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\begin{document}

\title[Research statement]{Research Statement}

\

\author[J. Starr]{Jason Starr}
%\address{Mathematics Department, M.I.T., Cambridge MA 02139, USA}
%\email{jstarr@math.mit.edu}


\date{\today}

\maketitle

%% \section{Rational curves on cubic threefolds} \label{sec-cubic}

%% There is an \emph{Abel map} from $\text{RatCurves}^e(X)$ to the
%% Griffiths intermediate Jacobian $J$ associated to the weight $3$ Hodge
%% structure on $X$ (more precisely, its dual).  

%% \begin{thm}[Harris, Roth, Starr ~\cite{HRS1}, ~\cite{HRS3}]\label{thm-HRS1}
%%   For every smooth cubic hypersurface $X$ in $\PP^4$ and each integer
%%   $e=1,2,3,4,$ and $5$,
%% each fiber of the Abel map
%%   $\text{RatCurves}^e(X) \rightarrow
%%   J$ containing a general point of $\text{RatCurves}^e(X)$
%% is irreducible and unirational.
%%   For every integer $e\geq 4$, the Abel map
%%   is dominant, and each
%%   general fiber is irreducible.
%% \end{thm}

%% This picture, though limited to the cases $e\leq 5$, is remarkably
%% similar to the picture of the Abel map from the symmetric powers of a
%% curve to the Jacobian of the curve: the Abel map becomes a
%% $\PP^N$-bundle over the Jacobian as the dimension of the symmetric
%% power grows.  The analogous conjecture here, due to Joe Harris, 
%% is that the Abel maps are
%% rationally connected fibrations for $e \geq 4$.  Theorem
%% ~\ref{thm-HRS1} is evidence for this conjecture; in fact, the only
%% evidence either for or against this conjecture.

%% Irreducibility of $\text{RatCurves}^e(X)$ is proved in ~\cite{HRS1},
%% and Theorem~\ref{thm-HRS1} is proved in ~\cite{HRS3}.
%% The cases $e=1,2$ and $3$ were known, and follow from the analysis of
%% lines on cubic threefolds by Clemens and Griffiths on their solution
%% of the L\"uroth problem.  For $e \geq 4$, irreducibility of the fibers
%% is new.  It turns on the bend-and-break theorem and deformation theory
%% techniques.  For $e=4$ and $5$, unirationality of the fibers follows
%% from analysis of liaison of quartic and quintic curves in the
%% intersection of $X$ with a cubic surface scroll.

\part{Research summary}

\section{Every rationally connected variety over the function field of
  a curve has a rational point} \label{sec-RCfib}

A nonempty,
smooth, projective variety over an uncountable, algebraically closed
field is \emph{rationally connected} if every pair of closed points
is in the image of a regular morphism from $\PP^1$ to the variety. A
smooth, projective variety over an arbitrary field $k$ is rationally
connected if its base-change to one (and hence every) uncountable,
algebraically closed field is rationally connected.

\begin{thm}[Graber, Harris, Starr ~\cite{GHS}]\label{thm-GHS}
Every rationally connected variety $X$ defined over the function field
$K$ of a curve $B$ over a characteristic $0$, algebraically closed field
$k$ has a 
$K$-rational point.  Equivalently, every projective, surjective
morphism $\pi:\mc{X}\rightarrow B$ whose general fiber is rationally
connected has an algebraic section.  
\end{thm}

\begin{thm}[de Jong, Starr ~\cite{dJS}]\label{thm-dJS}  
Replacing ``rationally connected'' by ``separably rationally
connected'', the previous theorem holds in arbitrary characteristic.
\end{thm}

Theorem~\ref{thm-GHS} was posed as a question by J. Koll\'ar, Y. Miyaoka and
S. Mori in their paper ~\cite{KMM}, and proved by them when
$\text{dim}(X)$ is $1$ or $2$.  Also,
Also, the special case  
of a smooth Fano hypersurface was proved by Tsen
~\cite[Thm IV.5.4]{K}.

\begin{cor}\label{cor-1}  Let $(R,m)$ be a complete DVR containing its
  residue field, assumed algebraically closed.
Let $\mathcal{X}$ be a regular, projective $R$-scheme.
If the geometric generic fiber $\mathcal{X} \otimes_R \overline{K}(R)$
is normal and separably rationally connected, then the closed fiber
$\mathcal{X} \otimes_R k$ is reduced on a nonempty open subset.
\end{cor}

This is the local version of Theorem~\ref{thm-dJS} from which it follows
by the Artin approximation theorem.

As another corollary of Theorem~\ref{thm-dJS}, Koll\'ar proved that every
smooth, connected, projective, separably rationally connected scheme
over an algebraically closed field has trivial algebraic fundamental
group ~\cite[Cor. 3.6]{DBX}.  
This was previously proved by Campana
~\cite{C91} and Debarre ~\cite[Cor. 4.18]{De}
in the special case that $k$ has characteristic zero.

Finally, Corollary~\ref{cor-3} connects two fundamental
conjectures regarding uniruled and rationally connected varieties.

\begin{conj}[Hard Dichotomy Conjecture, Conj
3.3.3~\cite{Matsuki}]\label{conj-1} Let
$\text{char}(k)=0$ and let $X$ be a smooth projective variety.  If
$h^0(X,\omega_X^{\otimes n})$ equals $0$ for all
positive $n$, then $X$ is uniruled.
\end{conj}

\begin{conj}[Mumford's Conjecture, Conj
IV.3.8.1~\cite{K}]\label{conj-2}  
Let
$\text{char}(k)=0$ and let $X$ be a smooth projective variety.  If 
$h^0(X,(\Omega^1_X)^{\otimes n}) = 0$ for all positive $n$, then $X$ is
rationally connected.
\end{conj}

\begin{cor}\label{cor-3}  Conjecture~\ref{conj-1} implies
Conjecture~\ref{conj-2}.
\end{cor}

\section{Rational connectedness and sections of families over curves}

Let $f\colon X \rightarrow B$ be a surjective morphism of projective
schemes of fiber dimension $d$ such that $B$ is irreducible and
normal.  For every smooth curve $C\subset B$, denote by $f_C \colon
X_C \rightarrow C$ the base-change of $f$ by $C\rightarrow B$.

\begin{thm}[Graber, Harris, Mazur, Starr ~\cite{GHMS}] \label{thm-GHMS}
If $f_C\colon X_C
\rightarrow C$ has a section for every smooth curve $C\subset B$,
then $X$ contains
a closed subvariety $Z$ whose geometric
generic fiber $Z\times_B \text{Spec}(\overline{K}(B))$ is 
rationally connected.
\end{thm}

By Theorem~\ref{thm-GHS}, 
if $X$ has a closed subvariety $Z$ whose geometric generic fiber is 
rationally connected, then every base-change $X_C\rightarrow C$ has a
section.    
Thus Theorem~\ref{thm-GHMS} is a converse
to part of Theorem~\ref{thm-GHS}.

Theorem~\ref{thm-GHMS} implied the first answer to
a question posed by Serre to Grothendieck.

\begin{ques}[Serre's Question~\cite{GtoS}]\label{ques-1}  
Does a variety over the function field of a curve
have a rational point when it is $\OO$-acyclic, i.e.,
when $h^i(X,\OO_X) = 0$ for every $i>0$?
\end{ques}

One can ask this question for any field.  Serre was motivated
by the case of a finite field, for which a positive answer was
proved by Katz ~\cite{Katz}.
Nevertheless the answer to Question~\ref{ques-1} is negative.

\begin{cor}[Graber, Harris, Mazur, Starr
  ~\cite{GHMS}]\label{cor-GHMS} 
There exists a smooth projective
  curve $C$ over $\CC$ with function field $K=K(C)$ and a smooth
  projective surface $X$ over $K$ such that $h^0(X,\Omega^1_X) =
  h^0(X,\Omega^2_X)=0$ and such that $X$ has no $K$-rational point.
  In fact $X$ is a polarized Enriques surface over $K$.  
\end{cor}

The corollary follows from Theorem~\ref{thm-GHMS} by considering
the universal Enriques surface $\mathcal{X}$
over a certain parameter space $B$ of
polarized Enriques surfaces.  By simple properties of the Chow groups of
$\mathcal{X}$, 
there is no subvariety $Z\subset \mathcal{X}$ as in
Theorem~\ref{thm-GHMS}.  Therefore there is a smooth curve $C\subset
B$ such that $f_C\colon \mathcal{X}_C \rightarrow C$ has no section.
The generic fiber of $\mathcal{X}_C$ gives a negative answer to
Question ~\ref{ques-1}.  

H\'el\`ene Esnault pointed out non-existence of $K$-rational points in
Corollary ~\ref{cor-GHMS} may possibly follow from an obstruction in
the Galois cohomology of $K$.  All known examples of obstructions
satisfy restriction and corestriction.  This means the order of the
obstruction class
divides the degree of $L/K$ for every residue field $L$ of a
closed point of $X$.  Equivalently, the order divides the
degree of every $K$-rational $0$-cycle on $X$.
However, by explicit construction, there exist
Enriques surfaces as in Corollary ~\ref{cor-GHMS} having a
$K$-rational $0$-cycle of degree $1$.  Therefore non-existence of
$K$-rational points is not explained by an obstruction satisfying
restriction and corestriction.

\begin{thm}[Starr ~\cite{S2}] \label{thm-Enr}
There exists a smooth projective
  curve $C$ over $\CC$ with function field $K=K(C)$ and a smooth
  Enriques surface $X$ over $K$ with no $K$-rational point but with a
  $K$-rational $0$-cycle of degree $1$.
\end{thm}

%% Another corollary of Theorem~\ref{thm-GHMS} concerns the \'etale
%% cohomology of an Abelian scheme $f\colon A \rightarrow B$ 
%% defined over a smooth, complex
%% projective scheme $B$.  Let $\mathcal{C}$ denote the set of all smooth
%% curves $C\subset B$.  For each element $C \in \mathcal{C}$, let
%% $f_C\colon A_C \rightarrow C$ be the base change of $f$ to $C$.
%% Define $R$ to be the homomorphism which is the product of restriction
%% of \'etale cohomology groups:
%% $$ R\colon H^1_{\text{et}}(B,A) \rightarrow \prod_{[C]\in \mathcal{C}}
%% H^1_{\text{et}}(C,A_C).$$
%% Elements of $H^1_{\text{et}}(B,A)$ classify equivalence classes
%% of torsors $g\colon T \rightarrow B$ for $A/B$.

%% \begin{cor}[Graber, Harris, Mazur, Starr ~\cite{GHMS}]\label{cor-GHMS2}
%% The homomorphism $R$ is injective.  Equivalently, for every nontrivial
%% torsor $T/B$ there exists a smooth curve $C\subset B$ such that
%% the restriction $T_C/C$ is a nontrivial torsor for $A_C/C$.
%% \end{cor}

\section{Rational points of a variety over the function field of a
  surface} \label{sec-R1C}

Let $k$ be an algebraically closed field of characteristic $0$, 
let $B$ be an algebraic surface over $k$ with function field $K$, and
let $X$ be a projective $K$-scheme such that $X\otimes_K \overline{K}$
is irreducible and smooth.  There is an obstruction to existence of
$K$-rational points on $X$ in the Brauer group of $K$.  We call this
obstruction a \emph{Brauer obstruction}, though it is often also
called the \emph{elementary obstruction}.  

\begin{thm}[de Jong, Starr ~\cite{dJS5}] \label{thm-R1C}
If the Brauer obstruction vanishes, if
$X\otimes_K \overline{K}$ is rationally connected, if for each $e$
one (hence
every) projective birational model of the parameter space
$\text{RatCurves}^e(X\otimes_K \overline{K})_{p,q}$ for
rational curves in $X\otimes_K \overline{K}$ containing a pair of
closed points $p,q$ is rationally connected, if $X\otimes_K
\overline{K}$ has a \emph{very twisting} family of pointed lines, and
if specific additional hypotheses hold, then $X$ has a $K$-rational point.
\end{thm}

Unfortunately, the specific additional hypotheses are quite strong.
We hope they can be removed, but have not yet proved this.
Nonetheless, all the hypotheses do hold when $X\otimes_K \overline{K}$
is a Grassmannian or certain other homogeneous spaces.  Consequently,
Theorem~\ref{thm-R1C} implies another proof of de Jong's Period-Index
Theorem. 

\begin{thm}[de Jong ~\cite{dJ}] \label{thm-dJ}
For every division algebra $D$ with center $K$, $\text{dim}_K(D)$
equals the square of the order of $[D]$ in the Brauer group of $K$.
\end{thm} 

\section{Rational curves on low degree hypersurfaces} \label{sec-hyper}

The parameter space $\text{RatCurves}^e(X)$ for degree $e$ rational
curves on a projective variety $X$ figures prominently in every
preceding theorem.  This research project probes more deeply the
structure of $\text{RatCurves}^e(X)$ for the simplest rationally
connected varieties: general hypersurfaces of degree $d\leq n$ in $\PP^n$.

%% For each closed subscheme $X$ of complex projective space $\PP^n$ and each 
%% integer $e>0$, 
%% $\text{RatCurves}^e(X)$ is the space parametrizing
%% smooth, degree $e$, rational
%% curves in $X$.  Closely related, $\text{RatCurves}^e(X)_{p,q}$ is the
%% subvariety parametrizing curves containing a fixed pair $p,q$ of
%% closed points of $X$.
%% These spaces, together with their compactifications, are
%% relevant to the following 2 problems.

%% \begin{prob}\label{prob-a}
%% When is a rationally connected variety unirational?
%% \end{prob}

%% Since a unirational variety contains rationally parametrized families
%% of rational curves of arbitrarily high dimension, to prove $X$ is not
%% unirational, it suffices to prove the dimension of unirational
%% subvarieties of $\text{RatCurves}^e(X)$ is bounded independently of
%% $e$.  

%% \begin{prob} \label{prob-b}
%% When does a rationally connected variety defined over the function
%% field of a surface have a rational point?
%% \end{prob}

%% This is the problem addressed by Theorem ~\ref{thm-R1C}.
%% The main hypothesis is
%% that every smooth, projective, birational model of
%% $\text{RatCurves}^e(X)_{p,q}$ is rationally connected.

%% \begin{ques}\label{ques-0}  
%% Most of the basic questions about $\text{RatCurves}^e(X)$ are open.
%% \begin{enumerate}
%% \item[(1)]
%% Is $\text{RatCurves}^e(X)$ irreducible?  
%% \item[(2)]
%% Is it reduced?
%% \item[(3)]
%% What is its dimension?  
%% \item[(4)]
%% Does it have a geometrically meaningful compactification?  
%% \item[(5)]
%% Is it normal?  
%% \item[(6)]
%% What are its singularities?  
%% \item[(7)]
%% What is the Kodaira dimension of a smooth, projective,
%%   birational model?
%% \item[(8)]
%% Is there a bound on the dimension of unirational or uniruled
%% subvarieties of $\text{RatCurves}^e(X)$?
%% \end{enumerate}  
%% \end{ques}

%% For a general hypersurface $X$ in complex projective space $\PP^n$,
%% many of these questions are answered.

%% \subsection{Irreducibility and dimension} \label{subsec-hyper1}

%% Let $X$ be a general hypersurface $X$ in $\PP^n$ of degree $d$.

\begin{thm}[Harris, Roth, Starr ~\cite{HRS2}]\label{thm-HRS3}
If
$d < \frac{n+1}{2}$,  
$\text{RatCurves}^e(X)$ 
is an irreducible, reduced, local
complete intersection scheme of dimension $(n+1-d)e + (n-4)$.
Moreover, it is a dense open subset of a geometrically meaningful
compactification which is also a local complete intersection space.
\end{thm}

%% The case $e=1$ of this theorem is well-known and has a long history.
%% The best result is due to Koll\'ar ~\cite[Theorem V.4.3]{K}.
%% The case $e>1$, requires completely different methods.  First of all,
%% $\text{RatCurves}^e(X)$ is not proper; it can be compactified  
%% via the Kontsevich stack of stable maps
%% $\overline{\mathcal{M}}_{0,0}(X,e)$, which is proper ~\cite{FP}.  But
%% this stack is not smooth, so irreducibility does not follow from
%% connectedness. 

%% The proof of Theorem~\ref{thm-HRS3} is by induction on $e$.  Mori's
%% bend-and-break lemma ~\cite{Miyaoka-Mori86} is used to reduce
%% properties of $\overline{\mathcal{M}}_{0,0}(X,e)$ to properties of the
%% boundary of this stack, which is built from Kontsevich
%% stacks for smaller values of $e$.  The base of the induction is
%% ``trees'' of lines in $X$, which are analyzed using Koll\'ar's theorem
%% and a new result of a classical flavor: the evaluation map
%% from the space of pointed lines to $X$ is flat.

%% \subsection{Rational connectedness} \label{subsec-hyper2}

\begin{thm}[Harris, Starr ~\cite{HS2}, \cite{Sr1C}, 
de Jong, Starr ~\cite{dJS4}]
\label{thm-HS1}
If $d^2 \leq n+1$, every
smooth, projective model of
$\text{RatCurves}^e(X)$ is rationally connected, therefore has
negative Kodaira dimension.  For $e\geq 2$, the same holds
for the space
$\text{RatCurves}^e(X)_{p,q}$ parametrizing rational curves containing a fixed,
but general, pair $p,q$ of points.  If $d^2 \leq n$, there exists a
\emph{very twisting} family of pointed lines in $X$.
\end{thm}

%% The proof of Theorem~\ref{thm-HS1} uses Koll\'ar's
%% criterion for rational connectedness~\cite[Thm IV.3.7]{K}: 
%% a scheme (resp. stack)
%% $V$ is rationally connected if there is a
%% rational curve $C$ in $V$ (resp. in the ``fine moduli locus'' of $V$) 
%% whose normal bundle is ample. 
%% The proof of Theorem~\ref{thm-HS1} 
%% is again by induction on $e$, gluing lines to degree
%% $e-1$ curves to obtain (reducible) degree $e$ curves.  The family of
%% lines must have a special property which we call \emph{very twisting}.  
%% Existence of very twisting families of lines follows from a
%% deformation theory computation.

%% To make the induction work we must
%% use \emph{reducible} curves in the moduli space for which the
%% normal bundle is not ample, so Koll\'ar's criterion does not apply.
%% But the normal bundle satisfies a weaker condition; it is
%% \emph{deformation ample}.  
%% We generalize Koll\'ar's criterion to the case of
%% reducible curves with deformation ample normal bundles, giving the
%% theorem.  

%% A simpler proof occurs in ~\cite{dJS4}.
%% Moreover, the hypothesis on the degree is generalized from $d^2 \leq n$
%% to $d^2 \leq n+1$.  
%% However, that proof does not give existence of very twisting families
%% of lines.  Since this is also a hypothesis in Theorem ~\ref{thm-R1C},
%% it is interesting independently of Theorem~\ref{thm-HS1}.

%% \subsection{General type models and $\QQ$-Fano models} \label{subsec-hyper3}

\begin{thm}[Starr ~\cite{S1}]
\label{thm-S1a}  
If $d^2 > n+1$, for every $e=\lfloor (n+1-d)/(d^2-n-1)
\rfloor,\dots,n-d$, every smooth, projective 
model of $\text{RatCurves}^e(X)$ is of general type, i.e., the Kodaira
dimension equals the usual complex dimension.  
\end{thm}

\begin{thm}[Starr ~\cite{S1}]
\label{thm-S1b}
If $d^2 < n+1$, for
every $e= 1,\dots,n-d$, $\text{RatCurves}^e(X)$ has a geometrically
meaningful compactification which is a normal, $\QQ$-Fano variety.
\end{thm}

\begin{thm}[de Jong, Starr ~\cite{dJS3}] \label{thm-cancomp}
There is an explicit, general formula for the canonical divisor class
on $\Kgnb{0,0}(X,e)$ whenever it is irreducible, reduced and normal of
the expected dimension.
\end{thm}

%% Theorem ~\ref{thm-S1a} is a converse to Theorem~\ref{thm-HS1}.  A
%% variety is of general type if the canonical bundle is big and the
%% singularities are, at worst, canonical.  
%% Theorem ~\ref{thm-cancomp}
%% implies the canonical bundle is big
%% when $d^2 > n+1$ and $e > (n+1-d)/(d^2-n-1)$.  An analysis of the
%% singularities of $\overline{\mathcal{M}}_{0,0}(X,e)$ implies the
%% singularities are canonical when $e\leq n-d$.  The analysis uses
%% Fulton and MacPherson's deformation to the normal cone of a particular
%% subvariety of $\overline{\mathcal{M}}_{0,0}(X,e)$.  Using an explicit
%% resolution of singularities, the normal cone has canonical
%% singularities when $e \leq n-d$.  Then an inversion-of-adjunction
%% theorem of Kawamata ~\cite{Kaw99} implies
%% $\overline{\mathcal{M}}_{0,0}(X,e)$ has canonical singularities.

%% The proof of Theorem~\ref{thm-S1b} is similar.  A variety is
%% $\QQ$-Fano if the anticanonical bundle is ample and the singularities
%% are, at worst, Kawamata log terminal.  Using Theorem
%% ~\ref{thm-CHS} and Theorem~\ref{thm-cancomp}, when $d^2 < n+1$, 
%% there is a contraction of the boundary in
%% $\Kgnb{0,0}(X,e)$ whose target has ample anticanonical bundle (the
%% anticanonical bundle of 
%% $\Kgnb{0,0}(X,e)$ is not ample). 
%% By the analysis of
%% singularities above, the target has Kawamata log terminal
%% singularities when $e \leq n-d$.  

%% \section{Nef and pseudo-effective cones of Kontsevich moduli spaces}
%% \label{sec-ample}  

%% Given a normal, projective variety $M$, the problem of classifying 
%% birational regular
%% morphisms from $M$ to a projective variety is essentially equivalent
%% to the problem of finding the set of eventually free, big
%% $\QQ$-Cartier divisors on $M$.  
%% This set is a cone in $\text{Pic}(M)\otimes \QQ$
%% whose closure is the nef cone.
%% Similarly, an important part of the problem of classifying birational
%% rational transformations from $M$ to a projective variety is the
%% problem of find the set of big $\QQ$-Cartier divisors on $M$.  This
%% set is a cone whose closure is the pseudo-effective cone.  Thus, the
%% problem of classifying birational maps
%% largely reduces to finding the nef and
%% pseudo-effective cones of $M$.

%% Thus, to classify
%% the projective birational models of $\text{RatCurves}^e(X)$, it is
%% useful to compute the nef and pseudo-effective cones of one projective
%% birational model, $\Kgnb{0,0}(X,e)$.  
%% The basic case is
%% when $X=\PP^n$: since every projective variety $X$ embeds
%% in some $\PP^n$, every nef line bundle on $\Kgnb{0,0}(\PP^n,e)$ gives
%% a nef line bundle on $\Kgnb{0,0}(X,e)$.  

%% \begin{thm}[Coskun, Harris, Starr ~\cite{CHS1}, ~\cite{CHS2}]
%%   \label{thm-CHS}
%% The problem of finding the nef cone of $\Kgnb{0,0}(\PP^n,e)$ reduces
%% to the problem of finding the nef cone of $\overline{M}_{0,e}$.
%% More generally, the problem of finding the nef cone of
%% $\Kgnb{0,r}(\PP^n,e)$ for every $r$, $n$ and $e$ is equivalent to the
%% problem of finding the nef cone of $\overline{M}_{0,r}$ for every
%% $r$.  Finally, the pseudo-effective cone of $\Kgnb{0,0}(\PP^n,e)$ is
%% an explicit rational polyhedral cone when $n \geq e$.  
%% \end{thm}

%% There is a well-known conjecture, the F-conjecture, for the nef cone
%% of $\overline{M}_{0,r}$.  By Theorem~\ref{thm-CHS}, this conjecture
%% implies an explicit conjecture for the nef cone of
%% $\Kgnb{0,0}(\PP^n,e)$.  

\section{Hilbert and Quot functors of Deligne-Mumford stacks}

Deligne-Mumford stacks occur naturally in the theory of moduli
spaces and parameter spaces.  This project aimed to construct
analogues for Deligne-Mumford stacks of some of the parameter spaces
useful in the study of schemes.  Hilbert schemes and Quot schemes are
very basic examples of parameter spaces, building blocks for other
important spaces such as Picard schemes and moduli spaces of vector
bundles and coherent sheaves.

%% Let $S$ be an algebraic space.  Let $\mathcal{X}$ be a
%% Deligne-Mumford stack and let $p:\mathcal{X} \rightarrow S$ be a
%% separated, locally finitely presented $1$-morphism.  Let $\mathcal{F}$
%% be a locally finitely presented, quasi-coherent sheaf on $\mathcal{X}$.
%% Following Grothendieck, denote by $\text{Quot}(\mathcal{F}/\mathcal{X}/S)$ the
%% Quot functor of flat families of locally finitely presented,
%% quasi-coherent quotients of $\mathcal{F}$ with proper support over the
%% base of the family.

\begin{thm}[Olsson-Starr~\cite{OS}] \label{thm-OS} 
Let $\mathcal{X}/S$ be a separated, locally finitely presented
Deligne-Mumford stack over an algebraic space $S$, and let
$\mathcal{F}$ be a locally finitely presented, quasi-coherent sheaf on
$\mathcal{X}$.  
\begin{enumerate}
\item 
The Quot functor $\text{Quot}(\mathcal{F}/\mathcal{X}/S)$ is represented by
an algebraic space separated and locally finitely presented
over $S$.  
\item  If $S$ is an affine scheme and if $\mathcal{X}$ is a tame,
global quotient stack whose coarse moduli space is a quasi-projective
$S$-scheme,  then the connected components of
$\text{Quot}(\mathcal{F}/\mathcal{X}/S)$ are quasi-projective
$S$-schemes.
\end{enumerate}
\end{thm}

%% Theorem~\ref{thm-OS} is the first generalization to Deligne-Mumford
%% stacks of theorems of Grothendieck~\cite{FGA} and Artin~\cite{Artin}
%% for the case that $\mathcal{X}$ is a scheme or an algebraic space.

%% Deligne-Mumford stacks were introduced by P. Deligne and
%% D. Mumford in ~\cite{DM} to study moduli of curves.  
%% They have also been applied to quantum cohomology by
%% Kontsevich ~\cite{Kv}, to nonabelian Hodge theory by Simpson
%% ~\cite{Simpson}, and to many other aspects of algebraic geometry.

%% Despite the many uses of stacks, only recently have the
%% foundations of the theory been developed.
%% G. Laumon and L. Moret-Bailly have written a book about the
%% foundations of stacks ~\cite{LM-B}.  
%% But the foundations are not yet fully developed.
%% Theorem~\ref{thm-OS} is a further development not
%% covered by ~\cite{LM-B}.
 
%% Theorem~\ref{thm-OS} was previously known for a finite quotient stack;
%% then the result follows easily from Grothendieck's theorem.  
%% For a general Deligne-Mumford stack, Theorem~\ref{thm-OS}
%% requires new ingredients:
%% an analogue for Deligne-Mumford stacks of
%% Grothendieck's formal existence theorem ~\cite[Thm IV.5.1.4]{EGA} and an
%% analogue of M. Artin's deformation/obstruction theory ~\cite[Section
%% 6]{Artin}.  Additionally, a new notion of Hilbert
%% polynomials for stacks is required, as well as a lemma about the 
%% K-theory of a
%% tame quotient stack with quasi-projective coarse moduli space.

%% \section{Irreducibility of Hurwitz schemes over a positive genus base}

%% Let $B$ be a smooth projective curve of genus $p$ over an
%% algebraically closed field $k$ of characteristic $0$.  Let $d>0$ be an
%% integer, and let $w\geq 0$ be an even integer such that $w \geq -2 +
%% 2(1-p)d$.  Define $\mathcal{H}^{d,w}_{S_d}(B)$ to be the scheme
%% parametrizing finite morphisms
%% $f\colon C\rightarrow B$ where $C$ is a smooth, irreducible,
%% projective curve of genus $g=d(p-1)+1+\frac{w}{2}$, all branch points of
%% $f$ are simple, and the monodromy group of $f$ is the whole symmetric
%% group $S_d$.  

%% \begin{thm}[Graber, Harris, Starr~\cite{GHS2}] \label{thm-hurw}
%% If $w\geq 2d$, then $\mathcal{H}^{d,w}_{S_d}(B)$ is 
%% connected, smooth, and finite-type over $k$.
%% \end{thm}

%% This is the first generalization to $p>0$ of the genus $0$ case
%% considered by Hurwitz ~\cite{Hurw}.  This result
%% was needed as a key step in our proof of Theorem~\ref{thm-GHS}.
%% The proof is topological in nature, and extends the usual genus $0$
%% proof via braid moves.  The new ingredient is a braid move which moves
%% a branch point around a nontrivial loop in the complete (i.e
%% unpunctured) Riemann surface.

\part{Research proposal}

\section{Brief overview}

The two main projects I am working on are: 
\begin{enumerate}
\item Existence of rational points of an variety defined over a 
non-algebraically-closed field.
\item Properties of varieties related to rationality.
\end{enumerate}

The first project aims to give some answers to the following
problem. 

\begin{prob}[Koll\'ar, Prob 6.1.2~\cite{K}]\label{prob-1}  
Let $F$ be a field
and $X_F$ a 
variety over $F$.  Find conditions on $F$ and $X_F$ which imply that
$X_F$ has a point in $F$.  
\end{prob}

The second project concerns several suggested 
properties of a variety 
generalizing rationality in characteristic
$0$.  
Three weak
generalizations are:
\begin{enumerate}
\item $X$ is \emph{ruled} if $X$ is birational to $Y \times \PP^1$,
i.e. $K(X) \cong K(Y)(t)$.
\item $X$ is \emph{uniruled} if there is a generically finite, dominant
morphism $f:Y \times \PP^1 \rightarrow X$, i.e. $K(Y)(t)$ is a finite
extension of $K(X)$.
\item $X$ has \emph{negative Kodaira dimension} if $h^0(X,\omega_X^{\otimes
n})$ equals $0$ for all $n>0$.
\end{enumerate}
Every ruled variety is uniruled, and every uniruled variety has
negative Kodaira dimension.  There are uniruled varieties which are
not ruled.  But it is unknown whether every variety of negative
Kodaira dimension is uniruled.  Conjecture ~\ref{conj-1} 
states that this is true.  There are several consequences
of Conjecture ~\ref{conj-1}, and one of the goals of the second project is to
prove or disprove one of these consequences.

Three strong generalizations of rationality are:
\begin{enumerate}
\item $X$ is \emph{unirational} if there is a generically finite, dominant
morphism $f:\PP^n \rightarrow X$, i.e. $k(t_1,\dots,t_n)$ is a finite
extension of 
$K(X)$. 
\item $X$ is \emph{rationally connected} if there exists a morphism
$f:\PP^1 \rightarrow X$ whose image is contained in the smooth locus
of $X$ and such that $f^* T_X$ is an ample vector bundle.
\item $X$ satisfies \emph{Mumford's condition} if
$h^0(X,(\Omega^1_X)^{\otimes n})$ equals $0$ for all $n>0$.
\end{enumerate}
Every unirational variety is rationally
connected.
And every
rationally connected variety satisfies Mumford's condition.
It has been conjectured that there are rationally connected
varieties which are not unirational.  And Conjecture ~\ref{conj-2} 
states that every variety satisfying Mumford's
condition is rationally connected.  The main goal of the
second project is to investigate these conjectures.
 
\section{Rational points of varieties over the function field of a surface}

Let $k$ be an algebraically closed field of characteristic $0$ and let
$K$ be a finitely generated extension of $k$ of transcendence degree
$r$.  
Theorem ~\ref{thm-GHS} is an answer to
Problem~\ref{prob-1} when $r=1$.
Theorem ~\ref{thm-R1C} is an answer to
Problem~\ref{prob-1} when $r=2$.  But the additional hypotheses of
that theorem are unreasonably strong, and limit the usefulness of the
theorem.  
A guiding result is the theorem of Tsen ~\cite{Tsen36} and Lang
~\cite{Lang52} that a hypersurface $X$ in $\PP^n_K$ has a $K$-rational
point if the degree $d$ satisfies $d^2 \leq n$.  Theorem ~\ref{thm-HS1}
implies the main hypotheses of Theorem ~\ref{thm-R1C}
for such a hypersurface.  Unfortunately, the
``additional hypotheses'' of Theorem ~\ref{thm-R1C} 
are not satisfied.  The goal is to replace these
unreasonable hypotheses by reasonable hypotheses, hypotheses satisfed
by the hypersurfaces in the Tsen-Lang theorem.

\section{Rational simple-connectedness} \label{sec-R1Ca}

Rational connectedness in algebraic geometry is strongly analogous to
path-connectedness in topology; replace continuous maps from the unit
interval by algebraic morphisms from $\PP^1$ to go from one to the
other.  Following this logic, simple-connectedness in topology should
have an algebraic geometry analogue which Barry Mazur introduced and
calls \emph{rational simple-connectedness}.  Since
simple-connectedness is path-connectedness of the space of based
paths, the algebraic geometry analogue is rational connectedness of
the spaces $\text{RatCurves}^e(X,d)_{p,q}$ parametrizing rational
curves in $X$, of fixed degree $e$, containing a pair $p,q$ of general
points.  This is one of the main hypotheses of Theorem ~\ref{thm-R1C}.
It is also the property proved for
hypersurfaces in Theorem ~\ref{thm-HS1}.

Unfortunately, this property is difficult to prove in general.  The
first hint at a general criterion for rational simple-connectedness is
Theorem ~\ref{thm-S1b}.  By a theorem of Qi Zhang
~\cite{Zhang} and Hacon-McKernan ~\cite{HMcK}, a variety with ample
anticanonical bundle and Kawamata log terminal singularities is
rationally connected.  Assume $\text{RatCurves}^e(X)_{p,q}$ is
irreducible, reduced and normal of the expected dimension.
Using Theorem ~\ref{thm-cancomp}
when both $c_1(T_X)$ and $c_1(T_X)^2 - 2c_2(T_x)$ are positive, one
birational model of $\text{RatCurves}^e(X)_{p,q}$ has ample
anticanonical bundle; namely the contraction of the boundary in
$\overline{\mathcal{M}}_{0,0}(X,e)$.  Thus, to prove it is rationally
connected, it suffices to prove it has Kawamata log terminal
singularities.  In principle, this can be proved locally using
deformation theory.  One goal is to analyze the singularities of this
contraction, hopefully leading to a proof that it has Kawamata log
terminal singularities.

Also, when $c_1(T_X)^2 - 2c_2(T_X)$ is nef, in a suitable sense, de
Jong and I have a bend-and-break argument to prove
$\text{RatCurves}^e(X)_{p,q}$ is uniruled, without using the
computation of the canonical bundle.  An alternative approach is to
push the bend-and-break argument further, hopefully leading to a
direct proof that $\text{RatCurves}^e(X)_{p,q}$ is rationally
connected. 

% \section{Characterization of rational chain connectedness in positive
%   characteristic}  

% Let $k$ be an algebraically closed field with $\text{char}(k) = p>0$.  

% \begin{conj}\label{conj-5}  Let $X$ be a smooth, projective $k$-scheme.
%   If the slope $[0,1)$ part of $H^*_{\text{cris}}(X)$ is
%   trivial, if $\pi_{1,\text{alg}}(X)$ is finite, and if the $p$-Sylow
%   subgroup of $\pi_{1,\text{alg}}(X)$ is trivial, then $X$ is
%   rationally chain connected.
% \end{conj}

% For $X$ a rationally chain connected $k$-scheme, H. Esnault proved
% that the slope 
% $[0,1)$ part of $H^*_{\text{cris}}(X)$ is trivial ~\cite{Esn}.
% F. Campana proved that $\pi_{1,\text{alg}}(X)$ is finite
% (c.f. ~\cite[Cor 4.18]{De}).  For $X$ a rationally chain connected
% surface, R. Crew and N. Nygaard proved that the $p$-Sylow subgroup of
% $\pi_{1,\text{alg}}(X)$ is trivial ~\cite{Crew}, ~\cite{Nyg}.

% If $X$ is a K3 surface, Conjecture~\ref{conj-4} was proved by Rudakov
% and Shafarevich (REFERENCE).  Cossec and Dolgachev used the case of a
% K3 surface to prove Conjecture~\ref{conj-4} for Enriques surfaces.  

\section{Mumford's Conjecture and the Hard Dichotomy Conjecture}

The goal of this project is to investigate
Conjectures ~\ref{conj-1} and ~\ref{conj-2}.
Both of these conjectures imply another, more accessible conjecture,
Conjecture ~\ref{cor-conj6} below.
First the Hard Dichotomy
Conjecture needs to be rephrased. 
Let $k$ be an algebraically closed field with
$\text{char}(k) = 0$,  let $X$ be a smooth, projective $k$-scheme and
let $f\colon X \rightarrow Q$ be the \emph{maximally rationally connected
fibration} of $X$, i.e., the unique dominant rational transformation
with rationally connected fibers and with $Q$ non-uniruled.  
For every $n>0$ and every section $s\in
H^0(X,(\Omega_X^1)^{\otimes n})$, let $\tilde{s}:T_X \rightarrow
(\Omega_X^1)^{\otimes (n-1)}$ be the map of locally free
sheaves defined by contracting $s$ with the given tangent vector.  
Define $\mathcal{F}\subset T_X$ to be the coherent subsheaf
which is the intersection over all $n>0$ and all $s$ of
$\text{Ker}(\tilde{s})$.  

\begin{conj}[Variant of Conjecture ~\ref{conj-1}]
\label{conj-6}
There exists a Zariski dense open set $U\subset X$ contained in the
domain of $f$ such that $\mathcal{F}|_U$ equals the kernel of $df\colon
T_X \rightarrow f^*T_Q$.  
\end{conj}

If $X$ satisfies Mumford's condition, then $\mathcal{F}=T_X$ so that
the maximally rationally connected fibration of $X$ is just the
constant map, i.e. $X$ is rationally connected.  Therefore
Conjecture~\ref{conj-6} implies Mumford's Conjecture.  Moreover the
proof of Corollary ~\ref{cor-3} also proves that
Conjecture~\ref{conj-6} is equivalent to Conjecture ~\ref{conj-1}.

One corollary of Conjecture~\ref{conj-6} would be the following result:

\begin{conj}~\label{cor-conj6}  There exists a dense open subset
$U\subset X$ such that $\mathcal{F}|_U$ is algebraically integrable.
In particular $\mathcal{F}|_U$ satisfies the Frobenius integrability
condition.  
\end{conj}

This conjecture is more accessible than Conjecture~\ref{conj-6}.  For
instance, the Frobenius integrability condition is a local rather than
global condition.  And Grothendieck has a conjecture predicting when
an algebraic foliation is algebraically integrable.  
The goal of this project is to prove 
Conjecture~\ref{cor-conj6}, or at least reduce it to Grothendieck's
conjecture, 
by directly determining
whether $\mathcal{F}$ satisfies the Frobenius integrability condition,
and whether the hypotheses of Grothendieck's conjecture hold.
This would provide indirect evidence for Conjectures ~\ref{conj-1} and
~\ref{conj-2}.   

\section{Unirationality and rational connectedness}

This project concerns a long-standing conjecture that there exist
non-unirational complex Fano manifolds.  General hypersurfaces in
$\PP^n$ of degree $d\leq n$ are complex Fano manifolds.  Koll\'ar
suggested a strategy in ~\cite{KollarSimple}: prove the Fano manifold
has few rational surfaces.  Equivalently, the strategy is to prove
$\text{RatCurves}^e(X)$ contains few rational curves.  The first step
is to prove $\text{RatCurves}^e(X)$ is not uniruled.

\begin{conj}[Starr]~\label{conj-fb}  Let $n \geq 6$ and let $1\leq d
\leq n-3$ be an integer such that $d< \frac{n+1}{2}$ and such that
$d^2 \geq n+2$.  
Let $X\subset \PP^n$ be a general hypersurface of degree $d$ and let
$e>0$ be an integer.  Then every smooth, projective model of
$\text{RatCurves}^e(X)$ is non-uniruled.  In fact it is of general type.
\end{conj}

The same ingredients as in Section ~\ref{sec-R1C} suggest
$\text{RatCurves}^e(X)$ is of general type: it has a projective
birational model whose canonical bundle is big.  Thus the conjecture
reduces to proving the model has canonical singularities.  As in
Section ~\ref{sec-R1C}, this problem is local in nature, and can be
approached using deformation theory.

\section{Cubic fourfolds and rational curves}

Let $k$ be an algebraically closed field with $\text{char}(k)=0$.
Let $X \subset \PP^5$ be a smooth cubic hypersurface.
A longstanding problem in algebraic geometry is to determine if
$X$ is rational when $X$ is very general.  Some cubic
fourfolds are known to be rational, in particular
Hassett has found new examples of rational cubic fourfolds in
~\cite{Has}, making use of his analysis in ~\cite{Has2} 
of Hodge structures of cubic fourfolds.
One key tool in analyzing the Hodge structure of a cubic fourfold is
the fact that $\text{RatCurves}^1(X)$ is a hyperK\"ahler manifold
which is a deformation of $\text{Hilb}^2(S)$ for some K3 surface $S$.
Considering the usefulness of this fact, it seems reasonable to ask if
$\text{RatCurves}^e(X)$ might be a hyperK\"ahler for some $e>0$.  The
goal of this project is to answer this question.

Of course $\text{RatCurves}^e(X)$ is not proper and may be singular.
The precise question considered is:

\begin{ques}  Let $X \subset \PP^5$ be a cubic hypersurface which is
  very general.  For $e\geq 5$ and odd, is $\overline{M}_{0,0}(X,e)$
  birational to a hyperK\"ahler manifold?  Is
  $\overline{M}_{0,0}(X,e)$ isomorphic to a Hilbert scheme
  $\text{Hilb}^f(S)$ for some K3 surface $S$?
\end{ques}

The
expectation is that the sequence of all the schemes $\text{RatCurves}^e(X)$
will give an invariant of $X$, roughly the monoid of
effective cycles in $\text{CH}_0(S)$ in the special case that
$\text{RatCurves}^1(X) \cong
\text{Hilb}^2(S)$.  The hope is to then extend this
invariant to all rationally connected fourfolds, find some factor of
this invariant which is birationally invariant, and use this
birational invariant to prove that $X$ is irrational.  

So far this project has led to the following theorem.

\begin{thm}[de Jong, Starr ~\cite{dJS2}]\label{thm-last}  
Let $X \subset \PP^5$ be a
  cubic hypersurface which is very general.  For every $e>0$, the
  Deligne-Mumford stack $\overline{\mathcal{M}}_{0,0}(X,e)$ is an
  integral, local complete intersection scheme and admits a regular
  $2$-form $\omega$.  
\begin{enumerate}
\item If $e\geq 5$ is odd, $\omega$ is 
  nondegenerate on a nonempty open subset.  
\item If $e\geq 6$ is even,
  $\omega$ is degenerate and the associated sheaf homomorphism $T_X
\rightarrow \Omega_X$ has a rank $1$ kernel $\mathcal{K}$.
\end{enumerate}
\end{thm}

The strategy for answering the first part of this question is to
determine the divisor
in $\overline{\mathcal{M}}_{0,0}(X,e)$ where $\omega$ is degenerate
and then to determine whether this divisor can be contracted.  
The divisor class of $\omega$ follows from Theorem ~\ref{thm-cancomp}.  
I do not yet 
know if this divisor can be contracted.

Another part of this project is to prove Conjecture~\ref{cor-conj6}
for $\overline{\mathcal{M}}_{0,0}(X,e)$ when $e\geq 6$ is even.  By
Theorem~\ref{thm-last}, the sheaf $\mathcal{F}$ is either zero or it is
$\mathcal{K}$.  Conjecture~\ref{cor-conj6} states that for
every even $e$ either $\mathcal{F}$ is zero, or $\mathcal{K}$ is
algebraically integrable and the leaves are rational curves.  Using a
result of Viehweg ~\cite{Viehweg77}, 
to prove this statement, it suffices to prove that $\mathcal{K}$ is
algebraically integrable.  So far I have proved this for the case $e=6$.

\bibliography{my}
\bibliographystyle{abbrv}

%% \begin{thebibliography}{99}

%% \bibitem[]{**} \textbf{My papers}

%% \bibitem{HRS1} Harris, J., Roth, M., and Starr, J., Curves of small
%% degree on cubic threefolds, submitted to \emph{Rocky Mount. J. of Math.},
%% preprint arXiv: math.AG/0202067.

%% \bibitem{HRS3} Harris, J., Roth, M., and Starr, J., Abel-Jacobi maps
%% associated to smooth cubic threefolds, submitted to \emph{J. Alg. Geom.},
%% preprint arXiv: math.AG/0202080.

%% \bibitem{HRS2} Harris, J., Roth, M., and Starr, J., Rational curves on
%% hypersurfaces of low degree, submitted to \emph{J. f\"ur die reine und
%% angewandte Math.}, preprint arXiv: math.AG/0203088.

%% \bibitem{HS2} Harris, J.,  and Starr, J., Rational curves on hypersurfaces
%% of low degree, II, submitted to \emph{Comp. Math.}, preprint arXiv:
%% math.AG/0207257.

%% \bibitem{OS} Olsson, M., and Starr, J., Quot functors for
%%   Deligne-Mumford stacks, to appear in \emph{Comm. in Alg.}, preprint
%%   arXiv: math.AG/0204307.

%% \bibitem{GHS2} Graber, T., Harris, J., and Starr, J., A note on
%%   Hurwitz schemes of covers of a positive genus curve, preprint arXiv:
%%   math.AG/0205056. 

%% \bibitem{GHS} Graber, T., Harris, J., and Starr, J., Families of
%%   rationally connected varieties, to appear in \emph{J. of AMS},
%%   preprint arXiv: math.AG/0109220.

%% \bibitem{dJS} de Jong, A. J., and Starr, J., Every rationally
%%   connected variety over the function field of a curve has a rational
%%   point, to appear in \emph{Amer. J. of Math.}

%% \bibitem{GHMS} Graber, T., Harris, J., Mazur, B., and Starr, J.,
%%   Rational connectivity and sections of families over curves,
%%   submitted to \emph{Ann. Sci. Ecole Norm. Sup.}, preprint arXiv:
%%   math.AG/0210225



%% \

%% \bibitem[]{*} \textbf{Other references}

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