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\title{Higher Fano manifolds and rational surfaces} 
\author[de Jong]{A. J. de Jong}
\author[Starr]{Jason Starr} 
%\date{Fall 2004}
\begin{document}

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\begin{abstract}
Let $X$ be a Fano manifold of pseudo-index $\geq 3$ such that
$c_1(X)^2-2c_2(X)$ is nef.  Irreducibility of some spaces of rational
curves on $X$ implies a general point
of $X$ is contained in a rational surface.
\end{abstract}

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\maketitle

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\section{Introduction} \label{sec-intro}

One consequence of the bend-and-break lemma is uniruledness of Fano
manifolds, \cite{Miyaoka-Mori86}.  In characteristic $0$,
Fano manifolds are even rationally connected, \cite{KMM92c}, \cite{Ca}.
We prove an analogous theorem with rational curves replaced by
rational surfaces for Fano manifolds satisfying positivity of the
second graded piece of the Chern character.

\begin{defn} \label{defn-1}
A Fano manifold is \emph{$2$-Fano} if $\text{ch}_2(T_X)$ is nef, where
$\text{ch}_2(T_X)$ is the second graded piece of the Chern character
$\frac{1}{2}(c_1(T_X)^2-2c_2(T_X))$.  In other words,
$\text{deg}(\text{ch}_2(T_X)|_S)$ is nonnegative for every surface $S$
in $X$.
\end{defn}

There is a space $\Kgnb{0,0}(X)$ parametrizing genus $0$ \emph{stable
maps} to $X$.  These are isomorphism classes of pairs $(C,f)$ of a proper,
at-worst-nodal, arithmetic genus $0$ curve $C$ and a morphism
$f:C\rightarrow X$ such that every irreducible component of $C$ on
which $f$ is constant intersects at least three other components.
To be precise, $\Kgnb{0,0}(X)$ is an algebraic stack with finite diagonal
(it is Deligne-Mumford in characteristic $0$).  It is proper and  
contains an open substack that
is isomorphic to the Hilbert scheme of smooth, embedded rational
curves in $X$, cf. \cite{FP}.  

Let $\mc{M}$ be a positive-dimensional, irreducible component of
$\Kgnb{0,0}(X)$ whose general point parametrizes a stable map with
irreducible domain, i.e., a morphism from $\PP^1$ to $X$.
Denote by $M$ the coarse moduli space of $\mc{M}$.  Denote by
$\Delta$ the locally principal closed substack of $\Kgnb{0,0}(X)$
parametrizing stable maps with reducible domain.  The closed substack
$\mc{M}\cap \Delta$ is a Cartier divisor.  The question we consider is
uniruledness of $M$.

\begin{thm} \label{thm-main}
If $X$ is $2$-Fano, every point of $M$ parametrizing a free curve and
contained in a proper curve in $M - M\cap \Delta$ is contained in a
rational curve in $M$.  

If a general point of $M$ parametrizes a birational, free curve and is
contained in a proper curve in $M-M\cap \Delta$, then a general point
of $X$ is contained in a rational surface.
\end{thm}

For a proper, smooth curve $C$ in $M- M \cap \Delta$
the deformation theory of
$\text{Hom}(C,M)$ is fairly simple and gives a lower bound on the
dimension of $\text{Hom}(C,M)$, cf. Lemma ~\ref{lem-3}.  Using the
hypothesis that $X$ is $2$-Fano, this lower bound can be made
arbitrarily large.  Then the bend-and-break approach of
\cite{Miyaoka-Mori86} implies $M$ contains a rational curve through
every point of $C$.

Fortunately there are nice sufficient conditions for $M-M\cap
\Delta$ to contain many proper curves.
Recall the \emph{pseudo-index} of $X$ is defined to be the minimal
$c_1(T_X)$-degree of any rational curve in $X$.

\begin{prop} \label{prop-cont}
If the pseudo-index of $X$ is $\geq 3$ and every irreducible component
of $\mc{M}\cap \Delta$ is an irreducible component of $\Delta$, then
$M - M\cap \Delta$ is a union of proper curves.

If the pseudo-index of $X$ is $\geq 3$, then every free morphism
deforms to a birational, free morphism.
\end{prop}

The proof uses a contraction of the locally principal closed subspace
$\Delta$ in $\Kgnb{0,0}(X)$ discovered independently by several
groups: Coskun, Harris and one of us, \cite{CHS1}; Adam Parker,
\cite{Parker}; and Musta\c{t}\v{a} and Musta\c{t}\v{a}, unpublished
(but see \cite{AAMus}).

Section ~\ref{sec-ex} gives some examples where the hypotheses of
Theorem ~\ref{thm-main} and Proposition ~\ref{prop-cont} hold.  
Section
~\ref{sec-sharp} shows Theorem ~\ref{thm-main} is sharp in two ways.
First, there are Fano manifolds that are not $2$-Fano where the
components $\mc{M}$ are not uniruled.  Second, there are $2$-Fano
manifolds where the components $\mc{M}$ are uniruled but not
rationally connected.  Finally Section ~\ref{sec-spec} speculates on
sufficient conditions for the components $\mc{M}$ to be rationally
connected.

\section{Proof of the theorem} \label{sec-pf}

For every point $x$, denote by $\text{Hom}(\PP^1,X,0\mapsto
x)_{\text{nc}}$ the open subscheme of $\text{Hom}(\PP^1,X,0\mapsto x)$
parametrizing nonconstant morphisms.

\begin{lem} \label{lem-1}
The dimension of every irreducible component of
$\text{Hom}(\PP^1,X,0\mapsto x)_{\text{nc}}$ is at least as large as
the pseudo-index of $X$.
\end{lem}

\begin{proof}
This follows from \cite[Theorem II.1.2, Corollary II.1.6]{K}.  
\end{proof}

\begin{proof}[Proof of Proposition ~\ref{prop-cont}]
First of all, the statement that a general deformation of a free
morphism is birational is very similar to \cite[Thm. II.3.14]{K}.
Let $\PP^1 \xrightarrow{g} \PP^1 \xrightarrow{h} X$ be a factorization
of a free morphism $f=h\circ g$ where $g$ has degree $m\geq 2$.  
Since $g^*h^*T_X$ is
ample, also $h^*T_X$ is ample.  Thus $h$ is free.  So the dimension of
$\Kgnb{0,0}(X)$ at $[h]$ is the ``expected dimension'', namely
$\text{deg}(h^*c_1(T_X)) + \text{dim}(X)-3$.  
The dimension of the stack of $m$-fold covers of
deformations of $h$ is 
$$
\text{deg}(h^*c_1(T_X)) + \text{dim}(X)-3 +
2(m-1).
$$  
On the other hand, the dimension of $\Kgnb{0,0}(X)$ at $[f]$ is
$$
\text{deg}(f^*c_1(T_X)) + \text{dim}(X)-3 = m\text{deg}(h^*c_1(T_X)) +
\text{dim}(X)-3.
$$
The difference of the first dimension from the second dimension is,
$$
(m-1)(\text{deg}(h^*c_1(T_X))-2).
$$
Thus, if $\text{deg}(h^*c_1(T_X)) \geq 3$ then a
general deformation of $f$ is not an $m$-fold cover of a
deformation of $h$.  

Next is the existence of proper curves in $M-M\cap \Delta$.  
Let $f:X\hookrightarrow \PP^r$ be a plurianticanonical embedding.
Denote by $\Kgnb{0,0}(f):\Kgnb{0,0}(X)\rightarrow \Kgnb{0,0}(\PP^r)$
the associated embedding.  Denote by
$\phi:\kgnb{0,0}(\PP^r)\rightarrow Y$ the contraction of the boundary
constructed in \cite{CHS1}. 
All that we will use about $\phi$ is the following.
\begin{enumerate}
\item[(i)]
Every connected component of $Y$ is a projective scheme.
\item[(ii)]
The restriction of $\phi$ to the open subset
$\kgnb{0,0}(\PP^r)-\Delta$ is an open immersion.
\item[(iii)]
For each pair of integers $i\leq j$,
the composition of $\phi$ with
$$
\Delta_{i,j}:\Kgnb{0,1}(\PP^r,i)\times_{\PP^r} \Kgnb{0,1}(\PP^r,j)
\twoheadrightarrow \Delta \subset \Kgnb{0,0}(\PP^r)
$$
factors through projection 
$$
\pi_j: \Kgnb{0,1}(\PP^r,i)\times_{\PP^r} \Kgnb{0,1}(\PP^r,j)
\rightarrow
\Kgnb{0,1}(\PP^r,j).
$$
\end{enumerate} 

Denote by $N$ the image of $M$ in $Y$.
Since the restriction of $\phi$ to $\kgnb{0,0}(\PP^r) - \Delta$ is an
open immersion, the restriction of $\phi\circ \Kgnb{0,0}(f)$ to
$M-\Delta$ is an immersion.  Since $\mc{M}_{\text{free}}$ is
dense in $\mc{M}$, $M$ has pure dimension equal to the expected
dimension, and $\mc{M}\cap \Delta$ is a Cartier divisor.  Therefore
$\text{dim}(N)$ equals $\text{dim}(M)$ and $\text{dim}(\mc{M}\cap
\Delta)$ equals $\text{dim}(M)-1$.

Denote
$\Delta_{i,j} \cap \Kgnb{0,0}(X)$ by $\Delta_{X,i,j}$.  Denote the
restriction of $\pi_j$ by $\pi_{X,j}:\Delta_{X,i,j} \rightarrow
\Kgnb{0,1}(\PP^N,j)$.  By Lemma ~\ref{lem-1}, every irreducible
component of every fiber of $\pi_{X,j}$ has dimension $\geq 1$, i.e.,
the difference of the pseudo-index and
$\text{dim}(\text{Aut}(\PP^1,0))$.  Therefore, for every irreducible
component $\Delta'$ of $\Delta$, the dimension of
$\phi(\Kgnb{0,0}(f)(\Delta'))$ is strictly less than the dimension of
$\Delta'$.  By hypothesis, every irreducible component $\Delta'$ of
$\mc{M}\cap \Delta$ is an irreducible component of $\Delta$.  Since
$\text{dim}(\Delta')$ equals $\text{dim}(\mc{M})-1$, the image of
$\Delta'$ in $N$ has dimension $\leq \text{dim}(N)-2$.

Since every connected component of $Y$ is projective, also $N$ is
projective.  Because $\text{dim}(\text{Image}(\Delta')) \leq
\text{dim}(N)-2$, a general intersection of $N$ with $\text{dim}(N)-1$
hyperplanes containing a point of $N-\text{Image}(\Delta')$ is a
complete curve that does not intersect $\text{Image}(\Delta')$.
Because there are only finitely many irreducible components of
$\mc{M}\cap \Delta$, a general intersection of $N$ with
$\text{dim}(N)-1$ hyperplanes containing a point of
$N-\text{Image}(\mc{M}\cap \Delta)$ is a complete curve that does not
intersect $\text{Image}(\mc{M}\cap \Delta)$.  The inverse image of
this curve in $\mc{M}-\mc{M}\cap \Delta$ is a complete curve
containing a given point of $\mc{M}-\mc{M}\cap \Delta$.  To be
completely precise, this curve is actually a proper, 1-dimensional
stack.  But every such stack is the image of a 
finite 1-morphism whose domain is a proper, smooth curve.
\end{proof}

Let $C$ be a smooth, proper, connected curve and let
$\zeta:C\rightarrow \Kgnb{0,0}(X)-\Delta$ be a nonconstant 1-morphism
whose general point parametrizes a free curve of $(-K_X)$-degree $e$.
This is equivalent to a pair $(\pi:\Sigma \rightarrow C,
F:\Sigma\rightarrow X)$ of a ruled surface $\Sigma$ over $C$ and a
morphism $F:\Sigma \rightarrow X$ mapping fibers of $\pi$ to curves in
$X$ of $(-K_X)$-degree $e$.  

\begin{lem} \label{lem-3}
The dimension at $[\zeta]$ of $\text{Hom}(C,\Kgnb{0,0}(X),\zeta|_B)$
is at least,
$$
\text{deg}(\text{ch}_2(T_X)|_{F(\Sigma)}) +
\frac{1}{2e}\text{deg}(c_1(T_X)^2|_{F(\Sigma)}) -
(e+\text{dim}(X)-3)(g(C)-1+\#(B)).
$$
\end{lem}

\begin{proof}
The argument will use Riemann-Roch on $\Sigma$.  Before beginning,
we recall one useful fact about the Chow ring of the ruled surface
$\Sigma$.  
Tsen's theorem implies existence of a section of $\Sigma$, 
i.e., a morphism
$\sigma:C \rightarrow \Sigma$ such that $\pi\circ \sigma =
\text{Id}_C$.  Every element in the Picard group of $\Sigma$ has the
form
$$
\alpha = e[\sigma(C)] + \pi^* D
$$
for a unique integer $e$ and a unique divisor class $D$ on $C$.  
On the one hand,
$$
-e(\omega_\pi \cdot \alpha) = -e^2(\omega_\pi\cdot [\sigma(C)]) +
2e\text{deg}(D).
$$
On the other hand,
$$
\alpha\cdot \alpha = e^2([\sigma(C)]\cdot [\sigma(C)]) +
2e\text{deg}(D).
$$
Finally, by the adjunction formula
$$
\omega_\pi \cdot[\sigma(C)] = -[\sigma(C)] \cdot [\sigma(C)].
$$
Altogether this gives the following equation for every divisor class
$\alpha$,
\begin{equation} \label{eqn-1}
-e(\omega_\pi\cdot \alpha) = \alpha\cdot \alpha.
\end{equation}
The particular case
$\alpha=\omega_\pi$ gives, 
$$
\omega_\pi\cdot \omega_\pi = 0.
$$  

To resume the proof,
the deformation theory of $\text{Hom}(C,\Kgnb{0,0}(X),\zeta_B)$ at
$[\zeta]$ is the same as the theory of deformations of the finite
morphism
$$
(\pi,F):\Sigma \rightarrow C \times X
$$
holding $C$, $X$ and $(\pi,F)|_{\pi^{-1}B}$ fixed.
The deformation theory of such
morphisms is controlled by the normal sheaf $\mc{N}$ of $(\pi,F)$,
$$
\mc{N}:= \text{Coker}(T_\Sigma \xrightarrow{d(\pi,F)} 
\pi^* T_C\oplus F^*T_X).
$$  
To be
precise, the Zariski tangent space to
$\text{Hom}(C,\Kgnb{0,0}(X),\zeta|_B)$ at $[\zeta]$ is isomorphic to
$$
H^0(C,\mc{N}(-\pi^{-1}B))
$$
and
$$
\text{dim}_{[\zeta]} \text{Hom}(C,\Kgnb{0,0}(X),\zeta_B) \geq
h^0(C,\mc{N}(-\pi^{-1}B)) - h^1(C,\mc{N}(-\pi^{-1}B)).
$$
For an excellent description of the infinitesimal theory of Hilbert
schemes leading to this formula we recommend
\cite[Chapter I]{K}.

By the Leray spectral sequence
$$
h^2(\Sigma,\mc{N}(-\pi^{-1}B)) = h^1(C,R^1\pi_* \mc{N}(-B)).  
$$
Because a general point of $C$ parametrizes
a free curve, the restriction of $\mc{N}$ to a general fiber of $\pi$
is generated by global sections; thus
it has no higher cohomology.  Therefore
$R^1\pi_*\mc{N}$ is a torsion sheaf on the curve $C$, which also has
no higher cohomology:
$$
h^1(C,R^1\pi_* \mc{N}(-B)) = 0.
$$
Because $h^2(\Sigma,\mc{N}(-\pi^{-1}B)) = 0$, 
$$
h^0(\mc{N}(-\pi^{-1}B))  - h^1(\mc{N}(-\pi^{-1}B)) 
=\chi(\Sigma,\mc{N}(-\pi^{-1}B)).
$$

Riemann-Roch gives
$$
\chi(\Sigma,\mc{N}(-\pi^{-1}B)) = \text{deg}(\text{ch}(\mc{N})\cdot
\pi^*\text{ch}(\OO_C(-B))\cdot \text{Todd}(\Sigma)).
$$
There are a couple of observations.  First,
$$
\text{ch}(\mc{N}) = \text{ch}(F^* T_X) -
\text{ch}(T_\pi) = F^*\text{ch}(T_X) - (1-\omega_\pi + \omega_\pi^2/2).
$$  
Second, since the Todd class is multiplicative,
$$
\text{Todd}(\Sigma) = \text{Todd}(\pi)\pi^*\text{Todd}(C) =
(1-\omega_\pi/2 + \omega_\pi^2/12)(1-\pi^*\omega_C/2).
$$
Combined with the contribution from $-\pi^{-1}B$, this gives,
$$
\pi^*\text{ch}(\OO_C(-B))\text{Todd}(\Sigma) = (1-\omega_\pi/2 +
\omega_\pi^2/12)(1-\pi^*\omega_C/2 - \pi^*B)
$$
which 
reduces to
$$
\pi^*\text{ch}(\OO_C(-B))\text{Todd}(\Sigma) = 
1 -(\omega_\pi + \pi^*\omega_C+2\pi^*B)/2
-(g(C)-1+\#B)[\text{point}]
$$
because $\omega_\pi^2 = 0$.  
Therefore
$\chi(\Sigma,\mc{N}(-\pi^{-1}B))$ equals
$$
\text{deg}(\text{ch}_2(T_X)|_{F(\Sigma)}) -\text{deg}(\omega_\pi\cdot
g^* c_1(T_X)) - (g(C)-1+\#B)(e+\text{dim}(X)-3).
$$
Finally, applying Equation ~\ref{eqn-1} to $\alpha = F^* c_1(T_X)$ gives
$$
-\omega_\pi\cdot F^* c_1(T_X) = \frac{1}{2e}F^*c_1(T_X)^2.
$$
\end{proof}

\begin{proof}[Proof of Theorem ~\ref{thm-main}]
Every proper curve in $M-M\cap \Delta$ is the image of a nonconstant
1-morphism $\zeta:C\rightarrow \mc{M}-\mc{M}\cap \Delta$ from a smooth
curve $C$.  The induced morphism $\text{Hom}(C,\mc{M}-\mc{M}\cap
\Delta) \rightarrow \text{Hom}(C,M)$ is finite.  By Lemma~\ref{lem-3},
$\text{dim}(\text{Hom}(C,M;\zeta|_B))$ behaves as if $M$ is smooth
along the image of $\zeta$ and the anticanonical degree of $\zeta(C)$
equals
$$
\text{deg}(\text{ch}_2(T_X)|_{F(\Sigma)}) +
\frac{1}{2e}\text{deg}(c_1(T_X)^2|_{F(\Sigma)}).
$$ 
Because $X$ is $2$-Fano, this degree is positive.  Therefore the usual
bend-and-break argument applies, cf. \cite[Theorem II.5.8]{K}.
\end{proof}

\section{Examples} \label{sec-ex}
We list below a number of Fano manifolds satisfying the hypotheses of
Theorem ~\ref{thm-main} and Proposition \ref{prop-cont}.  For further
work towards classifying such manifolds, please see the note
\cite{dJS6}.  But first there are a couple of observations about the
conclusion of Theorem ~\ref{thm-main}, i.e., existence of rational
surfaces containing a general point.  First of all, this is a
birational property.  Therefore it is reasonable to focus on
\emph{minimal} Fano manifolds, for instance those with Picard number
$1$.  

The second observation has to do with specialization for flat
families.  Let $R$ be a DVR whose residue field is algebraically
closed of characteristic $0$.  
Let $\mc{X}\rightarrow \SP R$ be a
projective smooth morphism.  Denote the closed fiber by $X_0$ and the
geometric generic fiber by $X_{\overline{\eta}}$.  Even assuming a
general point of $X_{\overline{\eta}}$ is contained in a rational
surface, it is not obvious that a general point of $X_0$ is contained
in a rational surface.  After all rational surfaces can specialize 
to non-rational surfaces. 
As we will explain, a general point of $X_0$ is contained in a
rational surface
if Theorem ~\ref{thm-main} applies to $X_{\overline{\eta}}$,
even if $X_0$ itself does not satisfy the
hypotheses of Theorem ~\ref{thm-main}.  This is one advantage of our
approach to studying rational surfaces on Fano manifolds.

Here is the argument.
Form the relative Kontsevich space
$$
\Kgnb{0,0}(\mc{X}/\SP R) \rightarrow \SP R.
$$
The closed fiber is
$\Kgnb{0,0}(X_0)$ and the geometric generic fiber is
$\Kgnb{0,0}(X_{\overline{\eta}})$.  
Consider those irreducible components $\mc{M}$ 
of $\Kgnb{0,0}(\mc{X}/\SP R)$ dominating $\SP
R$, i.e., such that $\mc{M} \rightarrow \SP R$ is flat.
In particular, the moduli point of 
each
free curve $C$ in $X_0$ is contained in a unique irreducible component
$\mc{M}$ dominating $\SP R$ (in fact $\mc{M}$ is smooth over $\SP R$
at $[C]$).
If the geometric generic fiber of $\mc{M}$ is
uniruled then also the closed fiber is uniruled.  Therefore, if
Theorem ~\ref{thm-main} applies to the geometric generic fiber
$X_{\overline{\eta}}$, then $C$ is contained in a rational surface in
$X_0$ (assuming $C$ is a free, \emph{embedded} curve).  
Since this also applies to deformations of $C$, a general point of
$X_0$ is contained in a rational surface.  Therefore it is reasonable
to focus on Fano manifolds $X_{\overline{\eta}}$ which are
\emph{general in moduli}.    

We know of two infinite sets of families of $2$-Fano
manifolds of Picard number $1$.  For each of these, the hypotheses of
Theorem ~\ref{thm-main} and Proposition ~\ref{prop-cont} hold for
every member of the family which is general in moduli (in the sense
above).  
First, the Grassmannian $\text{Grass}(k,n)$ 
of $k$-dimensional subspaces of a
fixed $n$-dimensional vector space is always Fano, and it is $2$-Fano if
and only if $k=1$, $n=2k$ or $n=2k+1$ (normalizing so that $2k\leq
n$).  
Second, a smooth complete intersection of type $(d_1,\dots,d_c)$
in an $n$-dimensional weighted projective space is Fano if and only if
$d_1 + \dots + d_c \leq n$.  
And it is 
$2$-Fano if and only if $d_1^2 + \dots + d_c^2 \leq n+1$.

For Grassmannians, and more generally for any projective homogeneous
space, Kim and Pandharipande have proved that every connected
component of $\Kgnb{0,0}(X)$ is irreducible, \cite{KP}.  Second, by
the same technique as in \cite{HRS2}, for a complete intersection of type
$(d_1,\dots,d_c)$ in $\PP^n$ which is \emph{general in moduli}, the
connected components of $\Kgnb{0,0}(X)$ are irreducible provided
$$
\sum_{i=1}^c d_i \leq \frac{n+c+1}{2}.
$$
This inequality is implied by the previous inequality $\sum d_i^2 \leq
n+1$.  There do exist complete intersections for which the connected
components
are reducible, e.g., any Fano hypersurface of degree $d\geq 4$ having
a conical hyperplane section.  However, the argument above proves that
a general point of every $2$-Fano complete intersection is contained
in a rational surface.

Two operations producing new $2$-Fano manifolds are worthy of comment,
although they do not produce minimal $2$-Fano manifolds.
First,
every product of $2$-Fano manifolds is $2$-Fano.
Second, let $X$ be a
smooth Fano manifold and let $L$ be a nef invertible sheaf.  The
$\PP^1$-bundle $\PP(\OO_X\oplus L^\vee)$ is Fano if and only if
$c_1(T_X)-c_1(L)$ is ample.  
And then it is $2$-Fano if
and only if $\text{ch}_2(T_X) + \frac{1}{2}c_1(L)^2$ is nef.  Notice
$\text{ch}_2(T_X)$ need not be nef, i.e., $X$ need not
be $2$-Fano.

Other operations on Fano manifolds produce $2$-Fano manifolds only
under strong hypotheses.  
For instance, a projective
bundle $\PP(E)$ of fiber dimension $\geq 2$ over a Fano manifold is
Fano if $E$ satisfies a weak version of semistability.  However, if
$\PP(E)$ is $2$-Fano then $E$ satisfies a very strong version of
semistability: the pullback to \emph{every} curve in $X$ is semistable.  
If $X$ is $\PP^n$ this condition implies
$\PP(E)$ is $\PP^m\times \PP^n$ for some $m$.

\section{The theorem is sharp} \label{sec-sharp}

The theorem is sharp in two ways.  First, let $X$ be a general cubic
hypersurface in $\PP^5$.  This is Fano, but it is not $2$-Fano.  By
the main theorem of \cite{dJS2}, there are infinitely many
non-uniruled irreducible components $\mc{M}$ of $\Kgnb{0,0}(X)$
satisfying the hypotheses of Theorem ~\ref{thm-main}.

Second, let $Y$ be the $\PP^1$-bundle over $X$, $Y = \PP(\OO_X \oplus
\OO_{\PP^5}(-2)|_X)$.  By the construction in the last section, $Y$ is
$2$-Fano.  Associated to the projection $\pi:Y\rightarrow X$, there is
a 1-morphism $\Kgnb{0,0}(\pi):\Kgnb{0,0}(Y)\rightarrow \Kgnb{0,0}(X)$.
Let $\mc{N}$ be an irreducible component
of $\Kgnb{0,0}(Y)$ containing a
very free curve.  It is easy to prove the restriction to $\mc{N}$ of the
contraction $\phi$ maps the Cartier divisor 
$\mc{N}\cap \Delta$ to a subvariety of codimension $\geq 2$.
(However it is not true that every component of
$\mc{N}\cap \Delta$ is a component of $\Delta$.)  Thus Theorem
~\ref{thm-main} implies $N$ is uniruled.  In fact, the restriction of
$\Kgnb{0,0}(\pi)$ to $\mc{N}$ is birational to a projective bundle
over the image component $\mc{M}$ of $\Kgnb{0,0}(X)$.  Choosing
$\mc{N}$ appropriately, $\mc{M}$ is one of the infinitely many
non-uniruled irreducible components of $\Kgnb{0,0}(X)$.  Therefore $N$
is not rationally connected, and the MRC quotient of $N$ is precisely
$M$.

\section{Speculation} \label{sec-spec}

For the counterexample $Y$ in the previous section, $\text{ch}_2(T_Y)$
is nef.  But it is not ``positive''.  It has intersection number $0$
with the surface $\pi^{-1}B$ for every curve $B$ in $X$.  If $X$ is
a Fano manifold such that $\text{ch}_2(T_X)$ has positive intersection
number with every surface, is $\mc{M}$ rationally connected?  We know
no counterexample.

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