\section{The tautological ring of the moduli space of pointed curves}

[copy in]

In Section \ref{background}, we briefly describe the objects under consideration for the sake
of non-experts, and in 
Section \ref{gor}, we describe the conjecture that motivates this result.

\subsection{Background on the tautological ring}
\label{background}
\remind{background}

A more detailed informal exposition of these well-known ideas is given in \cite{pv}.

When studying Riemann surfaces of some given genus $g$, one is naturally led to study
the moduli space $\cm_g$ of such objects.  This space has dimension $3g-3$, and has
a natural compactification due to Deligne and Mumford, \cite{dm}, the moduli space
$\cmbar$ of {\em stable} genus $g$ curves.  More generally, one can define a moduli
space of stable $n$-pointed genus $g$ curves, denoted $\cmbar_{g,n}$, over any given 
algebraically closed field
(or indeed over $\Spec \Z$).  We shall work over the complex numbers.  A stable
$n$-pointed genus $g$ complex curve is a compact curve with only nodes as singularities,
with $n$ distinct smooth points (labelled $1$ through $n$).
There is a stability condition:  each rational component has at least 3 special points,
and each component of genus 1 has at least 1 special point.
(A special point is a point on the normalization of the component that is either
a marked point, or a branch of a node.)  This stability condition is equivalent
to requiring that the automorphism group of the pointed curve be finite.  If a curve
is stable, then a short combinatorial exercise shows that $2g-2+n>0$.

The open subset of $\cmbar_{g,n}$ corresponding to smooth curves is denoted $\cm_{g,n}$.
The curves of "compact type" (those with compact Jacobian, or equivalently, with
a tree as dual graph) form a partial compactification, denoted $\cmbar_{g,n}^c$.

Even if one is interested primarily in nonsingular curves, i.e. $\cm_g$, one is
naturally led to consider the compactification, $\cm_{g,n}$; even if one is interested
primarily in unpointed curves, one is naturally led to pointed curve $\cmbar_{g,n}$ 
by the behaviour of the boundary, $\cmbar_{g,n} \setminus \cm_{g,n}$. 

The space $\cmbar_{g,n}$ is best considered as an orbifold or Deligne-Mumford stack;
it is nonsingular and proper (i.e. compact) of dimension $3g-3+n$, and hence has a good intersection theory.
Statements about $\cmbar_{g,n}$ translate to universal statements about families
of curves; for this reason, the topology of $\cmbar_{g,n}$ has been studied closely.

The cohomology of the moduli space is already very interesting.  (For example,
the cohomology of $\cm_g$ is the group cohomology of the mapping class group.) 
There is even further structure in the Chow ring of $\cmbar_{g,n}$ (an algebraic
version of the cohomology ring). 

The tautological ring, denoted $R^*(\cmbar_{g,n})$, is a subring of the Chow ring $A^*(\cmbar_{g,n})$.

Natural algebraic constructions typicall yield Chow classes lying in the tautological ring.
(In fact, the authors are unaware of a single Chow class known not to lie in the tautological
ring, although these classes certainly exist, see \remind{???}.)   A simple description
of the Chow ring is the following, due to Faber and Pandharipande (\cite{fp} Section 0.3).

Define the cotangent line class $\psi_i \in A^*(\cmbar_{g,n})$ ($1 \leq i \leq n$) as
the first Chern class of the line bundle with fiber $T_{p_i}^*(C)$ over
the moduli point $[C, p_1, \dots, p_n] \in \cmbar_{g,n}$.  The tautological system 
of rings $\{ R^* (\cmbar_{g,n}) \subset A^* ( \cmbar_{g,n} ) \}$ is defined as the
set of smallest $\Q$-subalgebras satisfying the following three properties:
\begin{itemize}
\item[(i)]  $R^*(\cmbar_{g,n})$ contains the cotangent lines $\psi_1$, \dots, $\psi_n$.
\item[(ii)]  The system is closed under push-forward via all maps forgetting markings:
$$
\pi_*:  A^*(\cmbar_{g,n}) \rightarrow A^*(\cmbar_{g,n-1}).
$$
\item[(iii)]  The system is close dunder push-forward via all gluing maps:
$$
\pi_*:  A^* ( \cmbar_{g, n_1 \cup \{ * \} } )  \otimes_{\Q}  A^*( \cmbar_{g_2, n_2 \cup \{ * \} } )
\rightarrow R^* ( \cmbar_{g_1 + g_2, n_1 + n_2} ),
$$
$$
\pi_*:  A^*( \cmbar_{g, n \cup \{ *, * \}} ) \rightarrow A^*( \cmbar_{g+1, n} ).
$$
\remind{In the first one, change the second * to a dot, as in fp.  In the second, the second * 
should also be a dot.}

The Hodge bundle $\E$ is the rank $g$ vector bundle with fiber $\om_C$ over the moduli point
$[C, p_1, \dots, p_n] \in \cmbar_{g,n}$.  The $\la$-classes are defined by $\la_k = c_k(\E)$ 
($0 \leq k \leq g$); $\la_i$ (and many other naturally defined classes) also lie in the
tautological ring.




\subsection{Is the tautological ring Gorenstein?}
\label{gor}  \remind{gor}


The study of the tautological ring was initiated in Mumford's foundational paper \cite{m}.
Recent interest in the tautological ring was sparked by Kontsevich's proof Witten's conjectures,
\cite{k}, which led to Faber's algorithm for computing all top intersections in
the tautological ring, \cite{falg}.  Based on his algorithm, Faber conjectured that
$R^*(\cm_g)$ is of a very special form, \cite{fconj}; it has the properties of the even-dimensional
cohomology of a complex projective variety of dimension $g-2$.   In paricular, it is a
Gorenstein ring, i.e. $R^{g-2}(\cm_g)=\Q$, and the intersection pairing
$$
R^i(\cm_g} \times R^{g-2-i}(\cm_g) \rightarrow R^{g-2}(\cm_g)
$$
is a perfect (for $0 \leq i \leq g-2$).  (Faber also gives a very specific description of
top intersections, which determines the structure of the ring.)

Theorem \remind{???} is motivated by the audacious speculation that {\bf MOVE TO ARTICLE}






Add to refs  \cite{dm}.  \cite{pv}.  \cite{m}, \cite{k}, \cite{falg}