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\title{Rational surfaces in Index-one Fano hypersurfaces}

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

\author[Starr]{Jason Michael Starr}
\address{Department of Mathematics \\
Massachusetts Institute of Technology \\
Cambridge MA 02139}
\email{jstarr@math.mit.edu} 

\date{\today}

\begin{document}

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\begin{abstract}
  We give the first evidence for a conjecture that a general,
  index-one, Fano hypersurface is not unirational: (i) a general point
  of the hypersurface is contained in no rational surface ruled,
  roughly, by low-degree rational curves, and (ii) a general point is
  contained in no image of a Del Pezzo surface.
\end{abstract}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Introduction
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\section{Introduction}

For complex, projective varieties a classical notion is
unirationality: A variety rationally dominated by projective space is
\emph{unirational}.  A modern notion is rational connectedness: A
variety is \emph{rationally connected} if every pair of points is
contained in a rational curve.  Every unirational variety is
rationally connected.  The two notions agree for curves and surfaces.
Conjecturally they disagree in higher dimensions.

\begin{conj} \label{conj-IM}
For every integer $n\geq 4$ there exists a non-unirational, smooth,
degree-$n$ hypersurface in $\PP^n$.
\end{conj}

A smooth hypersurface in $\PP^n$ of degree $d\leq n$ is an
\emph{index-$(n+1-d)$, Fano manifold}.  By ~\cite{C} and ~\cite{KMM},
every Fano manifold is rationally connected.  Versions of Conjecture
~\ref{conj-IM} have been around for decades.  The specific case $n=4$
is attributed to Iskovskikh and Manin, ~\cite{IskManin}.

In ~\cite{Ksimple}, Koll\'ar suggested an approach to proving
Conjecture ~\ref{conj-IM}.  A general point of an $n$-dimensional,
unirational variety is contained in a $k$-dimensional, rational
subvariety for each $k < n$.  Thus, Conjecture ~\ref{conj-IM} for
$n\geq 5$ follows from the next conjecture.

\begin{conj} \label{conj-surf}
For every integer $n\geq 5$, there exists a smooth, degree-$n$
hypersurface in $\PP^n$ whose general point is contained in no
rational surface.
\end{conj}  

In fact the conjecture fails for $n=4$.  The following argument was
related to us by Rahul Pandharipande and Joe Harris
and independently by Miles Reid.  
For every smooth degree $4$ hypersurface $X\subset
\PP^4$, for general $p\in X$, the set $C_p$ of lines $L$ osculating to
$X$ to order $3$ at $p$ is a smooth conic in the projective tangent
bundle $\PP T_p X\cong \PP^2$.  Of course $L\cap X = 3p + q_L$ for a
point $q_L$.  
Varying $L$ in $C_p$, the points $q_L$ sweep out a rational curve
$B_p$ (of degree $6$).
Varying $p$ in a rational curve $D$, the union of the curves $B_p$ is
a rational surface $B_D$.
Finally, varying $D$ among rational curves in $X$, a general point of
$X$ is contained in a rational surface $B_D$.

We give the first evidence for Conjecture ~\ref{conj-surf}.

\begin{thm} \label{thm-1}
For every integer $n\geq 5$, every smooth, degree-$n$ hypersurface $X$
in $\PP^n$ contains a countable union of closed, codimension-$2$
subvarieties containing the image of every generically-finite,
rational transformation $\PP^1 \times \PP^1 \dashrightarrow X$ mapping
a general fiber $\{t\}\times \PP^1$ isomorphically to an
$(n-1)$-normal, smooth, rational curve in $X$.
\end{thm}

\begin{thm} \label{thm-2}
For every integer $n\geq 5$, every smooth, degree-$n$ hypersurface in
$\PP^n$ contains a countable union of closed, codimension-$2$
subvarieties containing the image of every generically finite, regular
morphism from a Del Pezzo surface to $X$.
\end{thm}

A projective variety is \emph{$k$-normal} if every global section of
the restriction of $\OO_{\PP^n}(k)$ is the restriction of a global
section on $\PP^n$.  

We present two approaches here.  First, given a rational surface $S$ and a
regular morphism $f:S\rightarrow X$, to prove deformations of $f$ are
contained in a codimension-$2$ subvariety of $X$ it suffices to prove
$\bigwedge^{n-4}(f^*T_X/T_S)/\text{Torsion}$ has no nonzero
section.  In Section ~\ref{sec-normal} this is used to prove Theorems
~\ref{thm-1} and ~\ref{thm-2}.

Second, a rational surface with a pencil of rational curves gives a
morphism from $\PP^1$ to a parameter space of rational curves on $X$.
There is a construction of algebraic differential forms on
the parameter space. Since $\PP^1$ has only the zero form, these forms
impose restrictions on rational curves in the parameter space.  
In Section
~\ref{sec-forms}, these restrictions are used
to prove the following generalization
of Theorem ~\ref{thm-1}.

\medskip\noindent
\begin{thm} \label{thm-3}
For every integer $n\geq 5$, every smooth, degree-$n$ hypersurface $X$
in $\PP^n$ contains a countable union of closed, codimension-$2$
subvarieties containing the image of every generically-finite,
rational transformation $S \dashrightarrow X$ from a surface with a
pencil of curves mapping the general curve isomorphically to an
$(n-1)$-normal, smooth curve of genus $0$ or $1$, also assumed
non-degenerate if the genus is $1$.
\end{thm}

As the second approach does not apply to Theorem ~\ref{thm-2}, the
first approach is more productive.  However, further progress in
proving Conjecture ~\ref{conj-surf} will likely use both approaches,
as well as new ideas.

\bigskip

\paragraph{{\bf Acknowledgments.}} 
The authors thank Mike Roth for many illuminating discussions.  We
also thank the referee for helpful comments which improved the
exposition.  

\bigskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% First approach

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The first approach} \label{sec-normal}
 


Let $X$ be a smooth, degree-$n$ hypersurface in $\PP^n$, $n\geq 5$.  
Denote by $\Hilb(X)$ the Hilbert scheme of $X$.
Theorem ~\ref{thm-1} follows easily from the next theorem.

\begin{thm}\label{hmain}
Let $Z$ be an irreducible subscheme of
$\text{Hom}(\PP^1,\Hilb(X))$ satisfying,
\begin{enumerate}
\item[(i)]
the associated morphism $Z\times \PP^1 \rightarrow \Hilb(X)$
does not factor through the projection $Z\times \PP^1 \rightarrow Z$,
and
\item[(ii)]
the image of a general point of $Z\times \PP^1$ parametrizes a smooth,
$(n-1)$-normal, rational curve in $X$.
\end{enumerate}
Then there exists a codimension $\geq 2$ subvariety of $X$ containing
all curves parametrized by $Z\times \PP^1$.
\end{thm}

A morphism $\PP^1 \rightarrow \Hilb(X)$ is equivalent to a
closed subscheme $S'\subset \PP^1 \times X$, flat over $\PP^1$.
If a general point of $\PP^1$ parametrizes a
smooth rational curve, then $S'$ is an irreducible surface.  Any
desingularization $S$ of $S'$ is a surface fitting in a diagram,

\begin{equation} \label{S}
\xymatrix{ S \ar[r]^f \ar[d]^{\pi} & X \\ \P^1
}
\end{equation}

Associated to this diagram there is a derivative map
$$
df: T_S \rightarrow f^* T_X
$$
and a torsion-free sheaf
$$
\bigwedge^{n-4}(f^*T_X/T_S)/\text{Torsion}.
$$

\begin{prop} \label{sprop}
Let $Z$ be an irreducible subvariety of
$\text{Hom}(\PP^1,\Hilb(X))$ satisfying,
\begin{enumerate}
\item[(i)]
the associated morphism $Z\times \PP^1 \rightarrow \Hilb(X)$
does not factor through the projection $Z\times \PP^1 \rightarrow Z$,
\item[(ii)]
a general point of $Z\times \PP^1$ parametrizes a smooth curve in $X$,
and
\item[(iii)]
there is no codimension $2$ subvariety of $X$ containing all curves
parametrized by $Z\times \PP^1$.
\end{enumerate}
Then, for the morphism $\PP^1\rightarrow \Hilb(X)$ parametrized
by a general point of $Z$, the torsion-free sheaf
$\bigwedge^{n-4}(f^*T_X/T_S)/\text{Torsion}$ 
associated to the diagram in Equation ~\ref{S}
has a nonzero global
section.
\end{prop}

\begin{rmk} \label{rmk-P1}
This proposition 
holds if $\PP^1$ is replaced by any other curve, and
even if $\text{Hom}(\PP^1,\Hilb(X))$ is replaced by the scheme of all
maps from smooth projective connected curves to $\Hilb(X)$. 
\end{rmk}

\begin{proof}
Replacing $Z$ by a dense, Zariski open subset if necessary, we may
assume $Z$ is smooth.  Let $V^\prime \subset Z \times \P^1 \times X$ be the
pullback of  
the universal family to $Z \times \P^1$.

%% $$ \xymatrix{V \ar[d] \ar[r] & 
%% U \ar[d] \\
%% Z \times \P^1 \ar[r] & H.}
%% $$
%% \medskip

Let $\phi: V \to V^\prime$ be a desingularization of $V^\prime$. Denote by 
$g^\prime$ the projection map from $V^\prime$ to $Z \times X$, and 
denote by $p^\prime$ the projection map from $V^\prime$ to $Z$.  
Let $g= g^\prime \circ \phi$, and let $p = p^\prime \circ \phi$.

$$\xymatrix{V \ar@(ru, lu)[rr]^{g} 
\ar[r]^{\phi} \ar[ddr]_{p} & 
V^\prime \ar[d] \ar@(dr, ur)[dd]^{p^\prime} \ar[r]^-{g^\prime} & Z \times X \\ 
& Z \times \P^1 \ar[d] \\
& Z. }$$

Replacing $Z$ by a dense, Zariski open subset if necessary, we 
may assume $p$ is smooth, cf. ~\cite[Corollary III.10.7]{H}.  
Associated to the morphism $g$ is the derivative map,
$$
dg: T_{V} \rightarrow g^* T_X.
$$
Associated to the morphism $p$ is the derivative map,
$$
dp: T_{V} \rightarrow p^* T_Z.
$$
By hypothesis, $dp$ is surjective.  Denote by $T_{p}$
the kernel of $dp$.  Because $Z\times \PP^1 \rightarrow
\Hilb(X)$ does not factor through $Z$, the restriction of
$g$ to a general fiber of $p$ maps generically
finitely to its image.  Therefore the following sheaf homomorphism is
generically injective,
$$
dg: T_{p} \rightarrow g^* T_X.
$$
As $T_{p}$ is locally free and $V$ is integral,
the sheaf homomorphism $dg$ is injective on all of $X$.

Denote $\text{Coker}(dg)$ by $\mc{N}$,
$$
\mc{N} := \text{Coker}(dg:T_{p} \rightarrow
g^* T_X).
$$
The following is a 
commutative diagram with exact rows
$$
\xymatrix{
0 \ar[r] & 
T_{p} \ar[r] \ar[d]_{=} & 
T_{V} \ar[r]^{dp} \ar[d]^{dg} &
p^*T_Z \ar[r] %\ar[d]^{u} 
& 
0 \\
0 \ar[r] & 
T_{p} \ar[r] &
g^* T_X \ar[r] &
\mc{N} \ar[r] &
0}
$$
By the universal property of cokernels, there is a unique sheaf
homomorphism,
$$
u :p^* T_Z \rightarrow \mc{N},
$$
such that the following diagram commutes,
$$
\xymatrix{
0 \ar[r] & 
T_{p} \ar[r] \ar[d]_{=} & 
T_{V} \ar[r]^{dp} \ar[d]^{dg} &
p^*T_Z \ar[r] \ar[d]^{u} 
& 
0 \\
0 \ar[r] & 
T_{p} \ar[r] &
g^* T_X \ar[r] &
\mc{N} \ar[r] &
0}
$$

By generic smoothness, the rank of $dg$ at a general point of
$V$ equals the dimension of the closure of
$\text{Image}(g)$.  By hypothesis, this is $\geq n-2$.
Therefore the rank of $u$ at a general point is $\geq n-4$.  Thus, the
restriction of $u$ to a general $(n-4)$-plane in the fiber of
$p^* T_Z$ has rank $n-4$.  A general $(n-4)$-plane is the
tangent space of a general $(n-4)$-dimensional subvariety of $Z$.
We may replace $Z$ be the smooth locus inside a general
$(n-4)$-dimensional subvariety of $Z$.  
Therefore we may
assume $Z$ is
$(n-4)$-dimensional and $u$ is generically injective.  

Associated to $u$, there is an induced map,
$$
\bigwedge^{n-4}(u): p^* \bigwedge^{n-4} T_Z \rightarrow
\bigwedge^{n-4} \mc{N}/\text{Torsion}.
$$
Because $u$ is generically injective and $n\geq 5$, this map is
generically injective.  

Let $z$ be a general point of $Z$, and denote by $S^\prime$ and 
$S$ the fibers of $p^\prime$ and $p$ over $z$, respectively. 
Since $V$ is smooth, $S$ is a
smooth surface.  Let $f: S \to X$ be the restriction of $g$ 
to $S$. The restriction of $\mc{N}$ to $S$ is precisely $f^*
T_X/T_S$.  The restriction of $p^* T_Z$ to $S$ is precisely
the trivial vector bundle $T_{Z,z}\otimes_\CC \OO_S$.  Since $z$ is
general, the restriction of $\bigwedge^{n-4}(u)$ is generically
injective.  Therefore, it is
a nonzero map,
$$
\bigwedge^{n-4}(u)|_S: (\bigwedge^{n-4} T_{Z,z})\otimes_\CC \OO_S \rightarrow
(\bigwedge^{n-4} f^* T_X/T_S)/\text{Torsion}.
$$
By our assumption $T_{Z,z}$ is $(n-4)$-dimensional. 
Therefore this nonzero map is equivalent to a
nonzero global section of $(\bigwedge^{n-4} f^*
T_X/T_S)/\text{Torsion}$ (well-defined up to nonzero scaling).
\end{proof}

\begin{prop} \label{sprop2}
Let $\PP^1 \rightarrow \Hilb(X)$ be a morphism with associated
diagram as in Equation~\ref{S}.  If the curve parametrized by a general
point of $\PP^1$ is smooth, rational and $(n-1)$-normal then
$$
h^0(S,\omega_S\otimes \bigwedge^{n-2}f^* T_X) = 0.
$$
\end{prop}

\begin{proof}
Pulling back the short exact sequence of tangent bundles
$$ 0 \to T_X \to T_{\P^n}|_X \to N_{X/\P^n} \cong \O_X(n) \to 0$$
to $S$ and taking its $(n-1)^{\text st}$ exterior power gives 
another short exact sequence 
$$ 0 \to \bigwedge^{n-1} f^*T_X \to \w^{n-1} f^* T_{\P^n} \to 
f^*\O_X(n) \otimes \w^{n-2} f^*T_X \to 0.$$
Tensoring this sequence with $\omega_S \otimes 
f^*\O_X(-n)$ gives the following short exact sequence
$$
0 \to \omega_S \otimes f^*\O_X(-n) \otimes 
\bigwedge^{n-1} f^*T_X   \to 
\omega_S \otimes f^*\O_X(-n) \otimes 
\bigwedge^{n-1} f^* T_{\P^n} \to 
\omega_S \otimes \bigwedge^{n-2} f^*T_X \to 0.
$$

By applying the long exact sequence of cohomology, we conclude that
$$
h^0(S, \omega_S \otimes \bigwedge^{n-2} f^*T_X) = 0 
$$
if both
\begin{enumerate}
\item[(i)]
$h^0(S, \omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1} f^*T_{\P^n}) = 0$, and
\item[(ii)]
$h^1(S,  \omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1} f^*T_X) = 0$.
\end{enumerate}

\textbf{Proof of (i).} 
Consider the Euler exact sequence on $\P^n$
$$ 0 \to \O_{\P^n} \to \O_{\P^n}(1)^{n+1}\to T_{\P^n} \to 0.$$
Pulling this back to $S$, and taking its $n^{\text{th}}$ 
exterior power gives the following exact sequence
$$ 0 \to \w^{n-1} f^* T_{\P^n}   
\to f^*\O_X(n)^{\oplus(n+1)} \to f^*\O_X(n+1) \to 0.$$
Tensoring with $\omega_S \otimes 
f^*\O_X(-n)$ gives an injective map 
$$
\omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1} f^*T_{\P^n} \to 
\omega_S^{\oplus(n+1)}.
$$
Thus it suffices to prove $h^0(S,\omega_S) = 0$.  Because the
general fibers of $S\rightarrow \PP^1$ is a rational curve, $S$ is a
rational surface.  Therefore $h^0(S,\omega_S) = 0$.

\textbf{Proof of (ii).}
There is a canonical isomorphism
$$ \omega_S \otimes f^*\O_X(-n) \otimes \w^{n-1} f^*T_X \cong \omega_S
\otimes f^*\O_X(-n+1).$$  
So by Serre duality, it suffices to prove
$h^1(S, f^*\O_X(n-1)) = 0$. Let $C$ be a general fiber of the map 
$\pi: S \to \P^1$. There is a short exact sequence
\begin{equation}\label{eq0}
0 \to f^*\O_X(n-1) \otimes I_{C/S} \to 
f^* \O_X(n-1) \to f^* \O_X(n-1) |_C \to 0,
\end{equation}
where $\mathcal I_{C/S}$ is the ideal sheaf of $C$ in $S$. 
By hypothesis, the image of $C$ by $f$ is 
$(n-1)$-normal in $\P^n$, therefore the map 
$$
H^0(S, f^*\O_X(n-1)) 
\to H^0(C, f^*\O_X(n-1)|_C)
$$  
is surjective.  The long exact sequence 
of cohomology associated
to the sequence in Equation \ref{eq0} gives an isomorphism
\begin{equation} \label{eqhalf}
H^1(S, f^*\O_X(n-1) \otimes I_{C/S}) \cong 
H^1(S, f^*\O_X(n-1)).
\end{equation}

Recall a coherent sheaf $\mc{F}$ on $S$ is
called \emph{$\pi$-relatively globally generated} if the following sheaf
homomorphism is surjective,
$$
\pi^*\pi_* \mc{F} \rightarrow \mc{F}.
$$
The surface $S$ is smooth and the general fiber of $\pi$ is a
smooth, rational curve.  
Therefore
if $\mc{F}$ is $\pi$-relatively globally generated, 
then
$R^1\pi_* \mc{F}$ is the zero sheaf.

Of course, since $f^*\O_{X}(n-1)$ is globally generated, 
it is $\pi$-relatively globally generated.  Because
$I_{C/S} \cong \pi^* \O_{\P^1}(-1)$, the twist $f^*\O_X(n-1)\otimes
I_{C/S}$ is $\pi$-relatively globally-generated.
Thus the sheaves $R^1\pi_*(f^*\O_X(n-1))$ and $R^1\pi_*(f^*\O_X(n-1)\otimes
I_{C/S})$ are each zero.  So,
by the Leray spectral sequence, there are canonical isomorphisms
\begin{equation}\label{eq1}
H^1(S, f^*\O_X(n-1)) \cong 
H^1(\P^1, \pi_*(f^*\O_X(n-1))),
\end{equation}
\begin{equation}\label{eq2}
H^1(S, f^*\O_X(n-1) \otimes \mathcal I_{C/S}) \cong 
H^1(\P^1, \pi_*(f^*\O_X(n-1) \otimes \mathcal I_{C/S})).
\end{equation}

Taken together, Equations \ref{eqhalf}, \ref{eq1} and \ref{eq2} give a
canonical isomorphism 
$$ H^1(\P^1, \pi_*(f^*\O_X(n-1))) \cong 
H^1(\P^1, \pi_*(f^*\O_X(n-1)) \otimes \O_{\P^1}(-1)).$$
This is possible only if $h^1(\P^1, \pi_*(f^*\O_X(n-1))) = 0$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{hmain}] 
Let $Z$ satisfy the hypotheses of Proposition \ref{sprop}, and let $S$
and $f$ satisfy the conclusion of Proposition \ref{sprop}.  The
injective sheaf homomorphism 
$df:T_S \rightarrow f^* T_X$ induces a multiplication
map,
$$
\bigwedge^2 T_S \otimes \bigwedge^{n-4} f^* T_X \rightarrow
\bigwedge^{n-2} f^* T_X.
$$
Because $\bigwedge^3 T_S$ is the zero sheaf, the kernel of the
multiplication map contains the image of the following sheaf homomorphism
$$
\bigwedge^2 T_S \otimes T_S \otimes \bigwedge^{n-5} f^* T_X
\rightarrow
\bigwedge^2 T_S \otimes \bigwedge^{n-4} f^* T_X.
$$
This image is precisely the kernel of the following sheaf homomorphism
$$
\bigwedge^2 T_S \otimes \bigwedge^{n-4} f^* T_X \rightarrow
\bigwedge^2 T_S \otimes \bigwedge^{n-4}(f^* T_X/T_S).
$$
Therefore the multiplication map factors uniquely through this
surjective sheaf homomorphism, i.e., there is an induced sheaf
homomorphism,
$$
\bigwedge^2 T_S \otimes \bigwedge^{n-4}(f^* T_X/T_S) \rightarrow
\bigwedge^{n-2} f^* T_X.
$$
Because $\bigwedge^{n-2} f^* T_X$ is locally free and $X$ is integral,
every coherent subsheaf is torsion-free.  Thus the homomorphism
factors through the torsion-free quotient,
$$
\bigwedge^2 T_S \otimes \bigwedge^{n-4}(f^* T_X/T_S)/\text{Torsion} \rightarrow
\bigwedge^{n-2} f^* T_X.
$$

On the open dense subset of $S$ where $f$ is unramified, this
homomorphism is clearly injective.  Because the domain of the homomorphism
is torsion-free and $S$ is integral, the sheaf homomorphism is
injective on all of $X$.
Tensoring with the canonical bundle of
$\omega_S$, this gives an injective sheaf homomorphism
$$ 
(\w^{n-4}(f^*T_X/T_S))/\text{Torsion} 
\to \omega_S \otimes \w^{n-2} f^*T_X.
$$
By hypothesis, $(\w^{n-4}(f^*T_X/T_S))/\text{Torsion}$ has a nonzero
global section.  Therefore $\omega_S \otimes \w^{n-2} f^*T_X$ also has
a nonzero global section.

On the other hand, for $Z$ satisfying the hypothesis of
Theorem~\ref{hmain}, Proposition \ref{sprop2} implies,
$$
h^0(S,\omega_S \otimes \w^{n-2} f^*T_X) = 0.
$$
Thus $Z$ does not satisfy the hypothesis of Proposition \ref{sprop2},
i.e., it does not satisfy Hypothesis (iii).  Therefore there exists a
codimension $2$ subvariety of $X$ containing all the curves
parametrized by $Z\times \PP^1$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-1}]
For every generically-finite, rational transformation $\PP^1 \times
\PP^1 \dashrightarrow X$ restricting to a closed immersion on a
general fiber, there is an associated rational transformation
$$
\PP^1 \dashrightarrow \Hilb(X), \ \ t \mapsto
\text{Image}(\{t\}\times \PP^1).
$$
By properness of the Hilbert scheme and the valuative criterion, this
extends to a regular morphism.  Therefore, associated to each rational
transformation is an element in the Hom-scheme
$\text{Hom}(\PP^1,\Hilb(X))$.  Those rational transformations
satisfying the hypothesis of Theorem ~\ref{thm-1} give a locally
closed subset of $\text{Hom}(\PP^1,\Hilb(X))$.  As
$\text{Hom}(\PP^1,\Hilb(X))$ is a countable union of quasi-projective
varieties, this subset is also a countable union of quasi-projective
subvarieties.  By Theorem ~\ref{hmain}, for each such subvariety $Z$,
there is a codimension-$2$ subvariety of $X$ containing every curve
parametrized by $Z\times \PP^1$.  This subvariety contains the image
of each rational transformation $\PP^1 \times \PP^1 \dashrightarrow X$
giving a point in $Z$.  Therefore, there exists a countable union of
codimension-$2$ subvarieties of $X$ containing the image of every
rational transformation satisfying the hypothesis of
Theorem~\ref{hmain}.
\end{proof}

The proof of the Theorem \ref{thm-2} is similar to the proof of
Theorem \ref{hmain}.  There is a preliminary proposition.

\begin{prop} \label{DP}
Let $X$ be a smooth hypersurface of degree $n$ in $\P^n$.  For every
Del Pezzo surface $S$ and every generically finite morphism $f: S \to
X$, the only global section of
$\w^{n-4} (f^*T_X/T_S)/\text{Torsion}$ is the zero section.
\end{prop}

\begin{proof}
The proof is similar to the proof of Theorem \ref{hmain}. 
By the same type of argument, it suffices to prove,
\begin{enumerate}
\item[(i)]
$h^0(S, \omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1} f^*T_{\P^n}) = 0$, and
\item[(ii)]
$h^1(S,  \omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1} f^*T_X) = 0$.
\end{enumerate}

The proof of (i) is the same as in the proof of Theorem \ref{hmain},
since $h^0(S, \omega_S) = 0$.  As for (ii), there is a
canonical isomorphism
$$\omega_S \otimes f^*\O_X(-n) \otimes 
\w^{n-1}f^*T_X \cong (\omega_S^{-1} \otimes 
f^*\O_X(n-1))^{-1}.$$
Denote $\omega_S^{-1}\otimes f^*\O_X(n-1)$ by $L$.  The sheaf
$f^*\O_X(n-1)$ is globally generated.  By the hypothesis that $S$ is a
Del Pezzo surface, $\omega_S^{-1}$ is ample.  Thus $L$ is ample.  By
Kodaira vanishing, $h^1(S,L^{-1}) = 0$.  So, using the
canonical isomorphism, $h^1(S, \omega_S \otimes f^*\O_X(-n) \otimes
\w^{n-1}f^*T_X) = 0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm-2}]
By the same countability argument as at the beginning of the section,
it suffices to prove that for every flat family
$$ \xymatrix{ D \ar[d]^p \ar[r]^-{\phi} & X \times V \\
V }$$
such that $\phi$ is generically finite and a general fiber of $p$ is a
Del Pezzo surface, the image of $\phi$ is contained in a subvariety of
codimension $\geq 2$.  Let $S$ be the fiber of $p$ over a general
point of $V$, and let $f$ be the restriction $\phi|_S: S \to X$.  As
in the proof of the Proposition \ref{sprop}, if the image of $\phi$ is
contained in no subvariety of codimension $\geq 2$, then $H^0(S,
\w^{n-4} (f^*T_X/T_S)/\text{Torsion})$ has a nonzero global section.
Thus Proposition \ref{DP} proves the image of $\phi$ is contained in a
subvariety of codimension $\geq 2$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section 3: Second approach
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The second approach} \label{sec-forms}
%%\marginpar{sec-nonuni}

Let $n\geq 4$ be an integer.
Let $X$ be a smooth hypersurface in $\PP^n$ of degree $n$. Let $M$ be
a proper, smooth
variety parametrizing a family of genus $g$ curves in $X$.   

\begin{thm} \label{thm-nonuni1}
Assume $g=0$ or $1$.  Assume $\text{dim}(M)=n-3$ and the curves
parametrized by $M$ sweep an $(n-2)$-fold in $X$.  Assume a general
curve parametrized by $M$ is embedded, smooth, and $(n-1)$-normal.
Further, if $g=1$, assume the curve is nondegenerate.  Then
$h^{n-3,0}(M) \geq 1$.
In particular, $M$ is
not uniruled.
\end{thm}

Here is the outline of the proof.
\begin{enumerate}
\item[(i)]
There is a natural correspondence between $X$ and $M$ giving a map
$$
\phi:H^1(X,\Omega^{n-2}_X) \rightarrow H^0(M,\Omega^{n-3}_M).
$$  
It
suffices to prove $\phi\neq 0$.
\item[(ii)]
For a point $m$ of $M$, taking the fiber of a global section
of $\Omega^{n-3}_M$ at $m$ gives a map 
$$
\phi_m:H^1(X,\Omega^{n-2}_X) \rightarrow \Omega^{n-3}_M\otimes
\kappa(m).
$$
If $m$ is a general point of $M$, there is a description
of $\phi_m$ as the composition of a number of explicit ``component''
maps.  To prove $\phi\neq 0$, it suffices to prove $\phi_m$ is surjective.
\item[(iii)]
The hypotheses imply the component maps are surjective. 
For instance,
one of the component maps is
the ``restriction of sections'' map
$$
H^0(X,\OO_X(n-1))\rightarrow H^0(C,\OO_X(n-1)|_C)
$$
where $C$ is the curve parametrized by $m$.  Surjectivity of this map
is precisely the hypothesis that $C$ is $(n-1)$-normal.
\end{enumerate}

There is one preliminary reduction having to do with the \emph{type}
of family of curves parametrized by $M$.   
Embedded smooth curves in $X$ are parametrized by a scheme that is
simultaneously an open subset of three different proper parameter spaces:
the Chow variety of $X$, the Hilbert
scheme of $X$ and the Kontsevich space of stable maps to $X$.  Thus
there are (at least) three ways to generalize the notion of embedded
smooth curves. 
This
last one is most useful to us.  The Kontsevich (coarse moduli)
space is denoted by $\kgnb{g,0}(X)$.  

Because a general curve parametrized by $M$ is embedded and smooth,
there is a rational transformation $\zeta: M\dashrightarrow \kgnb{g,0}(X)$.
For smooth projective varieties $M$, $h^{n-3,0}(M)$ is a birational
invariant. 
So we are free to blow-up $M$ without changing $h^{n-3,0}(M)$.  
Therefore assume $\zeta$ is regular on all of $M$.

There is a universal family of curves over $\kgnb{g,0}(X)$ and a
map from this family to $X$.  The pullback family of curves 
is denoted $\pi:C_M\rightarrow M$.  The pullback of the map
is denoted $f_M:C_M\rightarrow X$.  

\subsection{Construction of the forms.} \label{subsec-forms}

First we construct the $(n-3)$-forms on $M$.  Then there will be some
work to prove some of these are nonzero.  The $(n-3)$-forms are
constructed using the obvious correspondence 
$$
M\xleftarrow{\pi} C_M
\xrightarrow{f_M} X
$$ 
between $M$ and $X$.
The first part of this correspondence is straightforward:  
For every pair of integers $(p,q)$
there is a pullback map
$$
f_M^*:H^{q+1}(X,\Omega^{p+1}_X) \rightarrow
H^{q+1}(C_M,\Omega^{p+1}_{C_M}).
$$
The other part of the correspondence is an ``integration along
fibers'' map 
$$
I:H^{q+1}(C_M,\Omega^{p+1}_{C_M}) \rightarrow H^q(M,\Omega^p_M).
$$
Naively it is clear what this is, but we include the construction
below for completeness.

\medskip\noindent
\textbf{Integration along fibers.}
Let $U\subset M$ be the (dense) open subset over which 
$C_U := \pi^{-1}(U)$ is a family of smooth curves.  There is a
canonical isomorphism
$$
\pi^*\Omega^p_U\otimes \omega_\pi|_U \cong
\Omega^{p+1}_{C_u}/\pi^*\Omega^{p+1}_U
$$
where $\omega_\pi|_U$ is the relative cotangent bundle
$\Omega_{C_U/U}$.  Composing with the quotient map gives
$$
\phi_p: \Omega^{p+1}_{C_U} \rightarrow
\pi^*\Omega^p_U\otimes \omega_\pi|_U
$$
As proved in \cite{dJS2} this extends to a sheaf homomorphism on all of $C_M$
$$
\phi_p:\Omega^{p+1}_{C_M} \rightarrow \pi^* \Omega^p_M \otimes
\omega_\pi.
$$
This sheaf homomorphism 
gives a map of cohomology
$$
\phi_{p,*}:H^{q+1}(C_M,\Omega^{p+1}_{C_M}) \rightarrow
H^{q+1}(C_M,\pi^*\Omega^p_M \otimes \omega_\pi).
$$

The morphism $\pi$ determines a Leray spectral sequence for cohomology
of sheaves on $C_M$.
The spectral sequence has an abutment map
$$
H^{q+1}(C_M,\pi^*\Omega^p_M \otimes \omega_\pi) \rightarrow
H^q(M,\Omega^p_M\otimes R^1\pi_*\omega_\pi).
$$
Of course there is a trace isomorphism 
$$
R^1\pi_*\omega_\pi
\xrightarrow{\cong} \OO_M.  
$$
So the abutment map can be written as
$$
a: H^{q+1}(C_M,\pi^*\Omega^p_M\otimes \omega_\pi) \rightarrow
H^q(M,\Omega^p_M).
$$
This gives the \emph{integration
  along fibers} map
$$
I = a\circ \phi_{p,*}:
H^{q+1}(C_M,\Omega^{p+1}_{C_M}) \rightarrow H^q(M,\Omega^p_M).
$$

\begin{lem}[\cite{dJS2}] \label{lem-nonuni3}
%\marginpar{lem-nonuni3}
For every pair of integers $(p,q)$
there is a natural map of $\mC$-vector spaces
$$
\phi_{p,q}=I\circ f_M^*:
H^{q+1}(X,\Omega^{p+1}_X) \rightarrow
H^q(M,\Omega^p_M).
$$
In particular, this gives a map of $\mC$-vector spaces
$$
\phi = \phi_{n-3,0}:H^1(X,\Omega^{n-2}_X)\rightarrow H^0(M,\Omega^{n-3}_M).
$$
\end{lem}

\subsection{Description of the forms -- overview}
\label{subsec-overview}

The goal is to prove $\phi\neq 0$.  Proving this directly from the
definition of $\phi$ seems difficult: the definition is fairly simple,
but not very explicit.

What would be
a more explicit description of $\phi$?  Because elements of
$H^0(M,\Omega^{n-3}_M)$ are sections of the sheaf $\Omega^{n-3}_M$,
one possibility is to try to describe the values of these sections at
some point $m\in M$.  The fiber $\Omega^{n-3}_M\otimes \kappa(m)$ is
the vector space of $(n-3)$-linear alternating forms on the Zariski
tangent space $T_m M$.  
Thus, given a point $m\in M$,
given an element
$\alpha \in H^1(X,\Omega^{n-2}_X)$ and given an
$(n-3)$-tuple of tangent vectors $\theta_1,\dots,\theta_{n-3} \in T_m
M$, the map $\phi$ is a rule 
$$
(\alpha,\theta_1,\dots,\theta_{n-3}) \mapsto \langle
\phi(\alpha)|_m, \theta_1\wedge \dots \wedge \theta_{n-3}
\rangle \in \mC.
$$
Equivalently, it is a map
$$
\phi_m: H^1(X,\Omega^{n-2}_X) \rightarrow (\bigwedge^{n-3} T_m
M)^\vee.
$$
An \emph{explicit description} of $\phi$
is quite simply an explicit description of the rule $\phi_m$ for
general $m$.    
In particular,
if $\phi_m\neq 0$ for some $m$
then $\phi\neq 0$.  

In fact there is such a description, at least when $m$ is in the
open subset $U$ parametrizing embedded smooth curves:
$\phi_m$ is the composition of a number of simple, explicit maps.  
To be honest, there are quite a number of these component maps.  For
this reason the composition $\phi_m$ is not very simple.  But it is
explicit.  

\begin{lem} \label{lem-nonuni5}
%\marginpar{lem-nonuni5}
Denote by $\mc{L}$ the invertible sheaf on $C$
$$
\mc{L} := \bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C.
$$
There is a commutative diagram 
%\marginpar{eqn-CD}
\begin{equation} \label{eqn-CD}
\xymatrix{
H^0(X,\Omega^n_{\PP^n}(2X)|X) \ar[d]_{r} \ar[r]^{\text{res}} &
H^1(X,\Omega^{n-2}_X) \ar[ddddd]^{\phi_m} \\
H^0(C,\Omega^n_{\PP^n}(2X)|_C) \ar[d]_{H^0(C,\psi)} \\
H^0(C,\omega_C\otimes \mc{L}^\vee) \ar[d]_{s} \\
H^1(C,\bigwedge^{n-2}N_{C/X}\otimes N_{X/\PP^n}^\vee|_C)^\vee
\ar[d]_{\delta^\dagger} \\
H^0(C,\bigwedge^{n-3}N_{C/X})^\vee \ar[d]_{w^\dagger} \\
(\bigwedge^{n-3} H^0(C,N_{C/X}))^\vee \ar[r]^{(\bigwedge
  d\zeta)^\dagger} & (\bigwedge^{n-3} T_m M)^\vee
}
\end{equation}
\end{lem}

Of course this is meaningless without an explicit description of the
component maps, which we give next.
But before
describing the component maps it is worth putting things in
perspective.  The proof that $\phi_m\neq 0$ uses only
Lemma~\ref{lem-nonuni5}, 
not the definition from Lemma~\ref{lem-nonuni3}.  So, the
reader is probably asking, why did we bother with Lemma~\ref{lem-nonuni3}?
Think of it this way:  We are about to give 
an explicit map from global
$(n-2,1)$-forms on $X$ to $(n-3,0)$-forms on $U$. Holomorphic
$(n-3,0)$-forms on $U$ are 
\emph{meromorphic} $(n-3,0)$-forms on $M$ (with poles contained in
$M-U$).
Now a uniruled variety has no nonzero holomorphic forms, but it has
\emph{many} nonzero meromorphic
forms. Therefore it is crucial that these meromorphic forms are
actually holomorphic on all of $M$.  This is precisely what
Lemma~\ref{lem-nonuni3} gives,
i.e., Lemma~\ref{lem-nonuni3} is a global regularity result.  

In order to make the rule $\phi_m$ explicit, we first must make
the \emph{inputs} of the rule explicit:
$\alpha\in H^1(X,\Omega^{n-2}_X)$ 
and the tangent vectors $\theta_1,\dots, \theta_{n-3}$.
This is what we do next.    

\subsection{Description of the inputs -- Griffiths
  residue calculus.} \label{subsec-GRC}

Griffiths described the primitive middle cohomology of a hypersurface
$X$ in $\PP^n$ as the residues of meromorphic $n$-forms on $\PP^n$
with poles along $X$, cf. \cite[Section 8]{GR}.  One can make the
description purely algebraic.  For the part we need, this is
particularly simple.  There is a short exact sequence of sheaves on $X$  
\begin{equation} \label{eqn-GRC}
\begin{CD}
0 @>>> \Omega_X^{n-2} @>>> \Omega^{n-2}_{\PP^n}(X)|_X @>>>
\Omega^{n-1}_X \otimes \OO_{\PP^n}(X)|_X @>>> 0.
\end{CD}
\end{equation}
The long exact sequence of cohomology gives a connecting map 
%\marginpar{eqn-Gr}
\begin{equation} \label{eqn-Gr}
\text{res}:
H^0(X,\Omega^n_{\PP^n}(2X)|_X) \rightarrow H^1(X,\Omega^{n-2}_X)
\end{equation}

By a small amount of diagram-chasing
$$
h^1(X,\Omega^{n-2}_{\PP^n}(X)|_X) = 0.
$$
Therefore $\text{res}$ is surjective, 
i.e., every element $\alpha \in H^1(X,\Omega^{n-2}_X)$ is the image
$\text{res}(\beta)$ of an element 
$$
\beta
\in H^0(X,\Omega^n_{\PP^n}(2X)|_X).
$$  
The usual way of saying this is
that an $(n-2,1)$-form $\alpha$ on $X$ is the ``residue'' of a meromorphic
$n$-form $\beta$ on $\PP^n$ with a double pole along $X$.  
(Of course
Griffiths describes all $(p,q)$-forms on $X$ in terms of residues, but
we only need the $(n-2,1)$-forms.)
Thus the linear map $\phi_m$
determines a linear map 
$$
\phi_m\circ \text{res}:
H^0(X,\Omega^n_{\PP^n}(2X)|_X) \rightarrow
(\bigwedge^{n-3} T_m M)^\vee.
$$
Since $\text{res}$ is surjective, 
$\phi_m$ is uniquely determined by $\phi_m\circ \text{res}$.

While we're at it, the component map $r$ is just restriction
of global sections of
$\Omega^n_{\PP^n}(2X)|_X$ to the curve $C$ parametrized by $m$,
$$
r: H^0(X,\Omega^n_{\P^n}(2X)|_X) \rightarrow
H^0(C,\Omega^n_{\PP^n}(2X)|_C).
$$

\subsection{Description of the inputs -- the tangent vectors.}
\label{subsec-theta} 

The inputs of the rule $\phi_m$ are an element $\alpha \in
H^1(X,\Omega^{n-2}_X)$ and an $(n-3)$-tuple of tangent vectors
$\theta_1,\dots, \theta_{n-3}\in T_m M$.  The previous section
describes $\alpha$ in terms of a more explicit element $\beta \in
H^0(X,\Omega^n_{\PP^n}(2X)|X)$.  Lemma~\ref{lem-nonuni5} asserts
that $\phi_m$ actually
depends only on the restriction $r(\beta)\in
H^0(C,\Omega^n_{\PP^n}(2X)|C)$, where $C$ is the curve in $X$
parametrized by $m$.  

In a similar way, Lemma~\ref{lem-nonuni5} asserts
that $\phi_m$ depends on the tangent
vectors $\theta_i$ only through certain associated elements, which are
more explicit.
Recall there is a morphism
$\zeta:U\rightarrow \Kgnb{g,0}(X)$.  The derivative of $\zeta$ is a map 
$$
d\zeta: T_m M \rightarrow T_{[C]} \Kgnb{g,0}(X)
$$
where $C$ is the embedded smooth curve parametrized by $m$.
Since
$$
T_{[C]} \Kgnb{g,0}(X) = H^0(C,N_{C/X})
$$
$d\zeta(\theta_i)$ is just a global section of the normal bundle
$N_{C/X}$ of $C$ in $X$.  

To be precise, the derivative $d\zeta$ gives
a map between exterior powers
$$
\bigwedge d\zeta: \bigwedge^{n-3} T_m M \rightarrow \bigwedge^{n-3}
H^0(C,N_{C/X}).
$$
The transpose is a map
$$
(\bigwedge d\zeta)^\dagger: (\bigwedge^{n-3} H^0(C,N_{C/X}))^\vee
\rightarrow (\bigwedge^{n-3} T_m M)^\vee.
$$
It will turn out there is a map
$$
\phi_{[C]}:H^0(C,\Omega^n_{\PP^n}(2X)|_C)
\rightarrow (\bigwedge^{n-3}
H^0(C,N_{C/X}))^\vee
$$
such that 
$$
\phi_m \circ \text{res} = \phi^{(1)}_m = \phi^{(2)}_m \circ r, \ \ \
\phi^{(2)}_m =  (\bigwedge d\zeta)^\dagger \circ \phi_{[C]}.
$$

There is another manipulation with the inputs $d\zeta(\theta_i)$.
Wedging global sections determines a map
$$
w:\bigwedge^{n-3} H^0(C,N_{C/X}) \rightarrow H^0(C,\bigwedge^{n-3}
N_{C/X}).
$$
The component map $w^\dagger$ is just the transpose
$$
w^\dagger: H^0(C,\bigwedge^{n-3} N_{C/X})^\vee \rightarrow
(\bigwedge^{n-3} H^0(C,N_{C/X}))^\vee.
$$

\subsection{Description of the rule -- the tangent bundle sequence.}
\label{subsec-tgt} 
The next component map relates $H^0(C,\bigwedge^{n-3} N_{C/X})^\vee$ to
$H^1(C,\mc{L})^\vee$ for a line bundle $\mc{L}$.  Then Serre
duality will relate this to $H^0(C,\omega_C\otimes \mc{L}^\vee)$.  The
final component map will come from a canonical isomorphism of
$\omega_C\otimes \mc{L}^\vee$ with another natural line bundle on $C$.  

The relation of $H^0(C,\bigwedge^{n-3} N_{C/X})^\vee$ to
$H^1(C,\mc{L})^\vee$ comes by way of the tangent bundle sequence of
$X$ in $\PP^n$ 
$$
\begin{CD}
0 @>>> T_X @>>> T_{\PP^n}|_X @>>> N_{X/\PP^n} @>>> 0.
\end{CD}
$$
This
gives
a sequence of normal bundles on $C$
$$
\begin{CD}
0 @>>> N_{C/X} @>>> N_{C/\PP^n} @>>>
N_{X/\PP^n}|_C @>>> 0. 
\end{CD}
$$
Taking $(n-2)^{\text{nd}}$ exterior powers turns this into a sequence 
$$
0 \rightarrow \bigwedge^{n-2} N_{C/X} \rightarrow \bigwedge^{n-2}
N_{C/\PP^n} \rightarrow 
(\bigwedge^{n-3} N_{C/X}) \otimes N_{X/\PP^n}|_C \rightarrow 0.
$$
Twisting each term by $N_{X/\PP^n}^\vee|_C$ gives an exact sequence,
\begin{equation} \label{eqn-ses}
0 \rightarrow \bigwedge^{n-2} N_{C/X} \otimes N_{X/\PP^n}^\vee|_C \rightarrow
\bigwedge^{n-2} N_{C/\PP^n} \otimes N_{X/\PP^n}^\vee|_C \rightarrow
\bigwedge^{n-3} N_{C/X} \rightarrow 0.
\end{equation}
%\marginpar{eqn-ses}
The long exact sequence of cohomology turns this into a connecting
map
$$
\delta: H^0(C,\bigwedge^{n-3} N_{C/X}) \rightarrow H^1(C,\bigwedge^{n-2}
N_{C/X} \otimes N_{X/\PP^n}^\vee|_C).
$$
The component map $\delta^\dagger$ is just the transpose
$$
\delta^\dagger: 
H^1(C,\bigwedge^{n-2} N_{C/X} \otimes N_{X/\PP^n}^\vee|_C)^\vee 
\rightarrow
H^0(C,\bigwedge^{n-3} N_{C/X})^\vee.
$$

Define $\mc{L}$ to be the line bundle 
$$
\mc{L} := \bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C.
$$
The component map $s$ is the isomorphism given by
Serre duality
$$
s: H^0(C,\omega_C\otimes \mc{L}^\vee) \xrightarrow{\cong}
H^1(C,\mc{L})^\vee =  
H^1(C,\bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C)^\vee.
$$

\subsection{Description of the rule -- the line bundle $\mc{L}$.}
\label{subsec-L} 
The last thing we need to 
understand the rule $\phi_m$ is an alternative description of the
line bundle
$$
\mc{L} := \bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C.
$$
To obtain this,
we will identify each of the two factors in this tensor product.

\medskip\noindent
\textbf{The first factor.}
The factor
$\bigwedge^{n-2}N_{C/X}$ is the determinant of $N_{C/X}$.  By 
adjunction for the embedding $C\hookrightarrow X$ the
determinant of $N_{C/X}$ is
canonically isomorphic to $\omega_C\otimes (\Omega^{n-1}_X)^\vee|_C$.

On the other hand, adjunction for the embedding $X\hookrightarrow
\PP^n$ gives an isomorphism $\Omega^{n-1}_X \cong
\Omega^n_{\PP^n}(X)|_X$.  These two adjunction isomorphisms together
give
$$
\bigwedge^{n-2}N_{C/X} \cong
\omega_C\otimes(\Omega^n_{\PP^n}(X))^\vee|_C.
$$

\medskip\noindent
\textbf{The second factor.}
Also, $N_{X/\PP^n}$ is isomorphic to
$\OO_{\PP^n}(X)|_X$.  Thus the factor, $N_{X/\PP^n}^\vee|_C$,
is isomorphic to $\OO_{\PP^n}(X)^\vee|_C$.  
Together these two identifications give an identification
\begin{equation} \label{eqn-Ns}
\mc{L} := \bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C \cong
\omega_C\otimes [\Omega^n_{\PP^n}(2X)]^\vee|_C.
\end{equation}

Taking the transpose and twisting by $\omega_C$ gives the isomorphism of
$\OO_C$-modules
$$
\psi:\Omega^n_{\PP^n}(2X)|_C \xrightarrow{\cong}
\omega_C \otimes [\bigwedge^{n-2} N_{C/X}\otimes
N_{X/\PP^n}^\vee|_C]^\vee = \omega_C\otimes \mc{L}^\vee.
$$
The final component map $H^0(C,\psi)$ is the map of global sections
associated to $\psi$
$$
H^0(C,\psi): H^0(C,\Omega^n_{\PP^n}(2X)|_C) \rightarrow
H^0(C,\omega_C\otimes \mc{L}^\vee).
$$

% \begin{lem} \label{lem-nonuni5}
% %\marginpar{lem-nonuni5}
% The map $\phi$ is induced by the isomorphism of $\OO_C$-modules
% $\phi^{(4)}_{[C]}$.  To be completely precise
% $$
% \phi_m \circ \text{res} = (\bigwedge d\zeta)^\dagger \circ
% w^\dagger\circ \delta^\dagger \circ s\circ
% H^0(C,\phi^{(4)}_{[C]})\circ r.
% $$
% \end{lem}

\begin{proof}[Proof of Lemma~\ref{lem-nonuni5}]
In the special case that $n=5$, this is 
\cite[Theorem 5.1]{dJS2}.  The proof in the general case is very
similar.  It amounts to much more of the same sort of diagram-chasing
above.  We will leave the details to the reader, referring to the
proof of \cite[Theorem 5.1]{dJS2} for the key ideas.

%% We wish there were a beautiful
%% proof of this.  However, the proof below is more of the same type of
%% diagram-chasing.  
%% Here are the main points.

%% There is a commutative diagram with exact rows and columns
%% %\marginpar{eqn-CD}
%% \begin{equation} \label{eqn-CD}
%% \xymatrix{
%% & 0 \ar[d] & 0 \ar[d] & \\
%%  & T_C \ar[d] \ar[r]^{\text{Id}} & T_C \ar[d] \\
%% 0 \ar[r] & T_X|_C \ar[d] \ar[r] & T_{\PP^n}|_C \ar[d] \ar[r] & 
%% N_{X/\PP^n}|_C \ar[d]^{\text{Id}} \ar[r] & 0 \\
%% 0 \ar[r] &  N_{C/X} \ar[d] \ar[r] & N_{C/\PP^n} \ar[d] \ar[r] & 
%% N_{X/\PP^n}|_C \ar[r] & 0 \\ 
%% &  0 &  0 &  
%% }
%% \end{equation}

%% The incarnation of $\phi^{(4)}_{[C]}$ is the collection of canonical
%% isomorphisms 
%% $$
%% \omega_C \otimes \bigwedge^{n-2} N_{C/X}^\vee \cong \Omega^{n-1}_X|_C
%% \cong \Omega^n_{\PP^n}(X)|_C.
%% $$
%% Denote any and all of these by $L$.  Twisting
%% Equation~\ref{eqn-CD} by $L$ gives a commutative diagram with exact
%% rows and columns
%% %\marginpar{eqn-CD2}
%% {\small
%% \begin{equation} \label{eqn-CD2}
%% \xymatrix@C=14pt{
%% & 0 \ar[d] & 0 \ar[d] \\
%% 0 \ar[r] & \bigwedge^{n-2} N_{C/X}^\vee \ar[d] \ar[r]^-{\cong} & 
%% \bigwedge^{n-1} N_{C/\PP^n}^\vee \otimes \OO_{\PP^n}(X)|_C \ar[d] 
%% \ar[r] &  0 \ar[d] \\
%% 0 \ar[r] & \Omega^{n-2}_X|_C \ar[d] \ar[r] & 
%% \Omega^{n-1}_{\PP^n}(X)|_C \ar[d] \ar[r] & 
%% \Omega^{n-1}_X|_C\otimes \OO_{\PP^n}(X)|_C \ar[d]^{\cong} 
%% \ar[r] & 0 \\
%% 0 \ar[r] & \omega_C \otimes \bigwedge^{n-3} N_{C/X}^\vee \ar[d] 
%% \ar[r] & \omega_C
%% \otimes \bigwedge^{n-2} N_{C/\PP^n}^\vee \otimes \OO_{\PP^n}(X)|_C 
%% \ar[d] \ar[r] & 
%% \omega_C \otimes \bigwedge^{n-2} N_{C/X}^\vee \otimes
%% \OO_{\PP^n}(X)|_C 
%% \ar[d] \ar[r] & 0 \\ 
%% & 0 & 0 & 0
%% }
%% \end{equation}
%% } 
%% (Just a word of explanation: For a sheaf $\mc{E}$ on $X$,
%% $\mc{E}(X)|_X$ denotes the tensor product 
%% $\mc{E}\otimes \OO_{\PP^n}(X)|_X$.  And for a sheaf $\mc{F}$ on $C$,
%% $\mc{F}(X)|_C$ denotes the tensor product, $\mc{F}\otimes
%% \OO_{\PP^n}(X)|_C$.)

%% \medskip\noindent
%% Consider the map 
%% $$
%% \Omega^{n-2}_X|_C \rightarrow \omega_C\otimes
%% \bigwedge^{n-3}N_{C/X}^\vee
%% $$
%% from the first column of Equation~\ref{eqn-CD2}.
%% There is an associated map of cohomology groups $H^1(C,-)$,
%% $$
%% H^1(C,\Omega_X^{n-2}|_C)\rightarrow H^1(C,\omega_C\otimes
%% (\bigwedge^{n-3} N_{C/X})^\vee).
%% $$
%% By Serre duality this is equivalent to a map
%% $$
%% t:H^1(C,\Omega_X^{n-2}|_C)\rightarrow
%% H^0(C,\bigwedge^{n-3}N_{C/X})^\vee
%% $$

%% There is a natural multiplication map
%% $$
%% w: \bigwedge^{n-3}H^0(C,N_{C/X})\rightarrow
%% H^0(C,\bigwedge^{n-3}N_{C/X}).
%% $$
%% The transpose is
%% $$
%% w^\dagger: H^0(C,\bigwedge^{n-3}N_{C/X})^\vee \rightarrow
%% (\bigwedge^{n-3} H^0(C,N_{C/X}))^\vee.
%% $$
%% Composing $w^\dagger\circ t$ gives
%% $$
%% w^\dagger\circ t:H^1(C,\Omega_X^{n-2}|_C)\rightarrow \bigwedge^{n-3}
%% H^0(C,N_{C/X})^\vee.
%% $$
%% Composing the restriction map 
%% $$
%% r':H^1(X,\Omega^{n-2}_X)\rightarrow H^1(C,\Omega_X^{n-2}|_C)
%% $$
%% with $w^\dagger\circ t$ gives a map,
%% $$
%% w^\dagger\circ t \circ r': H^1(X,\Omega^{n-2}_X) \rightarrow
%% (\bigwedge^{n-3} H^0(C,N_{C/X}))^\vee.
%% $$
%% By a similar argument to the one in the proof of ~\cite[Theorem
%% 5.1]{dJS2}, 
%% $$
%% \phi_{[C]} = w^\dagger\circ t\circ r'
%% $$
%% where $\phi_{[C]}$ is the map 
%% from Lemma ~\ref{lem-nonuni3}.  Therefore
%% $$
%% \phi^{(0)}_{[C]} = t\circ r'.
%% $$  

%% On the other hand, the second row 
%% \begin{equation} \label{eqn-2ndrow}
%% \begin{CD}
%% 0 @>>> \Omega_X^{n-2}|_C @>>> \Omega^{n-2}_{\PP^n}(X)|_C @>>>
%% \Omega^{n-1}_X|C\otimes \OO_{\PP^n}(X)|_C @>>> 0
%% \end{CD}
%% \end{equation}
%% of Equation~\ref{eqn-CD2} is precisely the restriction to $C$
%% is the short exact sequence giving rise to Equation~\ref{eqn-Gr} (from
%% the Griffiths residue calculus).
%% Therefore the map
%% $$
%% \phi^{(0)}_{[C]} = t\circ r'\circ \text{res}:
%% H^0(X,\Omega^n_{\PP^n}(2X)|_X) \rightarrow (\bigwedge^{n-3}
%% H^0(C,N_{C/X}))^\vee
%% $$
%% defining $\phi^{(1)}_{[C]}$ is obtained by taking the
%% pushout of Equation~\ref{eqn-2ndrow} 
%% by the last map
%% $$
%% \Omega^{n-2}_X|_C \rightarrow \omega_C\otimes
%% \bigwedge^{n-3}N_{C/X}^\vee
%% $$
%% in the first column of Equation~\ref{eqn-CD2}.  

%% This pushout is canonically isomorphic to the
%% third row 
%% \begin{equation} \label{eqn-3rdrow}
%% 0 \rightarrow \omega_C\otimes \bigwedge^{n-3} N_{C/X}^\vee \rightarrow
%% \omega_C\otimes \bigwedge^{n-2}N_{C/\PP^n}^\vee\otimes
%% \OO_{\PP^n}(X)|C \rightarrow \omega_C\otimes
%% \bigwedge^{n-2}N_{C/X}^\vee\otimes \OO_{\PP^n}(X)|_C \rightarrow 0
%% \end{equation}
%% of Equation~\ref{eqn-CD2}.  Therefore
%% $$
%% \phi^{(1)}_{[C]}  = t\circ r' \circ \text{res} 
%% = s'\circ \delta'\circ H^0(C,\phi^{(5)}_{[C]})\circ
%% r
%% $$
%% where $\delta'$ is 
%% the connecting map in cohomology 
%% $$
%% \delta': H^0(C,\omega_C\otimes
%% \bigwedge^{n-2}N_{C/X}^\vee\otimes\OO_{\PP^n}(X)|_C) \rightarrow 
%% H^1(C,\omega_C\otimes\bigwedge^{n-3}N_{C/X}^\vee)
%% $$
%% associated to Equation~\ref{eqn-3rdrow} and $s'$ is the Serre duality
%% isomorphism 
%% $$
%% s':H^1(C,\omega\otimes \bigwedge^{n-3}N_{C/X}^\vee) \rightarrow
%% H^0(C,\bigwedge^{n-3}N_{C/X})^\vee.
%% $$

%% Finally, Equation~\ref{eqn-3rdrow} is obtained from 
%% Equation~\ref{eqn-ses} 
%% $$
%% 0 \rightarrow \bigwedge^{n-2} N_{C/X} \otimes N_{X/\PP^n}^\vee|_C \rightarrow
%% \bigwedge^{n-2} N_{C/\PP^n} \otimes N_{X/\PP^n}^\vee|_C \rightarrow
%% \bigwedge^{n-3} N_{C/X} \rightarrow 0
%% $$
%% by dualizing and tensoring
%% with $\omega_C$.  In particular, using Serre duality and the
%% isomorphism $\phi^{(5)}_{[C]}$, the connecting
%% map $\delta'$ 
%% equals the transpose of the map
%% $$
%% \delta: H^0(C,\bigwedge^{n-3} N_{C/X}) \rightarrow H^1(C,\bigwedge^{n-2}
%% N_{C/X} \otimes N_{X/\PP^n}^\vee|_C).
%% $$
%% In other words
%% $$
%% s'\circ \delta' = \delta^\dagger\circ s.
%% $$

%% Putting these identities together finally gives
%% $$
%% \delta^\dagger \circ s \circ H^0(C,\phi^{(5)}_{[C]})\circ r = s'\circ
%% \delta'\circ H^0(C,\phi^{(5)}_{[C]})\circ r = 
%% \phi^{(1)}_{[C]}
%% =
%% \phi^{(0)}_{[C]}\circ \text{res}.
%% $$
\end{proof}

\subsection{Proof of the theorem.} \label{subsec-pfthm}
Finally we are ready to prove Theorem ~\ref{thm-nonuni1}. 
It suffices to prove the map $\phi$ from Lemma~\ref{lem-nonuni3} is
nonzero. 

\begin{proof}[Proof of Theorem~\ref{thm-nonuni1}]
Recall there is a morphism $\zeta:M\rightarrow \kgnb{g,0}(X)$.  The
subvariety of $X$ swept by curves in $M$ equals the subvariety swept
by curves in $\zeta(M)$.  Thus the dimension of the subvariety swept
by curves in $M$ has dimension $\leq 1 + \text{dim}(\zeta(M))$.  
By hypothesis, the curves parametrized by $M$ sweep an $(n-2)$-dimensional
subvariety of $X$. Thus $\text{dim}(\zeta(M)) = n-3$,
i.e., $\zeta$ is generically finite.  Because our base field is
characteristic $0$ and because $M$ is smooth, $\zeta$ is generically
unramified.  

The hypothesis that the curves of $M$ sweep an $(n-2)$-dimensional
subvariety of $X$ also implies the map $f_U:C_U\rightarrow X$ is
generically finite, where $U\subset M$ parametrizes only smooth curves
and $C_U\rightarrow U$ is this family of smooth curves.  Because $C_U$
is smooth and the characteristic is zero, $f_U$ is generically
unramified, i.e., 
the locus
where $f_U$ is ramified is a proper subvariety of $C_U$.  
Therefore,
for a general point $m\in U$ the map $f_U$ is unramified
at a general point of the fiber $C_m \subset C_U$.  

Let $V$ be
the maximal open subset of $M$ such that 
\begin{enumerate}
\item[(i)]
$\zeta|_V:V\rightarrow
\kgnb{g,0}(X)$ is unramified,
\item[(ii)]
every curve $C_m$ parametrized by $m\in V$ is embedded, smooth and
$(n-1)$-normal,
\item[(iii)]
if $g=1$, $C_m$ is also nondegenerate,
\item[(iv)]
and $f_U$ is unramified at a general point of $C_m\subset C_U$.
\end{enumerate}
The arguments above prove $V$ is dense in $M$.  
The claim is that for every $m\in V$, $\phi_m\neq 0$.  Because the
target of $\phi_m$ is $1$-dimensional this is equivalent to the claim
that $\phi_m$ is surjective.

Of
course it suffices to prove that $\phi_m\circ \text{res}$ is
surjective.  This is equivalent to the claim that the transpose map
$(\phi_m\circ \text{res})^\dagger$ is injective.  By
Lemma~\ref{lem-nonuni5}
$$
\phi_m \circ \text{res} = (\bigwedge d\zeta)^\dagger \circ
w^\dagger\circ \delta^\dagger \circ s\circ
H^0(C,\psi)\circ r.
$$
Thus the transpose is
$$
(\phi_m\circ \text{res})^\dagger = r^\dagger \circ
H^0(C,\psi)^\dagger \circ s^\dagger \circ \delta \circ w
\circ \bigwedge d\zeta.
$$
We will prove this in stages, by first proving injectivity of the maps
$$
(w\circ \bigwedge
d\zeta), s^\dagger,
H^0(C,\psi)^\dagger, \text{ and } r^\dagger.
$$
We reserve injectivity of $\delta$ for last, since this is the most
difficult to verify.

\medskip\noindent
\textbf{Injectivity of $w \circ \bigwedge d\zeta$.} 
It turns out this is precisely Item (iv) in the definition of $V$.
By hypothesis, there exists $p\in C_m$ such that $f_U$ is unramified
at $p$.  Consider the sheaf homomorphism
$$
df_U: N_{C_m/C_U} \rightarrow N_{C_m/X}
$$
and take its $(n-3)^\text{rd}$ exterior powers
$$
\bigwedge df_U: \bigwedge^{n-3} N_{C_m/C_U} \rightarrow
\bigwedge^{n-3} N_{C_m/X}. 
$$
Both sheaves are locally free, $C_m$ is integral, and the sheaf
homomorphism is injective at $p$ by hypothesis.  Therefore the sheaf
homomorphism 
is injective.  Thus the map of global sections
$$
H^0(C,\bigwedge df_U): H^0(C,\bigwedge^{n-3} N_{C_m/C_U}) \rightarrow
H^0(C,\bigwedge^{n-3} N_{C_m/X})
$$
is also injective.  
Because $C_m$ is a fiber of $C_U\rightarrow U$, 
$$
N_{C_m/C_U} = T_m M\otimes_k \OO_{C_m}.
$$
Therefore we have canonical isomorphisms
$$
H^0(C,\bigwedge^{n-3} N_{C_m/C_U}) = \bigwedge^{n-3} T_m M, \ \
H^0(C,\bigwedge df_U) = w\circ \bigwedge d\zeta.
$$
Because $H^0(C,\bigwedge df_U)$ is injective, $w\circ \bigwedge
d\zeta$ is injective.    

\medskip\noindent
\textbf{Injectivity of $H^0(C,\psi)^\dagger$ and $s^\dagger$.}
Because $\psi$ is an isomorphism, so is
$H^0(C,\psi)$, and thus also
$H^0(C,\psi)^\dagger$.  Similarly, the Serre duality map
$s$ is an isomorphism, and thus also $s^\dagger$ is an isomorphism.

\medskip\noindent
\textbf{Injectivity of $r^\dagger$.}
This is equivalent to proving surjectivity of $r$.
Recall $r$ is the restriction map
$$
r:H^0(X,\Omega^n_{\PP^n}(2X)|_X) \rightarrow
H^0(C,\Omega^n_{\PP^n}(2X)|_C).
$$
Now $\Omega_{\PP^n}(2X)$ is isomorphic to $\OO_{\PP^n}(-n-1)\otimes
\OO_{\PP^n}(2n) \cong \OO_{\PP^n}(n-1)$.  Therefore $r$ is the same as
the restriction map
$$
H^0(X,\OO_X(n-1))\rightarrow H^0(C,\OO_X(n-1)|_C).
$$
The hypothesis that $C$ is $(n-1)$-normal precisely says this
restriction map is surjective.  

\medskip\noindent
\textbf{Injectivity of $\delta$.}
Associated to the short exact sequence in Equation~\ref{eqn-ses},
there is an exact sequence of cohomology
\begin{equation} \label{eqn-ses2}
H^0(C,\bigwedge^{n-2} N_{C/\PP^n}\otimes N_{X/\PP^n}^\vee|_C)
\rightarrow H^0(C,\bigwedge^{n-3} N_{C/X}) \xrightarrow{\delta}
H^1(C,\bigwedge^{n-2} N_{C/X}\otimes N_{X/\PP^n}^\vee|_C).
\end{equation}
%\marginpar{eqn-ses2}

\begin{claim} \label{claim-nonuni1}
%\marginpar{claim-nonuni1}
$h^0(C,\bigwedge^{n-2} N_{C/\PP^n}\otimes
N_{X/\PP^n}^\vee|_C) = 0$; in particular $\delta$ is injective.
\end{claim}

\medskip\noindent
\textbf{Proof of Claim~\ref{claim-nonuni1}.}
We reduce the original claim to a second claim that is similar.  The
second claim 
will follow from the hypotheses on $g$ and on the curve $C$.

To begin with 
there is an isomorphism
$$
\bigwedge^{n-2} N_{C/\PP^n} \cong
N_{C/\PP^n}^\vee \otimes \bigwedge^{n-1}
N_{C/\PP^n}.
$$  
Adjunction for the embedding of $C$ in $\PP^n$ gives an
isomorphism
$$
\bigwedge^{n-1} N_{C/\PP^n} \cong
\omega_C \otimes (\Omega^n_{\PP^n})^\vee|_C \cong
\omega_C \otimes \OO_{\PP^n}(n+1)|_C.  
$$
Together with the isomorphism 
$N_{X/\PP^n}^\vee \cong \OO_{\PP^N}(-n)|_X$ this gives
$$
N_{C/\PP^n}\otimes N_{X/\PP^n}^\vee|_C \cong \omega_C\otimes
\OO_{\PP^n}(1)|_C \otimes
N_{C/\PP^n}^\vee. 
$$
In other words $N_{C/\PP^n}\otimes N_{X/\PP^N}^\vee|_C$ is obtained by
twisting the vector bundle $N_{C/\PP^n}^\vee$ by the line bundle
$\omega_C\otimes \OO_{\PP^n}(1)|_C$.

The vector bundle $N_{C/\PP^n}^\vee$ is a subsheaf of
$\Omega_{\PP^n}|_C$,
$$
\begin{CD}
0 @>>> N_{C/\PP^n}^\vee @>>> \Omega_{\PP^n}|_C @>>> \Omega_C @>>> 0.
\end{CD}
$$
Thus the twist $\omega_C\otimes \OO_{\PP^n}(1)|_C\otimes
N_{C/\PP^n}^\vee$ is a subsheaf of $\omega_C\otimes
\OO_{\PP^n}(1)|_C\otimes \Omega_{\PP^n}|_C$.  And so it goes with
global sections 
$$
H^0(C,\omega_C \otimes N_{C/\PP^n}^\vee \otimes
\OO_{\PP^n}(1)|_C) \subset
H^0(C,\omega_C \otimes
\Omega_{\PP^n}(1)|_C).
$$  
Thus Claim ~\ref{claim-nonuni1} follows from the following.

\begin{claim} \label{claim-nu}
%\marginpar{claim-nu}
$h^0(C,\omega_C\otimes \Omega_{\PP^n}(1)|_C)
= 0$.
\end{claim}

\medskip\noindent
\textbf{Proof of Claim~\ref{claim-nu}.}
The proof will ultimately boil down to injectivity of a certain
multiplication map $v$.  This injectivity follows from the hypotheses
that $g=0$ or $g=1$, that $C$ is embedded and smooth, and that $C$ is
nondegenerate if $g=1$.

There is an exact sequence on $\PP^n$.
$$
0 \to \Omega_{\PP^n}(1) \to H^0(\PP^n,\OO_{\PP^n}(1))\otimes_k
\OO_{\PP^n} \to \OO_{\PP^n}(1) \to 0.
$$
Pulling back and twisting with $\omega_C$ gives an isomorphism of
$H^0(C,\omega_C\otimes \Omega_{\PP^n}(1)|_C)$ with the kernel of a
multiplication map $v$,
$$
H^0(C,\omega_C\otimes \Omega_{\PP^n}(1)|_C) \cong
\text{Ker}(v),
$$
$$
v: H^0(\PP^n,\OO_{\PP^n}(1))\otimes_k H^0(C,\omega_C)
\rightarrow H^0(C,\omega_C\otimes \OO_{\PP^n}(1)|_C).
$$
Thus it suffices to prove $v$ is injective.

By hypothesis
$g=0$ or $g=1$.  
If
$g =0$ then already
$h^0(C,\omega_C) = 0$ and thus $\text{Domain}(v)=0$.  Thus $v$ is
injective if $g=0$.

If $g = 1$ then 
$\omega_C\cong \OO_C$.  Thus $v$ is simply the restriction map 
$$
H^0(\PP^n,\OO_{\PP^n}(1)) \rightarrow H^0(C,\OO_{\PP^n}(1)|_C).
$$
By hypothesis
$C$ is nondegenerate, i.e., the restriction map is injective.  Thus
again $v$ is injective if $g=1$.
This proves Claim~\ref{claim-nu} and thus also
Claim~\ref{claim-nonuni1}, 
both when
$g = 0$ and when $g = 1$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-3}]
This follows from Theorem~\ref{thm-nonuni1} by an argument similar to
the one in the proof of Theorem ~\ref{thm-1}.
\end{proof}

%\vspace {0.6 cm}


%\noindent Department of Mathematics and Statistics, Queen's University, 
%Kingston, Ontario, K7L 3N6.\\
%\noindent E-mail address : {\sf beheshti@mast.queensu.ca}


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\begin{thebibliography}{99}

{\small
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\end{thebibliography}

\end{document}
