X$ satisfying our second hypotheses
 such that the projection $\mu$
is simply branched --- that is, the branch divisor of $\mu$ consists of
$2d+2g-2$ distinct points in $B$ --- and such that each ramification
point $p \in C$ of $\mu$ maps to a smooth point of the fiber
$X_p$. Applying our first construction with $n = 2d+2g-2$, we arrive
at another stable map $\tilde{f}:\tilde C\rightarrow X$ satisfying our
hypotheses which is again simply branched over
$B$, with all ramification points mapping to smooth points of fibers of
$\pi$. But now the condition that $H^1(\tilde C,
\cN_{\tilde{f}}(-q_1-\dots-q_n)) = 0$ applied to the $n = 2d+2g-2$
ramification 
points of the map $\tilde{\mu}=\pi\circ\tilde{f}$
says that if we pick a
normal vector $v_i\in (\cN_{\tilde{f}})_{p_i}$ 
at each ramification point $p_i$ of
$\tilde \mu$ we can find a global section of the normal sheaf
$\cN_{\tilde{f}}$ with value $v_i$ at $p_i$. Moreover, since
ramification occurs at smooth points of fibers of $\pi$, for any
tangent vectors $w_i$ to $B$ at the image points $\tilde{\mu}(p_i)$ we can
find tangent vectors $v_i \in (\cN_{\tilde{f}})_{p_i}$ with $d\pi(v_i) =
w_i$. 
It follows that {\it as we deform the stable map $\tilde{f}:\tilde
  C\rightarrow X$, the branch
points of $\tilde \mu$ move independently}. A general deformation of
$\tilde{f}$ thus yields a general deformation of $\tilde
\mu$ --- in other words, the stable map $\tilde{\mu}$ is flexible. We thus make
the

\begin{defn}
Let $\pi : X \to B$ be as in hypothesis~\ref{hyp}, and let $f:C \subset X$ be a
stable map as in hypothesis~\ref{hyp2} 
such that the projection $\mu =\pi\circ f$ is 
simply branched. If each ramification point $p \in C$ of $\mu$ maps to a
smooth point of the fiber $X_p$ containing it, we will say the stable map
$f:C\rightarrow X$ is {\it pre-flexible}.
\end{defn}

\ni In these terms, we have established the


\begin{lm}\label{preflex}
Let $\pi : X \to B$ be as in~\ref{hyp}. If $X$ admits a pre-flexible
stable map, the map $\pi$ has a section.
\end{lm}

%\ni \underbar{Remark}. Note that we can extend the notion of
%pre-flexible and the statement of Lemma~\ref{preflex} to stable maps
%$f : C \to X$: we say that such a map is preflexible is the
%composition $\pi f$ is simply branched, and for each ramification
%point $p$ of $\pi f$ the image $f(p)$ is a smooth point of the map
%$\pi$, the statement of Lemma~\ref{preflex} holds.

\subsection{The second construction}

Our second construction is a very minor modification of the
first. Given a family $\pi : X \to B$ as in~\ref{hyp} and a stable map
$f:C \rightarrow X$ satisfying hypothesis~\ref{hyp2}, 
we pick a general fiber $X_b$
of $\pi$ and two 
points $p, q \in C$ such that $f(p),f(q)\in X_b$. 
We then pick a stable map $f'_0:C'_0
\rightarrow X_b$ with smooth, rational domain and two points $p',q'\in
C'_0$ such that $f'_0(p')=f(p)$ and $f'_0(q')=f(q)$ such that
$(f'_0)^*T X_b$ is an ample bundle.  

\ps

We also pick a large number $N$
of other general points $p_i \in C$ and stable maps $f'_i:C'_i \rightarrow
X_{p_i}$ in the corresponding fibers and points $p'_i\in C'_i$ such
that $f'_i(p'_i)=f(p_i)$, such that $(f'_i)^*TX_{p_i}$ are ample and
such that $df'_i(T_{p'_i} C'_i)$ is a general line in
$T_{f(p_i)}X_{p_i}$ (just as in the first construction).
Finally, we let $F:D\rightarrow X$ be the stable map obtained by
gluing $C$ and $C'_0,C'_1,\dots,C'_N$ by identifying $p$ with $p'$,
$q$ with $q'$ and for each $i$ identifying $p_i$ with $p'_i$.
As in the first construction, for $N$ large enough, there will be
deformations of $F:D\rightarrow X$ 
with smooth domain, and we choose
$\tilde{f}:\tilde C\rightarrow X$ a general smooth deformation of
$D$ (i.e. a general stable map which is a member of a $1$-parameter
family of stable maps specializing to $F$).
This process, starting with the stable map $f:C \rightarrow X$ and
arriving at the new stable map $\tilde{f}:\tilde C\rightarrow X$, 
is our second construction. It has the properties that

\begin{enumerate}
\item $\tilde{f}:\tilde{C}\rightarrow X$ satisfies hypothesis~\ref{hyp2};
\item the degree $d$ of $\tilde{mu} := \pi\circ\tilde{f}$ is the same
  as the degree of $\mu:=\pi\circ f$;
\item the genus of the new curve $\tilde C$ is one greater than the
genus of the curve $C$ we started with;
\item for any $n$ points $q_1,\dots,q_n \in \tilde C$ we have
$H^1(\tilde C, \cN_{\tilde{f}}(-q_1-\dots-q_n)) = 0$; and
\item the branch divisor of $\mu$
has two new points: it consists of a small
deformation of the branch divisor of $\mu$,
together with a pair of simple branch points $b', b'' \in B$ near $b$,
each having as monodromy the transposition exchanging the sheets of
$\tilde C$ near $p$ and $q$.
\end{enumerate}

In effect, we have simply introduced two new simple branch points to
the cover $C \to B$, with assigned (though necessarily equal)
monodromy. Note that we can apply this construction repeatedly, to
introduce any number of (pairs of) additional branch points with
assigned (simple) monodromy; or we could carry out a more general
construction with a number of curves $C_0$.

\section{Proof of the main theorem}\label{mainproof}

\subsection{The proof in case $B = \P^1$}\label{mainarg}

We are now more than amply equipped to prove the theorem. We start
with a morphism $\pi : X \to B$ as in~\ref{hyp}. To begin with, by
hypothesis $X$ is projective; embed in a projective space and take the
intersection with $\dim(X) - 1$ general hyperplanes to arrive at a
smooth curve $C \subset X$. The inclusion $f:C\hookrightarrow X$
satisfies hypothesis~\ref{hyp2} and is the stable map we start with.

\ps

What do $C$ and the associated map $\mu : C \hookrightarrow X \to B$
look like? To answer this, start with the simplest case: suppose that
the fibers $X_b$ of $\pi$ do not have multiple components, or in other
words that the singular locus $\pi_{\rm sing}$ of the map $\pi$ has
codimension 2 in $X$. In this case we are done: $C$ misses
$\pi_{\rm sing}$ altogether, so that all ramification of $\mu : C \to
B$ occurs at smooth points of fibers; and simple dimension counts
show we can choose $C$ so that the branching of $\mu$ is simple. 
In other words, $f:C\hookrightarrow X$ is
pre-flexible already.

\ps

The problems start if $\pi$ has multiple components of fibers. If $Z
\subset X_b$ is such a component, then each point $p \in C \cap Z$
will be a ramification point of $\mu$, and no deformation of $C$ will
move the corresponding branch point $\pi(p) \in B$. The curve $C$ can
not be flexible. And of course it's worse if $\pi$ has a multiple
(that is, everywhere-nonreduced) fiber: in that case $\pi$ cannot
possibly have a section (a postiori we will see this cannot occur).

\ps

To keep track of such points, let $M \subset B$ be the locus of points
such that the fiber $X_b$ has a multiple component. Outside of $M$,
the map $\mu : C \to B$ is simply branched, and all ramification
occurs at smooth points of fibers of $\pi$.

\ps

Now here is what we're going to do.  First, pick a base point $p_0 \in
B$, and draw a cut system: that is, a collection of real arcs joining
$p_0$ to the branch points $M \cup N$ of $\mu$, disjoint except at
$p_0$. The inverse image in $C$ of the complement $U$ of these arcs is
simply $d$ disjoint copies of $U$; call the set of sheets $\Gamma$
(or, if you prefer, label them with the integers 1 through $d$). Now,
for each point $b \in M$, denote the monodromy around the point $b$ by
$\sigma_b$, and express this permutation of $\Gamma$ as a product of
transpositions:
$$
\sigma_b = \tau_{b,1}\tau_{b,2}\dots\tau_{b,{k_b}} 
$$
so that in other words
$$
\tau_{b,{k_b}}\dots\tau_{b,2}\tau_{b,1}\sigma_b =  I
$$
is the identity. For future reference, let $k = \sum k_b$. We will
proceed in three stages.

\ps

\ni \underbar{Stage 1}: We use our second construction to produce a
new stable map $\tilde{f}:\tilde C\rightarrow X$ such that
$\tilde{f}$ is unramified at $\tilde{f}^{-1}(M)$, and such that
in a neighborhood of each point $b \in M$ we have 
$k_b$ new pairs of simple branch points of $\tilde{\mu}=\pi\circ
\tilde{f}$, say $s_{b,i}, t_{b,i} \in B$, with
the monodromy around $s_{b,i}$ and $t_{b,i}$ equal to
$\tau_{b,i}$. Note that $\tilde C$ will have genus $g(C) + k$, and
that the branch divisor of the projection $\tilde \mu $
will be the union of $M$, of a small deformation $\tilde N$ of $N$,
and of all the points
$s_{b,i}$ and $t_{b,i}$. In particular we can find disjoint
discs $\Delta_b \subset B$, with $\Delta_b$ containing the points $b$
and $t_{b,1}, t_{b,2},\dots,t_{b,{k_b}}$, so that the monodromy around
the boundary $\partial \Delta_b$ of $\Delta_b$ is trivial.

\ps

Now, for any fixed integer $n$ this construction can be carried out so
that the stable map $\tilde{f}$ has the property that $H^1(\tilde C,
\cN_{\tilde{f}}(-q_1-\dots-q_n)) = 0$ for any $n$ points $q_i \in
\tilde C$. Here we want to choose
$$
n = \#N +  2k
$$
so that there are global sections of the normal sheaf
$\cN_{\tilde{f}}$ 
with arbitrarily assigned values on the ramification points of
$\tilde C$ over $N$ and the points $s_{b,i}$ and $t_{b,i}$.  This
means in particular that we can deform the stable map
$\tilde{f}:\tilde C\rightarrow X$ so as to
deform the branch points of $\tilde \mu$ outside of $M$
independently. What we will do, then, is

\newpage

\ni \underbar{Stage 2}: We will vary $\tilde{f}:\tilde{C}\rightarrow
X$ so as to {\em keep all 
the branch points $b \in N$ and all the points $s_{b,i}$ fixed; and
for each $b \in M$ specialize all the branch points $t_{b,i}$ to $b$
within the disc $\Delta_b$}.

%\vspace{3.5in}
\label{splat}
\begin{picture}(330,270)
\put(10,10){\makebox(330,240){\includegraphics{splat.eps}}}
\end{picture}
%\hspace{.2in}
%\special{picture splat}

\

\ni To say this more precisely, let $\beta \in N_1(X)$ be the class of
the stable map $\tilde{f}:\tilde{C}\rightarrow X$, and consider the maps
$$
\overline M_{g',0}(X, \beta) \longrightarrow \overline M_{g',0}(B,
d) \longrightarrow B_{2d+2g'-2}
$$
with the second map assigning to a stable map $C \to B$ its branch
divisor (we only need the \emph{branch morphism} as a set map, but in
fact it is a regular morphism ~\cite[FaP]).  
What we are saying is, starting at the branch divisor
$$
D_1 =  \tilde N + \sum s_{b,i} +  \sum t_{b,i} + \sum_{b \in M} k_b\cdot b
$$
of the map $\tilde \mu$, draw an analytic arc $\gamma = \{D_\l\}$ in
the subvariety
$$
\Phi \, = \, \tilde N + \sum s_{b,i} + \sum_{b \in M} k_b\cdot b +
\sum(\Delta_b)_{k_b} \, \subset \, B_{2d+2g'-2}
$$
tending to the point
$$
D_0 = \tilde N + \sum s_{b,i} +  2\sum_{b \in M} k_b\cdot b .
$$ 
Since the image of the composition
$$
\overline M_{g',0}(X, \beta)  \longrightarrow  B_{2d+2g'-2}
$$
contains $\Phi$, we can find an arc $\delta = \{f_{\nu}\}$ in
$\overline M_{g',0}(X, \beta)$ that maps onto $\gamma$, with $f_1$ the
inclusion $\tilde C \hookrightarrow X$.

\ps

\ni \underbar{Stage 3}: Let $f_0 : C_0 \to X$ be the limit, in
$\overline M_{g',0}(X, \beta)$, of the family of curves constructed in
Stage 2: that is, the point of the arc $\delta$ over $D_0 \in \Phi
\subset B_{2d+2g'-2}$. Let $A \to C_0$ be the normalization of any
irreducible component of $C_0$ on which the composition $\pi f_0$ is
nonconstant (that is, whose image is not contained in a fiber), and
let $f_A : A \to X$ be the stable map where $f_A$ is the restriction of 
$f_0$ to $A$.  In particular, $f_A:A\to X$ is a stable map satisfying
hypothesis~\ref{hyp2}.  

\ps

By construction, the composition $\mu_A :=\pi \circ f_A$ is unramified over a
neighborhood of $M$: the monodromy around the boundary $\partial
\Delta_b$ of each disc $\Delta_b$ is trivial, and it can be branched
over at most one point $b$ inside $\Delta_b$, so it can't be branched
at all over $\Delta_b$. Indeed, it is (at most) simply branched over
each point of $N$ and each point $s_{b,i}$, and unramified
elsewhere. Moreover, since we can carry out the specialization of
$\tilde{f}:\tilde{C}\rightarrow X$ 
above with the entire fiber of $\tilde{\mu}$ over the points
of $N$ and the $s_{b,i}$ fixed, the ramification of $\mu_A$ 
over these points will occur at smooth points of the corresponding
fibers of $\pi$. In other words, {\em the map $f_A : A \to X$ is
preflexible}, and we are done.  \qed

\subsection{The proof for arbitrary curves $B$}\label{barb}

As we indicated at the outset, there are two straightforward ways of
extending this result to the case of arbitrary curves $B$.

\ps

For one thing, virtually all of the argument we have made goes over
without change to the case of base curves $B$ of any genus $h$. The
one exception to this is the statement that the space $\overline
M_{g,0}(B,d)$ of stable maps $f : C \to B$ of degree $d$ from curves
$C$ of genus $g$ to $B$ has a unique irreducible component whose
general member corresponds to a flat map $f : C \to B$. This is false
in general---consider for example the case $g = d(h-1) + 1$ of
unramified covers. It is true, however, if we restrict ourselves to
the case $g \gg h,d$ (that is, we have a large number of branch
points) and look only at covers whose monodromy is the full symmetric
group $S_d$. Given this fact, and observing that our second
construction allows us to increase the number of branch points of our
covers $C \to B$ arbitrarily, the theorem can be proved for general
$B$ just as it is proved above for $B \cong \P^1$.

\ps

Alternatively, Johan de Jong showed us a simple way to deduce the
theorem for general $B$ from the case $B \cong \P^1$ alone. We argue
as follows: given a map $\pi : X \to B$ with rationally connected
general fiber, we choose any map $g : B \to \P^1$ expressing $B$ as a
branched cover of $\P^1$. We can then form the ``norm" of $X$: this is
the (birational isomorphism class of) variety $Y \to \P^1$ whose fiber
over a general point $p \in \P^1$ is the product
$$
Y_p = \prod_{q \in g^{-1}(p)} X_q .
$$
Since the product of rationally connected varieties is again
rationally connected, it follows from the $\P^1$ case of the theorem
that $Y \to \P^1$ has a rational section, and hence so does $\pi$.

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\end{document}
