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\begin{document}
\pagestyle{plain}
\title{\large{The moduli space of curves and its tautological ring}}
\author{Ravi Vakil}
%\address{Dept. of Mathematics, Stanford University, Stanford CA~94305--2125}
%\email{vakil@math.stanford.edu}
\thanks{Ravi Vakil is assistant professor in the Department of Mathematics at Stanford University.
His work was supported in part by NSF grant DMS-0098769, an AMS
Centennial Fellowship, and an Alfred P. Sloan Research Fellowship.
2000MSC:  14H10, 14H81, 30Fxx.}
\date{Friday, March 14, 2003.}
%\begin{abstract}
%
%\end{abstract}
\maketitle
%\tableofcontents

{\parskip=12pt % closing bracket is just before the bibliography 
 
The moduli space of curves has proven itself a central object in
geometry.  The past decade has seen substantial progress in
understanding the moduli space of curves, involving ideas,
for example,  from
geometry (algebraic, symplectic, and differential), physics, topology,
and combinatorics.  Many of the new ideas are related to the
tautological ring of the moduli space, a subring of the cohomology (or
Chow) ring that (i) seems highly structured,
and (ii) seems to include all the geometrically natural classes.

The goal of this article is to give the reader an introduction to the
moduli space of curves and some intuition into it and its structure.
We do this by focusing on its tautological ring, and in particular on
its beautiful combinatorial structure.  As motivation, we consider the
prototypical example of a moduli space, that of the Grassmannian
(parameterizing $k$-dimensional subspaces of an $n$-dimensional space).
Its cohomology ring has a elegant structure, and Mumford suggested
studying the moduli space of curves in the same way.\oremind{\cite[p.~298]{m}}
We introduce the moduli space of (genus $g$, $n$-pointed) curves, with
enough information to give a feel for its basic geography.  We next
introduce the tautological cohomology (or Chow) classes.  Finally, we
describe some of what is known about the tautological ring,
emphasizing combinatorial aspects.  We will assume as little as
possible, although some notions from geometry and topology will be
used.

The selection of material is, of course, a personal one, and there are
necessarily omissions of important work.  However, we hope that enough
has been given for the reader new to the field to appreciate the deep
structure of this important geometrical object.  For the expert, we
hope that the view will be a refreshing one with perhaps some
combinatorial surprises.  Many important recent developments in this
highly interdisciplinary field are beyond the scope of this article.
For example, the most exciting news in recent times is  Madsen
and Weiss' proof\oremind{\cite{mw}} of Madsen's generalization 
of Mumford's conjecture,\oremind{\cite[p.~272]{m}}
based on earlier work of Madsen and Tillmann; the methods are
topological and singularity-theoretic.  Other
relevant fields include Teichm\"{u}ller theory, geometric
topology, conformal field theory, and arithmetic geometry
(Grothendieck's {\em dessins d'enfants}).  Even within algebraic
geometry, this discussion 
leaves out important recent work of Farkas, Gibney-Keel-Morrison, and many
others.


We will use the algebro-geometric language of schemes, but readers
should feel free to interpret these loosely as a certain type of
(ringed, topological) space, and to think instead in their category of
choice (such as algebraic varieties, analytic spaces, manifolds,
topological spaces, etc.).  All dimensions will be complex (=
algebraic), not real, unless otherwise specified.

The perspective here on moduli spaces is that of Grothendieck,
Mumford, and Deligne.  The highlights discussed are informed most
heavily by the books of Arbarello, Cornalba, Griffiths, Harris, and
Morrison, and by work of Faber, Getzler, Looijenga, Pandharipande,
Pikaart, and others.  In particular, the author is deeply grateful to
Joe Harris for teaching him how to think about this beautiful subject.
Suggestions from D. M. Jackson and many others have greatly
improved the exposition of this article.


%{\em Does the typical reader know about pullbacks?}

\bigheading{Prototypical example of a moduli space:  The Grassmannian}

A moduli space is, informally, a parameter space for a certain
type of object.  This is best understood in terms of a prototypical
example, that of the Grassmannian.

Fix a dimension $n$ vector space $\com^n$.  (We work over the complex
numbers.  However, the construction works over an arbitrary base
field, and in much greater generality.)  The Grassmannian $G(k,n)$ is
the set of dimension $k$ subspaces of $\com^n$.  If $V$ is a dimension
$k$ subspace, denote the corresponding point of $G(k,n)$ by $[V]$.
The Grassmannian clearly has more structure than just that of a set;
for example, there is an intuitive notion of ``nearby $k$-spaces'', so
$G(k,n)$ should be a topological space.  There is a natural way to
give $G(k,n)$ the structure of a manifold (and indeed the structure of
a variety), as follows.

Fix $k$ general subspaces of dimension $n-k+1$, $W_1$, \dots, $W_k$.
``Most'' dimension $k$ subspaces $V$ intersect each $W_i$ in a
one-dimensional subspace (i.e. a point of $\proj W_i$).  Conversely,
for ``most'' choices of points of $[\ell_1] \in \proj W_1$, \dots, $[\ell_k]
\in \proj W_k$, the $\ell_i$ are linearly independent and hence span a
$k$-dimensional subspace $V$, and $V \cap W_i = \ell_i$ for all $i$.
Thus we have given a ``large subset'' of $G(k,n)$ the structure of a
complex manifold (or a smooth variety).  The choice of a different
$k$-tuple $(W_i)_{1 \leq i \leq k}$ will give a different large
subset.  One may quickly check that these subsets cover $G(k,n)$, and
that the manifold (or algebraic) structures are compatible.  Hence
$G(k,n)$ is a complex manifold, and indeed a smooth complex algebraic
variety.  (Exercise: if $k=1$, $G(k,n) \cong \proj^{n-1}$ as manifolds,
and the ``usual'' cover by $n$ affine spaces $\com^{n-1}$ is of the
form described above.)

There is a ``universal family'' over $G(k,n)$:
\lremind{universal}
\begin{equation}
\label{universal}
\begin{array}{ccc}
U & \hookrightarrow & G(k,n) \times \com^n \\
\pi \downarrow  & \swarrow \\
G(k,n).
\end{array}
\end{equation}
Here $U$ is a complex algebraic variety, and 
$\pi$ is a morphism (of varieties).
Furthermore, $\pi^{-1}[V] =
V \subset \com^n$.   In fact, $U$ is a vector bundle over $G(k,n)$, sometimes
called the {\em tautological bundle}.

There is an explicit sense in which this algebraic structure on the
set $G(k,n)$ is the ``right'' one. (This is the rigorous definition of 
a moduli space.)   A family of $k$-spaces $\rho: V
\rightarrow B$ parameterized by a variety $B$ induces a map of sets $B
\rightarrow G(k,n)$ (taking a point $b \in B$ to the class of the
fiber $\rho^{-1}(b)$).  This map of sets is in fact an algebraic
morphism.  Moreover, the pullback of the universal family $U
\rightarrow G(k,n)$ to $B$ by this morphism is our original family $V
\rightarrow B$.  Hence we have a bijection between 
the set of families of $k$-planes
parameterized by a variety $B$ and morphisms $B \rightarrow G(k,n)$.
This uniquely specifies the  algebraic structure on $G(k,n)$.
(Caveat:  The notion 
of a ``family of $k$-spaces parameterized by a base'' needs
to be appropriately defined.  The appropriate algebraic version
of continuity of the family is ``flatness''.)

In short, we have defined $G(k,n)$ in a categorical manner as follows. 
We have a ``moduli functor'' that is a contravariant functor from 
the category of varieties to the category of sets, that takes 
a variety $B$ to the set of families $\Fam(B)$ of $k$-planes parameterized
by $B$.
A morphism of varieties $B' \rightarrow B$ induces
a map of sets $\Fam(B) \rightarrow \Fam(B')$:  a family
over $B$ pulls back to a family over $B'$.
This functor is isomorphic to another functor, sending 
$B$ to $\Hom(B,G(k,n))$. 
However, the Grassmannian can (and should) be thought of rather
concretely.  A point of the Grassmannian should be thought of as a
$k$-plane; ``nearby'' points of the Grassmannian should be thought of as ``nearby''
$k$-planes in $\C^n$.  

\mediumheading{Cohomology classes of moduli spaces}

If $\cm$ is a moduli space of a certain kind of object (or structure), then
facts about $\cm$ can be interpreted as universal statements about 
all families of such objects.   
In particular, cohomology classes of the moduli space are universal
cohomology classes.  For example, the classifying space of rank $k$ complex vector bundles
can be interpreted as a moduli space $\cm_{\VB}$, and 
\begin{equation}
\label{chern}
H^*(\cm_{\VB}) = \Z[x_1, \dots, x_k],
\end{equation} 
where $x_i$ has degree (i.e. real codimension) $2i$.  Any vector
bundle $V$ over a base $B$ induces a map $B \rightarrow \cm_{\VB}$,
and the pullback of $x_i$ is the $i$th Chern class $c_i(V)$ of the
bundle.  Hence \eqref{chern} can be interpreted as showing that the
only universal cohomology classes (in the appropriate sense) coming
from vector bundles are Chern classes, and there are no relations
among the classes.  (To make this discussion precise in the algebraic
category, one needs the notion of Artin stacks.)

In analogy with Chern classes, the cohomology ring of a moduli space parameterizing
a certain kind of object is of interest for many reasons, including the following.
\begin{enumerate}
\item[(i)]  
Specific questions about the objects in question can 
be reduced to calculations
on the moduli space.
\item[(ii)] 
To prove general facts 
about about all families of objects, it suffices to prove a
specific fact about a single (hopefully finite-dimensional) space.
\end{enumerate}

The cohomology ring of the Grassmannian has a rich structure,
connected to combinatorics, representation theory, and the theory of
symmetric functions.  (See \cite{tableaux} for a detailed
description.)  We observe first that $G(k,n)$ is smooth and compact.
Hence we can associate cohomology classes with their
\Poincare duals, and will do so.

It is a fundamental fact that the cohomology ring is generated (as a
ring) by the Chern classes of a naturally defined vector bundle on
$G(k,n)$: the universal family $U \rightarrow G(k,n)$ of
\eqref{universal}. 

\mediumheading{A specific question:  How many lines meet four given lines in three-space?}

An example of a specific question about $k$-planes that can be reduced
to calculations in $H^*(G(k,n))$ is the following.  Fix four general
lines $L_1$, $L_2$, $L_3$, $L_4$ in $\com^3$.  How many lines meet all
four?  Because this question is in some sense linear algebra, one
might suspect its
answer to be  $0$, $1$, or $\infty$.  In fact it is $2$.  
(The reader may enjoy
showing this.)

This is a favorite question of many mathematicians, because they will
all agree that it is the baby case of an important idea.  However,
they won't agree on what the important idea is!  (Their answer will depend
on whether their primary interest is linear algebra, geometry, topology,
representation theory of $S_n$ and $GL(n)$, Gromov-Witten theory and
quantum cohomology, combinatorics, symmetric functions, or something
else.)

This question can be reduced to intersection theory on the
Grassmannian as follows.  Interpret $G(2,4)$ projectively, as the
space of projective lines in projective three-space.  It has (complex)
dimension four.  Given a fixed line $L_i$ in $\proj^3$, the locus of
lines meeting $L_i$ is (complex) codimension 1.  Let $D \in
H^2(G(2,4))$ be the cohomology class corresponding to this locus.  (In fact 
$D=c_1(U)$.)
Then this question translates to: compute $D^4$.

\bigheading{The moduli space of $n$-pointed genus $g$ curves}

We next apply the machinery of moduli spaces to algebraic curves.
(For a more detailed but still informal introduction, see
\cite[Part~4]{mirror}.  For a more complete introduction, see
\cite{hmo}.  For excellent summaries of recent research, see \cite{hl, fl}.)

We take smooth curves to be Riemann surfaces (connected compact
complex manifolds of dimension 1), although the following discussion
applies over an arbitrary field (and indeed over an arbitrary base
scheme).  An $n$-pointed smooth curve is a smooth curve with $n$ {\em
  distinct} points labeled $p_1$ through $p_n$.  (In real dimension
2, there is a natural identification among Riemannian structures up to
scaling, conformal structures, almost complex structures, complex
structures, and complex algebraic structures.  These equivalences do
not hold for higher dimensional spaces.)

The genus of a smooth curve is its genus as a surface.  For example,
Figure~\ref{gn63} is a 3-pointed genus 6 curve.  Perhaps the best way
to get a handle on a general genus $g$ curve is as a high-degree
branched cover of the projective line $\proj^1$.  This was the
perspective of Riemann,\oremind{\cite{riemann}} and perhaps surprisingly, many
of the results proved in the last decade involve understanding the
moduli of curves in terms of branched covers.

\begin{figure}
\begin{center}\include{gn63}\end{center}
\caption{A 3-pointed genus 6 curve \lremind{gn63}}
\label{gn63}
\end{figure}

Fix a genus $g$, and a non-negative integer $n$.  Exclude the cases
$(g,n) = (0,0)$, $(0,1)$, $(0,2)$, and $(1,0)$ for technical reasons
that will be described later; the cases $(1,1)$ and $(2,0)$ are also
somewhat special.  Let $\cm_{g,n}$ be the set of smooth genus $g$
$n$-pointed curves.   (When $n=0$, the notation $\cm_g$ is often used.) 
As in the case of the
Grassmannian, $\cm_{g,n}$ has additional structure.  Unfortunately, it
cannot be given the structure of a complex manifold, or a variety.
(More precisely: the moduli functor is not representable in these
categories.)  However, it can be given the structure of an orbifold,
or more precisely (to avoid vagaries of definition of orbifold that
cause problems with $(1,1)$ and $(2,0)$), a smooth Deligne-Mumford
stack.  The general reader may think of $\cm_{g,n}$ as a complex
manifold without going too far wrong; the main caveat is that
cohomology should be taken with $\Q$-coefficients, not
$\Z$-coefficients.  In order to camouflage this issue, we will call
$\cm_{g,n}$ a ``space''.  Note for experts: each point of an orbifold
has an associated finite group.  For $\cm_{g,n}$, the group
corresponding to the pointed curve $(C, p_1, \dots, p_n)$ is the
automorphism group of the pointed curve. 
If the group is $\{ e \}$, $\cm_{g,n}$ is a manifold near that point.
For example, most genus $3$
curves have only the trivial automorphism, but the general
{\em hyperelliptic curve} given by the compactification of the locus in $\com^2$
$$y^2 =
\prod_{i=1}^8 (x-a_i) \quad \quad \text{($a_i \neq a_j$ for $i \neq j$)}$$ 
has a
nontrivial automorphism (given by $(x,y) \mapsto (x,-y)$), and the
curve $$x^3 y + y^3 z + z^3 x = 0$$ in the projective plane has the most
automorphisms of any genus 3 curve, 168 (exercise: find them all!).

We will need one fact about $\cm_{g,n}$.
Remarkably, in some sense this was known to Riemann.\oremind{\cite[p.~134]{riemann}}  

\noindent
{\bf Fact 1.} The space $\cm_{g,n}$ is smooth and irreducible
(i.e.  connected), of dimension $3g-3+n$.  

The geometry of this space is rather subtle, and very few general
statements are known.  One example which is elementary to state is the
following.  Any complex curve of genus at most 10 can be arbitrarily
well approximated by a rationally definable curve (i.e. one defined
over $\Q$, described by equations with rational coefficients).
However, an appropriate generalization of a conjecture of Lang would
imply that in genus at least 24, the set of such curves has measure 0.
(This follows from three facts: (i) rationally definable curves
correspond to $\Q$-points of $\cm_g$; (ii) $\cm_{g,0}$ is
``unirational'' for $g \leq 10$; and (iii) $\cm_{g,0}$ is of ``general
type'' for $g \geq 24$, a celebrated result of Harris, Mumford,
and\oremind{\cite{eh}} Eisenbud.)

Another fact suggesting that the topology of this
space has some elegant structure is 
Harer and Zagier's result\oremind{\cite{hz}} that 
the (orbifold) Euler characteristic of $\cm_{g,n}$ is given by: \lremind{hzeq}  
\begin{equation}
\label{hzeq}\chi(\cm_{g,n}) = (-1)^n \frac  
{ (2g+n-3)! B_{2g}} { 2g (2g-2)! }
\end{equation}
%\begin{equation}
%\label{hzeq}\chi(\cm_{g,n}) = (-1)^{n-1} \frac { (2g+n-3)!} { (2g-2)!} \zeta(1-2g) = (-1)^n \frac  
%{ (2g+n-3)! B_{2g}} { 2g (2g-2)! }
%\end{equation}
if $g >0$,  where
$B_{2g}$ denotes the Bernoulli 
number.\oremind{Mention real curves?}
(If $g=0$, $\chi(\cm_{g,n}) = (-1)^{n+1}(2g+n-3)!$.)

\mediumheading{Deligne and Mumford's compactification $\cmbar_{g,n}$ of $\cm_{g,n}$}

Deligne and Mumford \cite{dm} introduced a compactification
$\cmbar_{g,n}$ of the moduli space $\cm_{g,n}$.  The additional
points of the moduli space  correspond to {\em nodal curves}.
The points of a nodal curve are either smooth, or
analytically isomorphic to a neighborhood of the origin of
$xy=0$ in
$\com^2$.  The curve is still required to be connected and compact.
An $n$-pointed nodal curve is a nodal curve with $n$ distinct smooth
points labeled $p_1$ through $p_n$.

Informally, the {\em arithmetic genus} of a curve with nodes is the
genus of its smoothing.  For example, a smoothing of
Figure~\ref{gnbar63} is Figure~\ref{gn63}; hence Figure~\ref{gnbar63}
has arithmetic genus 6.  (Warning: the picture of the node is
misleading, due to the difficulties of depicting a node in 
real two- or three-dimensional space.  In fact, at each node the two
branches are glued together transversely, and are not tangent.)
 
\begin{figure}
\begin{center}\include{gnbar63}\end{center}
\caption{A 3-pointed genus 6 curve with three nodes \lremind{gnbar63}}
\label{gnbar63}
\end{figure}

The {\em geometric genus} of a {\em component} of a nodal curve is
found by ``ungluing'' the nodes, and finding the genus of the corresponding
component.  For example, the components of the curve of Figure~\ref{gnbar63} have
genus 0, 2, and 3.

To each nodal curve with labeled points, we associate a {\em dual
  graph}.  Vertices correspond to components; they are labeled with
their geometric genus.   For each node, we draw an edge joining the
appropriate vertices.  For each marked point, we draw a ``half-edge''
incident to the appropriate vertex, with the same label as the point.
For example, the dual graph corresponding to Figure~\ref{gnbar63} is
shown in Figure~\ref{dual}.  (Combinatorial exercise: if a nodal curve
$C$ has $k$ components of geometric genera $g_1$, \dots, $g_k$, and
$\delta$ nodes, then the arithmetic genus of $C$ is $\sum_{i=1}^k
(g_i-1) + 1 + \delta$.)

\begin{figure}
\begin{center}\include{dual} \end{center}
\caption{The dual graph corresponding to Figure~\ref{gnbar63} \lremind{dual}}
\label{dual}
\end{figure}

A nodal curve is said to be {\em stable} if it has finite automorphism
group.  This is equivalent to a combinatorial condition:
(i) 
each genus 0 vertex of the dual graph has valence at least three, and (ii) each
genus 1 vertex has valence at least one.
For example, the curve of Figure~\ref{gnbar63} is stable.

Now let $\cmbar_{g,n}$ be the moduli space of $n$-pointed genus $g$
{stable} curves.  (The reader may verify that, as we have assumed
$(g,n) \neq (1,0)$, one need only
check condition (i).)  


\noindent
{\bf Fact 2.}  The moduli space of stable curves $\cmbar_{g,n}$
is a smooth, compact, irreducible space (Deligne-Mumford stack), of dimension $3g-3+n$.

Moreover, the moduli space has a natural stratification.

\noindent
{\bf Fact 3.}
$\cmbar_{g,n}$ is stratified by topological type, or equivalently, by  dual graph.

In other words, to each stable dual graph there corresponds a stratum.
For example, the stratum given by Figure~\ref{dual} consists of
(points corresponding to) all curves looking like Figure~\ref{gnbar63}.
To give such a curve is the same as giving a 3-pointed genus 0 curve,
a 3-pointed genus 2 curve, and a 3-pointed genus 3 curve, and gluing
them together as shown in Figure~\ref{gnbar63}.  Thus this stratum is
isomorphic to $\cm_{0,3} \times \cm_{2,3} \times \cm_{3,3}$.  In the
same way, one may show that each stratum is isomorphic to a product
of moduli spaces of smooth pointed curves, modulo a finite group.
(For an example with a nontrivial finite group: the stratum
corresponding to the dual graph of Figure~\ref{S3} is isomorphic to
the quotient of $\cm_{1,2} \times \cm_{1,2} \times \cm_{1,2}$ by the
symmetric group $S_3$.)  In particular, the next fact follows (after
mild combinatorics) from the first three.

\begin{figure}
\begin{center}\include{fun} \end{center}
\caption{The stratum corresponding to this graph is isomorphic to
$\left( \cm_{1,2} \times \cm_{1,2} \times \cm_{1,2} \right) / S_3$
\lremind{S3}}
\label{S3}
\end{figure}

\noindent
{\bf Fact 4.}  The codimension of a stratum is the number of edges of the
corresponding dual graph.    $\cmbar_{g,n}$ contains $\cm_{g,n}$ as a dense open stratum.
(The other strata are called {\em boundary strata}.)
The 0-dimensional strata, or 0-strata, correspond to dual graphs where all vertices
are  trivalent of genus 0.

\mediumheading{Examples:  $\cmbar_{0,3}$, $\cmbar_{0,4}$, and $\cm_{1,1}$}

It is a straightforward fact that any three points on $\proj^1$
may be sent to $0$, $1$, $\infty$ by a unique automorphism of $\proj^1$.
Hence $\cm_{0,3}$ consists of a single point.  One may quickly check
that the only stable 3-pointed genus 0 graph is the obvious one (with one
genus 0 vertex), so $\cmbar_{0,3}$ is a point as well.

If $p_1$, $p_2$, $p_3$, $p_4$ are distinct points on $\proj^1$, then
the automorphism of $\proj^1$ of the previous paragraph
(sending the first three points to $0$, $1$, and $\infty$
respectively) will send $p_4$ to some number $\la \neq 0, 1, \infty$.  
This map is the cross-ratio map:
$$
\la = \frac { (p_4 - p_1) (p_2-p_3)}
{ (p_4 - p_3) (p_2 - p_1)}.
$$
Hence $\cm_{0,4} \cong \proj^1 - \{ 0, 1, \infty \}$.  
(In its modern guise of the map to $\cm_{0,4}$, the lowly
cross-ratio remains an important tool in geometry.)
The
compactified space $\cmbar_{0,4}$ has three boundary strata, the last three in
Figure~\ref{mbar04}.  In fact $\cmbar_{0,4} \cong \proj^1$; the reader
should try to associate the three boundary strata with the three
degenerate values ($0$, $1$, $\infty$) of $\la$ .  The three boundary
strata are trivially rationally equivalent, as they are points on $\proj^1$.
This relation goes by many names; we will refer to it as the
cross-ratio relation.

\begin{figure}
\begin{center}\include{mbar04}\end{center}
\caption{The strata of $\cmbar_{0,4}$ \lremind{mbar04}}
\label{mbar04}
\end{figure}

The reader may check that no stable $n$-pointed genus 0 curve has a
non-trivial automorphism.  Thus $\cmbar_{0,n}$ is a manifold (and
indeed an algebraic variety).

The space $\cm_{1,1}$ parameterizes genus 1 curves with a marked point,
i.e. elliptic curves.  To each elliptic curve is associated a
$j$-invariant (a complex number), and to each complex number there is
an associated elliptic curve with that $j$-invariant, so $\cm_{1,1}$
looks a lot like $\C$.  However, $\cm_{1,1}$ is not a manifold (as 
every elliptic curve has an involution, so $\cm_{1,1}$
has non-trivial stack structure).  In a concrete sense, $\cm_{1,1}$,
not $\C$, is the right moduli space for elliptic curves. 
For example,
modular forms are sections of a line bundle on the space
$\cmbar_{1,1}$.

\mediumheading{Natural morphisms between moduli spaces of curves}

There are natural morphisms between moduli spaces of curves.  For example, given
an $n$-pointed genus $g$ curve ($n>0$, $(g,n) \neq (0,3), (1,1)$), one may
forget point $p_n$ to obtain an $(n-1)$-pointed  genus $g$ curve.  Hence
there is a ``forgetful morphism'': \lremind{forgetful}
\begin{equation}\text{{\em Forgetful morphism:}}  \quad \quad \quad \cmbar_{g,n} \rightarrow \cmbar_{g,n-1}. \label{forgetful}
\end{equation}
(Caution:  when a point is forgotten, the resulting curve may not be stable.
However, there is a natural way to ``stabilize'' an unstable curve.
Exercise:  if point $p_3$ is forgotten in Figure~\ref{gnbar63}, what should
be the resulting stable 2-pointed genus 6 curve?)

Also, given an $(n_1+1)$-pointed genus $g_1$ curve $C_1$ and an
$(n_2+1)$-pointed genus $g_2$ curve $C_2$, one may glue one of the marked points of $C_1$
 to one of the marked points of $C_2$ to obtain a new curve $C$ of genus $g_1+g_2$ 
with $n_1+n_2$ marked points (two of the previously labeled points have disappeared).
Similarly, 
 given an $n+2$-pointed genus $g$ curve $C$, one may glue two  of the marked points of $C$
together, obtaining a new curve  of genus $g+1$ 
with $n$ marked points.  Hence we have: \lremind{gluing}
\begin{equation}
\text{{\em Gluing morphisms:}} \quad \quad \quad
\cmbar_{g_1, n_1 +1} \times 
\cmbar_{g_2, n_2 +1} \rightarrow \cmbar_{g_1+g_2, n_1+n_2},
\quad 
\cmbar_{g, n+2}
\rightarrow \cmbar_{g+1, n}.
\label{gluing}
\end{equation}
The gluing morphisms came up implicitly in our earlier discussion of boundary strata as products of 
smaller moduli spaces of curves.

Call the forgetful and gluing morphisms \eqref{forgetful}, \eqref{gluing} the {\em natural morphisms}
between moduli spaces of curves.


\bigheading{The tautological ring}

%\mediumheading{Definition of the tautological ring}

In algebraic geometry, rather than considering the cohomology ring, we
consider the algebraic version of it, the Chow ring $A^*(\mgnbar)$.
The Chow groups are generated by algebraic cycles.  Relations in Chow
groups are generated by the atomic relation that two points in
$\proj^1$ are ``homologous'' (the analogous terminology in Chow is
{\em rationally equivalent}; we reserve the word ``homologous'' for
cohomology), and the condition that relations pull back and push forward
under appropriate morphisms.  As both the generators and relations of
the Chow ring are a subset of those generating 
cohomology, Chow is neither stronger nor weaker than simplicial
cohomology.  For example, as the generators of Chow groups are
algebraic and hence of even real dimension, Chow groups do not see
odd-graded cohomology.  On the other hand, two distinct points on a
smooth curve of positive genus are not equivalent in Chow, while they
are in homology.  \oremind{For a rigorous introduction to intersection theory
in algebraic geometry, see \cite{it}.}  The reader may think instead
about cohomology, although
some information will be lost, as will be explained soon.
There is a natural map $A^k \rightarrow H^{2k}$.  (The index
for Chow corresponds to algebraic (= complex) codimension, hence
its doubling in the map to cohomology.)
Because we are considering Deligne-Mumford stacks or orbifolds, 
we take rational rather than integral coefficients.

In some sense, it appears that the Chow ring $A^*(\mgnbar)$ is not the
right object to study. Instead, attention has focused on a certain
``tautological'' subring, denoted by $R^*(\mgnbar) \subset A^*(\mgnbar)$,
for three admittedly vague reasons: (i) many interesting questions in
geometry boil down to the tautological subring; (ii) no natural
questions seem to reduce to questions in the non-tautological part
of the Chow ring; and (iii) the tautological subring has, at least
conjecturally, a great deal of structure.
For example, $A^*(\mgnbar)$ is probably usually huge except in 
special cases (see the discussion of $\cmbar_{1,11}$ below).
$R^*(\mgnbar)$ is much smaller.

However, in complex codimension 1, Arbarello and 
Cornalba showed\oremind{\cite{acharer}} using results 
of Harer\oremind{\cite{harer1, harer2}} that the
tautological ring is equal to the cohomology ring.  

As a first informal definition, $R^*(\mgnbar)$ consists of {\em all
  the classes naturally coming from geometry}.  By this
``definition'', any geometric calculation that can be translated to
the Chow ring will require only knowledge of the tautological ring.
(For an example of how straightforward facts in the tautological ring
can lead to solutions of longstanding classical problems, see
\cite{quartic}.)  An easy example of a class coming from geometry is
the class of a boundary stratum.

A second example of classes are Chern classes of certain natural
vector bundles on the moduli space.  For example, there are $n$
natural line bundles on $\mgnbar$, denoted by $\LL_1$, \dots, $\LL_n$.
The fiber of $\LL_i$ at the point $[C, p_1, \dots, p_n] \in
\cmbar_{g,n}$ is the (one-complex-dimensional) cotangent space to $C$ at
$p_i$.  Define $\psi_i = c_1(\LL_i)$; these are called the {\em
  $\psi$-classes}, and will be central players in the rest of this
exposition.

Another natural vector bundle on $\cm_{g,n}$ is the {\em Hodge
bundle}:  to a smooth curve $C$ one associates the $g$-dimensional
vector space of differentials.  This definition can be extended
over the boundary, giving a rank $g$ vector bundle $\EE$ over $\cmbar_{g,n}$.
Define $\la_k = c_k (\EE)$ ($1 \leq k \leq g$).  These
{\em $\la$-classes} will come up again later.

There are many other natural classes, including the
Miller-Morita-Mumford $\ka$-classes, $\ka_1$, $\ka_2$, \dots (where
$\ka_i$ is codimension $i$).  For
brevity, we omit their definition, although they will arise again
in Faber's conjecture.  Mumford's original definition\oremind{\cite{m}}
of the tautological ring of $\cm_g$ was in terms 
of $\kappa$-classes.\oremind{\cite[p.~299]{m}}
The $\ka$-classes were also independently defined by Morita\oremind{\cite{morita}}
and Miller.\oremind{\cite{miller}}

The generalization of the tautological ring to $\cmbar_{g,n}$ is
due to\oremind{[HL]} Hain and Looijenga.\oremind{The 
appropriate generalization to $\cmbar_{g,n}$ is due to Arbarello and Cornalba \cite[Sec.~1]{ackont}.)  {\em Apparently that's false.}}
The following description is 
due to Faber and Pandharipande.\oremind{\cite[p.~216]{fpconj}}

\noindent
{\em Definition.}  The system of tautological rings $\left( R^*(\mgnbar)
\subset A^*(\mgnbar) \right)_{g,n}$ is the smallest system of $\Q$-algebras
satisfying:
\begin{enumerate}
\item[(i)]  $\psi_1, \dots, \psi_n \in R^*(\mgnbar)$, and
\item[(ii)] the system is closed under pushforwards by natural morphisms
(\eqref{forgetful} and \eqref{gluing}).
\end{enumerate}
For example, the boundary strata are obtained by pushing forward the
fundamental class by gluing morphisms \eqref{gluing}.
All other ``natural'' geometric classes are in this ring, including
$\la$-classes, $\ka$-classes, and pullbacks of tautological classes
under natural morphisms.

\noindent
{\em Definition.} Define the tautological ring on any open subset of
$\mgnbar$ by its restriction from $\mgnbar$.

The striking (often conjectural) combinatorial structure of the
tautological ring comes in two related flavors: \begin{itemize}
\item structure related to boundary
strata (and hence graphs); and
\item structure of top
intersections in the tautological ring, often called {\em Hodge integrals};
especially top intersections of 
 $\psi$-classes.
\end{itemize}
Before describing what is known about these
two structures in arbitrary genus,
we will first recapitulate what is known in low genus.
In genus 0, the structure is exceptionally elegant and combinatorial.
In higher genus, odder behavior starts to happen; examples
often involve $\cmbar_{1,11}$.

\mediumheading{Genus zero}

The gluing morphisms \eqref{gluing} combined with the cross-ratio relation give
equivalences among boundary strata (in cohomology), depicted in
Figure~\ref{crossratio}.  In the Figure, $A$, $B$, $C$, and $D$ are
taken to be graphs with one ``root''.  We will also refer to these as
{\em cross-ratio relations}.  The equivalence shown in
Figure~\ref{crossratioeg} is an example of a cross-ratio relation.  

\begin{figure}
\begin{center} \include{crossratio} \end{center}
\caption{The cross-ratio equivalences of boundary strata \lremind{crossratio}}
\label{crossratio}
\end{figure}

\begin{figure}
\begin{center} \include{crossratioeg} \end{center}
\caption{An example of a cross-ratio equivalence
(in $\cmbar_{25}$)
\lremind{crossratioeg}}
\label{crossratioeg}
\end{figure}

Keel has proved\oremind{\cite{keel}} that in genus 0 (for any number of
points), the cohomology and Chow groups are generated by boundary
strata, and that the relations are generated by the cross-ratio
relations.  In particular, $H^* = A^{*/2} = R^{*/2}$.  (In fact, the
cohomology ring is generated as an algebra by complex codimension 1
strata.)  Multiplication of boundary strata in cohomology can be
described combinatorially as well, so the cohomology ring of
$\cmbar_{0,n}$ has a beautiful combinatorial structure.

A different aspect of the combinatorial structure of the
cohomology/Chow/tautological ring of $\cmbar_{0,n}$ is visible 
in terms of $\psi$-classes:
$$
\int_{\cmbar_{0,n}} \psi_1^{a_1} \cdots \psi_n^{a_n} = \binom {n-3} {a_1, \dots, a_n}.
$$
We are using 
ordinary multiplication to represent cup product.
Define $\int_X y$ (where $y \in H^{2i}(X)$ or $A^i(X)$)  
to be the degree of $y$ if $i=\dim X$,
and to be $0$ otherwise.
As $\dim \cmbar_{0,n} = n-3$, $\sum_i a_i$ must be $n-3$ in
order to produce a $0$-cycle.  

This result is even more beautiful
in the language of generating functions.  Let \lremind{Egn}
\begin{equation}
\label{Egn}
E_{g,n} = \sum_{a_1, \dots, a_n} \left(
\int_{\cmbar_{g,n}} \psi_1^{a_1} \cdots \psi_n^{a_n} \right) t_1^{a_1} \cdots t_n^{a_n}
\end{equation}
%\begin{equation}
%\label{Egn}
%E_{g,n} = \sum_{a_1, \dots, a_n} \left(
%\int_{\cmbar_{g,n}} \psi_1^{a_1} \cdots \psi_n^{a_n} \right) t_1^{a_1} \cdots t_n^{a_n}
% = \sum_{a_1, \dots, a_n} \left(
%\int_{\cmbar_{g,n}} (\psi_1 t_1)^{a_1} \cdots (\psi_n t_n)^{a_n} \right)
%\end{equation}
where $t_1$, \dots, $t_n$ are formal variables.  
This is a homogeneous
polynomial of degree $\dim \cmbar_{g,n} = 3g-3+n$.  Then \lremind{iz0}
\begin{equation}
\label{iz0}
E_{0,n}=(t_1
+ \dots + t_n)^{n-3}.  
\end{equation}

\mediumheading{Genus one:  The strange case of $\cmbar_{1,11}$}

Before we discuss how well genus 1 is understood, we begin with a
pathology: $\cmbar_{1,11}$.  In genus 0, the map $A^{*/2}
\rightarrow H^*$ is an isomorphism.  One blunt reason for $A^{*/2}(X)
\rightarrow H^*(X)$ {\em not} to be an isomorphism for a complex
projective manifold $X$ is if $X$ has odd cohomology: the algebraic
classes are generated by algebraic (hence complex analytic) cycles, and
hence are even-dimensional.  
(But the Euler characteristic of the moduli space of curves may be 
negative by \eqref{hzeq}, odd cohomology definitely arises.)
There is a more refined obstruction as
well:  Hodge theory dictates that there is a sub-vector space
of the even-dimensional cohomology where the algebraic cohomology
must lie; we will call this the cohomology of {\em Tate type}.

The 22-real-dimensional space $\cmbar_{1,11}$ has odd cohomology, in
its middle dimension.  The existence of this cohomology is a
consequence of the existence of a non-zero 11-form on this
11-complex-dimensional orbifold.  (This uses the existence of the
famous weight 12 modular form $\Delta$.)\oremind{\cite{deligne}}  Hence
the natural map from the Chow ring of $\cmbar_{1,n}$ to its
cohomology ring $H^*(\cmbar_{1,n})$ cannot be a
surjection for $n=11$, and indeed for $n \geq 11$.  \oremind{I feel
  like Pikaart should be credit here.  For what?  What ref?  I
asked Carel Jan. 6.}

Also, the existence of this non-zero 11-form implies that
$A_0(\cmbar_{1,n})$ is huge (for $n \geq 11$) --- for example, it is
uncountably generated. (A more precise statement and proof of this
folklore fact appear in \cite{socle}.)  As $R_0(\cmbar_{1,n})$ (i.e.
$R^{\dim \cmbar_{1,n}}(\cmbar_{1,n})$) is countably generated (indeed
one-dimensional, see below), we see that $R^0(\cmbar_{1,n})
\rightarrow A^0(\cmbar_{1,n})$ cannot be an isomorphism.
On the other hand, for $n \leq 10$, the maps $R^{*/2}(\cmbar_{1,n}) \rightarrow
A^{*/2}(\cmbar_{1,n}) \rightarrow H^*(\cmbar_{1,n})$ are isomorphisms,
by work of Belorousski. \oremind{pasha}

% This is attested to in Faber-Looijenga

Hence the cohomology ring has more information than the Chow ring (for
example, it has odd cohomology), and the
Chow ring has more information than the cohomology ring (for example,
it can better distinguish 0-cycles).  In genus 1, the tautological
ring is well-understood, and it is ``better-behaved'' in that it sees
neither of these two extra types of information, as we shall soon see.

As in genus 0, the tautological groups are generated by boundary
strata.  (This follows from the definition of the tautological ring
using $\psi$-classes, and the fact that on $\cmbar_{1,n}$, $\psi_i$ is
rationally equivalent to a sum of boundary strata.)  Intersections
between boundary strata are quite straightforward.  There is a new
relation in codimension 2 on $\cmbar_{1,4}$ due to 
Getzler;\oremind{\cite{m14}}
the relation is complicated, and omitted here.  It is perhaps
surprising that this relation was not discovered earlier, given the
fact that $\cmbar_{1,4}$ is a compact complex fourfold, parameterizing
simple objects of long-standing interest (essentially, four points on
an elliptic curve).  Furthermore, Getzler has announced that
his relation and the
cross-ratio relation generate {\em all} relations on $\cmbar_{1,n}$,
and that the map from the tautological ring to the cohomology ring is an
injection.\oremind{\cite{ezranew}}  (More precisely, the map $R^{*/2} \rightarrow
H^*$ gives an isomorphism of the tautological ring with the even cohomology.
As a consequence, for example, any even-dimensional class
is homologous  to a tautological class.)

There is an analog to (\ref{iz0}), involving the logarithm
of a certain generating function, divided by 24. \oremind{Delete?}

\mediumheading{Genus greater than one}

Starting in genus 2, the boundary strata do not generate the
tautological ring: $\psi_1$ on the threefold $\cmbar_{2,1}$ is not
homologous to boundary strata.  (This is in some sense the only
example in genus 2; all others are derived from this via the gluing
morphism.)  The Chow ring of $\cmbar_{2,n}$ is known for small $n$,
but many open questions remain.  Belorousski and Pandharipande have
proved a new relation in codimension 2 on
$\cmbar_{2,3}$.\oremind{\cite{bp}} Polito showed in her thesis that
there are no additional codimension 2 relations for $\cmbar_{g,n}$,
but surely there are more to be found in higher codimension.  An explicit
algebraic cycle, not homologous to a tautological class, defined over
$\Q$, was found on $\cmbar_{2,22}$ by Graber and
Pandharipande;\oremind{\cite{nontaut}} it is the first in a family of
examples, and its construction uses $\cmbar_{1,11}$.

In genus greater than 2, the full Chow ring is well-understood in a
few cases, including: $A^* (\cmbar_3)$ (work of Faber;
it contains only tautological elements),\oremind{\cite{fm3}} $A^*(\cm_4) \cong \Q[\la_1]
/ (\la_1^3)$ (Faber\oremind{\cite{fm4}} again), and $A^*(\cm_5) \cong
\Q[\la_1] / (\la_1^4)$ (work of Izadi\oremind{izadi} combined with
work of Faber).  Pikaart has shown\oremind{\cite[Cor.~2.4.7]{p}; also
  see p. 480 Cor. 4.7 of the Texel volume} that for large $g$ (for
given $n$), the cohomology of $\cmbar_{g,n}$ is not all of Tate type
and hence not all algebraic (let alone tautological); his argument
involves $\cmbar_{1,11}$.  There is also a higher-genus version of
\eqref{iz0}, given by a ``genus-expansion'' ansatz of the physicists
Itzykson and Zuber.\oremind{\cite[(5.32)]{iz}} \oremind{For a proof,
  see \cite[Thm.~3.1]{gjvk}.}

\bigheading{The tautological ring in arbitrary genus: top
  intersections of $\psi$-classes}

The reason for interest in top intersections of $\psi$-classes is that
knowing these values allows us to find the top intersections of any
classes in the tautological ring, by appropriately unwinding the
definition of the tautological ring.  (Based on his earlier work,
Faber has\oremind{falg} written a remarkable {\sc Maple} program that will
explicitly perform intersections of many different types of classes in
the tautological ring \cite{faberprog}.)  Furthermore, these top
intersections of $\psi$-classes have a striking structure which
allows them to be computed, using Witten's conjecture or Hurwitz
numbers.

Witten's conjecture\oremind{\cite{w}} (proved by
Kontsevich,\oremind{\cite{kont}} and later by\oremind{\cite{op}} 
Okounkov and Pandharipande) was the catalyst for much of
the interest in the tautological ring in recent years.  
(See \cite{lkont} for a detailed exposition.)
Define
$F_g$ by 
$$
F_g := \sum_{n \geq 0}
\frac 1 {n!} \sum_{k_1, \dots, k_n} 
\left( \int_{\cmbar_{g,n}} \psi_1^{k_1} \cdots \psi_n^{k_n} \right)
t_{k_1} \cdots t_{k_n} 
,$$
the generating function for all 
top intersections of $\psi$-classes in genus $g$.
(Compare this to \eqref{Egn}.)
Define a generating function for all such intersections
in {\em all} genus by
$$ 
F :=  \sum_g F_g \la^{2g-2}.$$
This is {\em Witten's free energy} or 
the {\em Gromov-Witten potential of a point}.
For convenience, we will set $\la=1$.

Witten's conjecture (Kontsevich's Theorem) gives a 
recursion for top intersections of $\psi$-classes in the form of
a partial differential equation satisfied by $F$.  Using this differential
equation along with
a straightforward geometric fact known as the {\em string
  equation} and the trivial fact (the ``initial condition'')
$\int_{\cmbar_{0,3}} 1 = 1$ (as $\cmbar_{0,3}$ is a point), all top
intersections are quickly recursively determined.  

\noindent {\bf Witten's Conjecture (Kontsevich's Theorem).}
$$
(2n+1) 
 \frac {\partial^3} {\partial t_n \partial t_0^2 } F
= 
\left(  
\frac {\partial^2 } {\partial t_{n-1} \partial t_0} F \right)
\left(  \frac {\partial^3} {\partial t_0^3}
 F 
\right) 
+ 2
\left(  \frac {\partial^3 } {\partial t_{n-1} \partial t_0^2} 
F \right)
\left(  \frac {\partial^2} {\partial t_0^2} 
F 
\right)
+ \frac 1 4
 \frac {\partial^5} {\partial t_{n-1} \partial t_0^4} 
F.
$$

Kontsevich's proof\oremind{\cite{kont}} of Witten's conjecture 
is by combinatorializing these top intersections,
expressing them in terms of sums over ``ribbon-graphs''.  
Okounkov and Pandharipande's more recent proof\oremind{\cite{op}}
is by way of Hurwitz numbers, which is a second way of combinatorializing
top intersections of $\psi$-classes, and is described below.

Witten's conjecture indicates a deep connection between 
moduli spaces such as these, and integrable systems.  This 
link is still mysterious.
Another link to integrable systems is via the Virasoro Lie algebra.
Dijkgraaf, Verlinde, and Verlinde showed that Witten's conjecture
(along with the string equation) implies that $e^F$ is annihilated by a sequence of
differential operators $L_{-1}$, $L_0$, $L_1$, \dots, corresponding to
part of the Virasoro algebra.  The first differential equation $L_{-1}
e^F=0$ is the string equation, and the second is geometrically
obvious, but the geometric meaning of the rest is obscure.  One of the
fundamental open questions in Gromov-Witten theory is a generalization
of this formulation of Witten's conjecture, and deals with maps of
curves to an arbitrary (smooth projective) target manifold: the {\em
  Virasoro conjecture} (due to the physicists Eguchi, Hori, and Xiong,
as well as S. Katz).\oremind{, see \cite[Sect.~10.1.4]{ck}).}  This question has
recently been settled in two important open cases: that of projective
spaces (by 
\oremind{\cite{givental}}
Givental) and that of smooth curves (by\oremind{\cite{op2, op3, op4}}
Okounkov and Pandharipande).


\mediumheading{Hurwitz numbers}

Another combinatorial description of top intersections of
$\psi$-classes, which has been alluded to above, 
 is via {\em Hurwitz numbers}. 
Hurwitz numbers count, essentially, ordered
factorizations of a permutation into transpositions, and these 
factorizations may be regarded as combinatorial objects.
More precisely, fix a positive integer $d$, a non-negative integer
$r$, and a conjugacy class of $S_d$ given by a partition $\al$
(i.e. $\al_1 + \dots + \al_n=d$).  Define $g$ by $2-2g = d+n-r$.  
Then define \lremind{hdef}
\begin{equation}
\label{hdef}\tilde{H}^g_{\al} = \# \left\{ \si_1, \dots, \si_r \in
\operatorname{tran}(S_d) : \si_1 \cdots \si_r \in C(\al) \right\} / d!
\end{equation}
where $\operatorname{tran}(S_d)$ is the set of transpositions in $S_d$ (elements
of the form $(ij)$), and $C(\al) \subset S_d$ is the conjugacy class
defined by $\al$.

This number can be interpreted as the answer to a problem in enumerative
geometry, as follows.  Fix $r+1$ distinct points $\{ p_1, \dots,
p_r, \infty \}$ on the sphere, interpreted as the complex projective
line $\com \proj^1$, and count the number of degree $d$ covers of the
sphere, unbranched away from the $r+1$ points, with the simplest
possible branching above each of the first $r$ points (i.e. two of the
sheets come together, analytically the projection of the parabola
$y^2=x$ to the $x$-axis in $\com^2$; the remaining $d-2$ sheets are unbranched), and
with branching above $\infty$ given by $\al$.  Then the cover
also has a complex structure by the Riemann existence theorem,
and hence is a Riemann surface, although possibly with many components.

Define the Hurwitz number $H^g_{\alpha}$  as the number of such covers where the cover
has {\em one} component.  In this case, the covering curve has genus
$g$.  The analog in \eqref{hdef} is to require that the $\si_i$
generate $S_d$.  Equivalently, if one constructs a graph with vertices
labeled $1$ through $d$, and draws an edge between $(jk)$ if $\si_i =
(jk)$, then this ``monodromy graph'' must be connected.  The simplest way to describe
the relationship between the numbers $H^g_{\al}$ and
$\tilde{H}^g_{\al}$ is to construct generating functions $H$ and
$\tilde{H}$ for these numbers in a straightforward way; then
$\tilde{H} = e^H$.

Hurwitz numbers were first studied by Hurwitz.\oremind{ \cite{hurwitz}.}  In
recent times, they have been studied by combinatorialists, representation
theorists, physicists, the Arnol'd school of geometers, symplectic and
algebraic geometers interested in Gromov-Witten theory, and more.  For
a summary of this rich history, see \cite{elsv2,gjvk} 
\oremind{Find some good source!
  Perhaps ask Misha Shapiro too}.

Hurwitz numbers are obviously easy to compute.  If you play with
them for a while, you may soon come to the following conjecture:
\lremind{gjeq}
\begin{equation}
\label{gjeq}
H^g_{\al} = \frac {r!} {\# \Aut ( \al)} \prod_{i=1}^n \frac {\al_i^{\al_i}}
{\al_i!} P(\al_1, \dots, \al_n)
\end{equation}
where $P$ is some mysterious symmetric polynomial of total degree between
$2g-3+n$ and $3g-3+n$.
This conjecture is a baby form of a much more sophisticated polynomiality
conjecture\oremind{\cite[Conj.~1.2]{gj}} of the combinatorialists 
Goulden and Jackson, which
can be interpreted as a ``structure theorem'' for Hurwitz numbers.
The number $3g-3+n$ is extremely suggestive:  it is the dimension
of $\cmbar_{g,n}$, and there is indeed a genus $g$ curve with $n$ marked
points in the picture:  the cover has genus $g$, and has $n$ pre-images
over $\infty$.

Equation \eqref{gjeq} is explained by the following truly remarkable formula:

\noindent
{\bf Theorem (The ``ELSV formula'', 
Ekedahl, Lando, Shapiro, and Vainshtein).}\oremind{\cite{elsv1, elsv2}}
\lremind{elsvf}
\begin{equation}
\label{elsvf}
P(\al_1, \dots, \al_n) = \int_{\cmbar_{g,n}} \frac { 1 - \la_1 + \cdots + (-1)^g \la_g } { ( 1 - \al_1 \psi_1 ) \cdots (1 - \al_n \psi_n) }.
\end{equation}
This equation should be interpreted as follows:  formally invert the 
denominator, expand everything out, and keep only the
terms of codimension $\dim \cmbar_{g,n} = 3g-3+n$.  Hence for
fixed $g$ and $n$, 
$$
P(\al_1, \dots, \al_n) = \sum_{ \atp {k_1 + \dots + k_n + j = 3g-3+n}{0 \leq j \leq g}}
\left(  
(-1)^g 
\int_{\cmbar_{g,n}} 
\psi_1^{k_1} \psi_2^{k_2} \cdots \psi_n^{k_n} \la_j
\right)  \al_1^{k_1} \al_2^{k_2} \cdots \al_n^{k_n},
$$
immediately identifying the mysterious polynomial $P$.
Note in particular that the highest-degree portion of $P$
is a generating function for top intersections of $\psi$-classes
(and is, upon replacement of $\al_i$ by $t_i$, equ. \eqref{Egn}).
This gives a straightforward way of computing these
abstractly-defined numbers:  it is not hard to recover the coefficient
of a polynomial $P$ of known degrees by computing enough values,
and Hurwitz numbers are easy to compute.  
Hence, remarkably, the combinatorics of the symmetric group
lead inexorably to the tautological ring of the moduli 
space of curves!

There are many other consequences; see \cite{elsv2, gjvk} for  surveys.
For example, (i) Goulden and Jackson's full polynomiality conjecture
can be proved, (ii) conjectured recursions for Hurwitz numbers
can be proved, and new ones found, 
(iii) after a non-trivial
change of variables, the generating function $H$ becomes
(a mild generalization of) Witten's free energy of a point, and
(iv) this combinatorial description of intersections of
$\psi$-classes leads to Okounkov and Pandharipande's 
proof of Witten's conjecture via asymptotic combinatorics.

We remark in passing that the ELSV formula is an equality not just of
numbers, but in Chow (in $R_0(\cmbar_{g,n}):= R^{3g-3+n}(\cmbar_{g,n})$).  Interpreted properly,
this yields:

\noindent {\bf Theorem \cite{socle}.}
$R_0(\cmbar_{g,n}) \cong \Q$.

In other words, any two tautological 0-cycles on $\cmbar_{g,n}$
are commensurate.  This is trivial in cohomology, but
far from obvious in the Chow ring (where $A_0$, as described
earlier, can be huge).    Also, Witten's conjecture can in some sense
be interpreted as a recursion not just of numbers, but in Chow.

The ELSV formula can be proved in Gromov-Witten
theory;\oremind{\cite{gvelsv};} this interpretation leads to
connections with the second, related, source of combinatorial
structure in the tautological ring, the boundary strata.

\bigheading{The tautological ring in arbitrary genus:  cycles
of arbitrary dimension}

In some sense we have a handle on 0-cycles.  By Faber's 
algorithms,\oremind{\cite{falg, faberprog},} 
we also know how to intersect any two tautological
cycles.  In cohomology, this would be enough for us to determine the
full structure of the ring, thanks to \Poincare duality.  It has been
conjectured that ``\Poincare duality'' also holds for $\cmbar_{g,n}$:

\noindent
{\bf \Poincare duality conjecture (or speculation) for 
stable curves (Hain-Looijenga).}\oremind{ \cite[p.~108]{hl}}
$R^*(\cmbar_{g,n})$ satisfies \Poincare duality.

In other words, if $d=\dim \cmbar_{g,n}= 3g-3+n$:
\begin{enumerate}
\item[(I)]  $R^{i}(\cmbar_{g,n})=0$ for $i>d$, which is obvious
for dimensional reasons;
\item[(II)] $R^{d}(\cmbar_{g,n}) \cong \Q$;
\item[(III)] the natural pairing $R^i(\cmbar_{g,n}) \times R^{d-i}(\cmbar_{g,n}) \rightarrow R^d(\cmbar_{g,n}) \cong \Q$ is perfect.
\end{enumerate}
For convenience, we will call such a ring a {\em dimension $d$
  \Poincare duality ring}, and (I) and (II) the {\em socle} conditions.
(This terminology is from the theory of Gorenstein rings.)

There is no {\em a priori} reason to expect the tautological ring to
satisfy \Poincare duality.  (For example, the Chow ring certainly
doesn't, as the hugeness of $A_0(\cmbar_{1,11})$ shows.)  At this
point, there is a great deal of low-genus evidence for this
conjecture; however, the only genus-free evidence to date
is (II), described earlier.

This conjecture was motivated by an earlier conjecture by Faber.

\noindent
{\bf Faber's conjecture (for smooth curves) \cite[Conj.~1]{fconj}.}
\begin{enumerate}
\item[(a)] $R^*(\cm_g)$ is a dimension $g-2$ \Poincare duality ring.
\item[(b)] The $[g/3]$ classes $\ka_1$, \dots, $\ka_{[g/3]}$ generate
the ring, with
no relations in degrees $[g/3]$.
\item[(c)] There exist explicit formulas for the proportionalities in 
degree $g-2$, which may be given as follows.  
$$
\psi_1^{d_1+1} \psi_2^{d_2+1} \cdots \psi_k^{d_k+1} = 
\frac { (2g-3+k)! (2g-1)!!} { (2g-1)! \prod_{j=1}^k (2 d_j+1)!!} \ka_{g-2}
= 
\sum_{\si \in S_k} \ka_{\si}.$$
\end{enumerate}
Here $(2a-1)!!$ is a shorthand for $1 \cdot 3 \cdots \cdot (2a-1)$;
$\ka_{\si} = \ka_{|\al_1|} \ka_{|\al_2|} \cdots
\ka_{|\al_{\nu(\si)}|}$ for a decomposition $\si = \al_1 \al_2 \cdots
\al_{\nu(\si)}$ of the permutation $\si$ into disjoint cycles; and
$|\al|$ is the sum of the elements in the cycle $\al$, where $S_k$
acts on the $k$-tuples with entries $d_1$, \dots, $d_k$.  (Faber
conjectured furthermore that $R^*(\cm_g)$ ``behaves like'' the
algebraic cohomology ring of a smooth projective manifold of dimension
$g-2$; the additional properties required will not be 
discussed here.)  Faber verified the conjecture for genus up to 15
--- a highly nontrivial calculation on difficult-to-handle spaces.
Part (c) in particular highlights the combinatorial structure
of the tautological ring.
Significant progress has been made on this conjecture, in the form
of three theorems.


\noindent
{\bf Theorem.}  
\begin{enumerate}
\item[(a)] (Looijenga) $R^i(\cm_g) = 0$ for $i>g-2$, 
$R^{g-2}(\cm_g) = \Q$.\oremind{\cite{lthm}}
  (The latter statement uses a calculation of Faber \cite[Thm.~2]{fconj}.)
\item[(b)] (Morita, Ionel) Part (b) of Faber's conjecture is true in
cohomology.\oremind{\cite{ifaber}} \oremind{Morita's
proof is announced in \verb+http://arxiv.org/PS_cache/math/pdf/991/9911258.pdf+
p. 387, available at math.GT/9911258.}
Ionel's proof should extend to the Chow ring without difficulty.
\item[(c)] (Getzler, Pandharipande, Givental)
Part (c) of Faber's conjecture is true. 
\end{enumerate}
(A word of explanation of (c): 
\oremind{In \cite[Sect.~1.4]{fabvir}, it was shown}
Getzler and Pandharipande showed that the
Virasoro conjecture for $\proj^2$ implies this part of Faber's
conjecture.  As described earlier, Givental has since proved the
Virasoro conjecture for all projective spaces.  This proof for (c) is
quite round-about and complicated; it would be good to have a better
proof, and this seems within reach.)

Hence all that remains is the ``perfect pairing'' part of the conjecture;
while there is some prospect for proving cases of this open problem
(for example, the case $i=1$ or possibly $2$), there are
no current hopes for approaching the general problem.

Faber and Pandharipande have proposed two other generalizations of
Faber's conjecture, to curves with ``rational tails'' (curves whose
dual graph contains a vertex of genus $g$), and curves of ``compact
type'' (curves whose dual graph has no loops).  For example, the curve
of Figure~\ref{gn63} is of both types, but the curve of
Figure~\ref{gnbar63} is neither.

%
%\begin{figure}
%\caption{The locus of stable curves not of compact type
%is the closure of this stratum \lremind{Delta0}}
%\label{Delta0}
%\end{figure}
%
%\noindent
%{\bf   \Poincare duality conjecture 
%for 
%compact-type curves
%\cite[Conj.?]{fpconj}.}  
%$\R^*(\cm_{g,n}^{c})$ is a dimension $2g-3+n$ \Poincare duality ring.

\mediumheading{Boundary strata and vanishing of tautological classes}

These conjectures are closely related to the combinatorial structure
of the tautological ring involving boundary strata, by way of the following
result.

\noindent
{\bf Tautological vanishing theorem \cite{theoremstar}.}
Any tautological class of codimension $i$ 
is trivial away from strata satisfying
$$
{\text {\# genus 0 vertices  }} \geq i-g+1.
$$

For example, the ``vanishing'' part of Looijenga's theorem (a) above
(that $R^i(\cm_g)=0$ for $i>g-2$) is a consequence as follows.  If $i
\geq g$, then any codimension $i$ class vanishes on the open set
corresponding to curves with no genus 0 vertices, and in paricular on
$\cm_g$.  If $i=g-1$, then the theorem does not directly apply.
However, from the definition of the tautological ring, we see that any
tautological class on $\cm_g$ is pushed forward from $\cm_{g,1}$;
hence a codimension $g-1$ class on $\cm_g$ is pushed forward from a
codimension $g$ class on $\cm_{g,1}$, which vanishes by the same
argument.  The reader may like to extend this argument to address the
case of ``curves with rational tails'' described above (part of 
the ``rational tail''
conjecture of Faber and Pandharipande).

As another example, the reader may enjoy proving the analog of part
(a) of Faber's conjecture for the moduli space of ``compact type''
curves $\cm^c_{g,n} \subset \cmbar_{g,n}$ by showing the following
sequence of facts.  If $i>2g-3+n$, there is no compact type (i.e.
no-loop) stable dual graph of genus $g$ with $n$ points.  If
$i=2g-3+n$, then there are: they are strata containing only trivalent
genus 0 vertices, $g$ genus 1 ``leaves'', and $n$
``half-edges''.\oremind{Lie!  $\cmbar_{2,0}$} These strata are of
codimension precisely $2g-3+n$, so $R^{2g-3+n}(\cm_{g,n}^c)$ is
generated by these strata.  Any two of these strata are related by
cross-ratio relations.  It is a straightforward geometric fact that
any one of these strata is non-trivial in cohomology (and hence the
Chow ring), so we have shown that $R^i(\cmbar_{g,n}^c)$ vanishes for
$i>2g-3+n$, and is one-dimensional for $i=2g-3+n$.

The analogous theorem for stable curves, the fact 
$R_0(\cmbar_{g,n}) \cong \Q$ described earlier,
follows similarly; there are many extensions along these lines.

We conclude with another remarkable consequence,  which indeed
partially motivated the vanishing theorem.  As described earlier,
in genus 1, $\psi_i$ is rationally equivalent to a sum of boundary strata.  In
genus 2, Mumford and Getzler  \oremind{\cite{gconj}}
proved that on $\cmbar_{2,n}$, $\psi_i
\psi_j$ is also supported on the boundary (i.e. is trivial on
$\cm_{2,n}$).  Based on this, Getzler conjectured the
following statement, which  was recently proved by Ionel.  (Ionel
states the result in cohomology, but her proof should carry through in the
Chow ring.)

\noindent
{\bf Ionel's theorem (formerly Getzler's 
conjecture).}\oremind{\cite{igetz}}\oremind{\cite[footnote~1]{gconj}}
If $g>0$, all degree $g$ monomials in $\psi_1$, \dots, $\psi_n$ vanish
on $\cm_{g,n}$.  

\oremind{Note to self:  Tautological boundary.  Do I want to mention
it?}

\bigheading{Conclusion} 

Since Kontsevich's proof of Witten's conjecture, there
has been a great deal of progress in understanding the geometry and
topology of the moduli space of curves, and in particular its
tautological ring.  The objects in question are quite classical, and
the fundamental perspective behind the proofs of the results described
above, that of understanding curves as covers of the projective line,
is very much so.  However, the recent progress is highly motivated by
work in other fields (such as string theory, combinatorics,
representation theory, topology, symplectic geometry, integrable
systems and more), and some of the techniques are quite modern
(such as those of Gromov-Witten theory).  As new ideas continue
to flow into algebraic geometry from these neighboring fields, 
there should be even more exciting developments in the near future.

\oremind{* Browse through Gibney-Keel-Morrison (printed out)?    Look over AC1 (* grab online)?.Quick at department:  get good ref for Hurwitz.

Length:  108 words/8 lines is 13.5 words per line.  35 lines per page,
 18.5 pages, for a total of roughly 8700 words.

People I want to send it to:
{\em First round (done):}  Carel, Rahul, Eduard, Getzler, Ian M, Tom, Boas, Pasha, Bini.
{\em Second round:}   Harris, Cornalba, Shapiro, Arbarello, Elly.
Also Yasha, Ralph,
Alex Gamburd.  
} 

} % end of parskip; it started just before the introduction

\begin{thebibliography}{[MirSym]}
\oremind{\bibitem[AC1]{acharer} E. Arbarello and M. Cornalba, {\em The
Picard groups of the moduli spaces of curves}, Topology {\bf 26} (1987), 153--171.
\lremind{acharer} \cited}
\oremind{\bibitem[AC2]{ackont}  E. Arbarello and M. Cornalba, {\em Combinatorial and
algebro-geometric cohomology classes on the moduli spaces of curves},
J. Alg. Geom. {\bf 5} (1996), 705--749. \cited
\lremind{ackont} \oremind{Read through.  The combinatorial classes
are conjectured to be tautological, but there was no place to discuss this!}}
\bibitem[ACGH]{acgh} E. Arbarello, M. Cornalba, P. Griffiths,
and J. Harris, {\em Geometry of Algebraic Curves I}, 
Grundlehren der math. Wiss., {\bf 267}, Springer, Berlin, 1985. \lremind{acgh}
\oremind{Keep!}
\oremind{\bibitem[B]{pasha} P. Belorousski, 
{\em Chow rings of moduli spaces of pointed elliptic curves},
 Ph.D. thesis,
University of Chicago.  \lremind{pasha} \cited}
\oremind{\bibitem[BP]{bp} P. Belorousski and R. Pandharipande, {\em A
descendent relation in genus 2}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) {\bf
29} (2000), no. 1, 171--191.   \oremind{Nothing to
extract.  I've also looked at the mathreview}  \lremind{bp}  \cited}
\oremind{\bibitem[CK]{ck} D. Cox and S. Katz, {\em Mirror Symmetry and 
Algebraic Geometry}, Mathematical surveys and Monographs {\bf 68}, Amer.
Math. Soc., Providence, RI, 1999.
\cited \lremind{ck}}
\oremind{\bibitem[D]{deligne} P. Deligne, {\em Formes modulaires et
repr\'{e}sentations $l$-adiques}, S\'{e}m. Bourbaki, Expos\'{e}
{\bf 355}, (F\'{e}vrier 1969), 139--172. \cited \lremind{deligne}}
\bibitem[DM]{dm} P. Deligne and D. Mumford, {\em The
irreducibility of the space of curves of given genus},
 Publ. Math. IHES 36 (1969), 75--109.  \lremind{dm} \cited
\oremind{Keep}
%\bibitem[E]{e} D. Edidin, his little note on the moduli space of curves.
\oremind{\bibitem[EH]{eh} D. Eisenbud and J. Harris, {\em The Kodaira
dimension of the moduli space of curves of genus $\geq 23$},
Invent. Math. {\bf 90} (1987), no. 2, 359--387.  \lremind{eh}
\cited }
\oremind{\bibitem[ELSV1]{elsv1} T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein,
{\em On Hurwitz numbers and Hodge integrals}, C. R. Acad. Sci. Paris S\'{e}r. I Math. {\bf 328} (1999), 1175--1180. \lremind{elsv1} \cited}
\bibitem[ELSV]{elsv2} T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein,
{\em Hurwitz numbers and intersections on moduli spaces of curves},
Invent. Math. {\bf 146} (2001), 297--327.
\lremind{elsv2} \cited  \oremind{Currently keeping.}  \oremind{Everything 
is fully extracted from both ELSV papers.}
\oremind{\bibitem[Fa1]{fm3} C. Faber, {\em Chow rings of mduli spaces of curves I:
The Chow ring of $\cmbar_3$}, Ann. of Math. {\bf 132} (1990), 331--419.
\cited  \lremind{fm3}  \oremind{Mathscinet:  describes $\cmbar_3$ completely.}
\bibitem[Fa2]{fm4} C. Faber, {\em Chow rings of mduli spaces of curves II:
Some results on the Chow ring of $\cmbar_4$}, Ann. of Math. {\bf 132} (1990), 421--449.  \cited  \oremind{Take a look at this, to make sure that
he describes the full Chow ring of $\cmbar_4$.  Mathscinet
says no:  $A^*(\cm_4) = \Q[\la]/(\la^3)$, and $\dim A^2(\cmbar4)=13$.}}
\bibitem[F1]{fconj} C. Faber, {\em A conjectural description
of the tautological ring of the moduli space of curves}, in 
{\em Moduli of Curves and Abelian Varieties}, 109--129, Aspects. Math.,
{\bf E33}, Vieweg, Braunschweig, 1999.  \lremind{fconj} \cited
\oremind{I've extracted everything from this article.}
\oremind{\bibitem[Fa4]{falg} C. Faber, {\em Algorithms for computing
intersection numbers on moduli spaces of curves,
with an application to the class of the locus of
Jacobians}, in {\em New Trends in Algebraic Geometry (Warwick, 1996)},
093--109, London Math. Soc., Lecture Note Ser. {\bf 264}, Cambridge Univ.
Press, Cambridge, 1999. \cited \lremind{falg}
\oremind{I've extracted all useful things from this article.}}
\bibitem[F2]{faberprog} C. Faber, {\sc Maple} program
for computing Hodge integrals, personal communication.  Available
at \verb+http://math.stanford.edu/~vakil/programs/+.  \cited 
\lremind{faberprog}
\bibitem[FL]{fl} C. Faber and E. Looijenga, {\em Remarks on moduli of curves},
in {\em Moduli of curves and abelian varieties}, 23--45, Aspects.
Math. {\bf E33}, Vieweg, Braunschweig, 1999.  \cited \lremind{fl}
\oremind{Extracted (both article and math review).}
\oremind{Keep!}
%\bibitem[FaP1]{lambdag} C. Faber and R. Pandharipande, {\em Hodge integrals, partition
%matrices, and the $\la_g$ conjecture}, math.AG/9908052.  
% \oremind{This is where the $\la_g$ conjecture is proved.  Look over this.}
%\lremind{lambdag}
\oremind{\bibitem[FaP]{fpconj} C. Faber and Pandharipande, {\em Logarithmic series
and Hodge integrals in the tautological ring}, Michigan Math.
 J. {\bf 48} (2000), 215--252.
 \lremind{fpconj}  \oremind{Info extracted.}}
%\bibitem[FaP3]{fp3} C. Faber and R. Pandharipande, {\em Hodge
%integrals and Gromov-Witten theory}, Invent. Math. {\bf 139} (2000), no. 1,
%173--199.  {\bf Not yet cited --- take a look at it.} \lremind{fp3}
% NOT RELEVANT
%\bibitem[Fn]{fantechi} B. Fantechi, Stacks for Everybody
\oremind{\bibitem[Fu1]{it} W. Fulton, {\em Intersection Theory}, 2nd ed., Ergebnisse
der Mathematik und ihrer Grenzgebiete {\bf 2}, Springer-Verlag, Berlin, 1998.
\cited \lremind{it}}
\bibitem[Fu]{tableaux} W. Fulton, {\em Young Tableaux with Applications
to Representation Theory and Geometry}, Cambridge U.P., New York, 1997.
 \cited \lremind{tableaux}
% \bibitem[Fetal]{fetal} Fulton et al., their stacks book.
%Will I refer to this?  \lremind{fetal}
% K. Behrend, D. Edidin, W. Fulton, A. Kresch, B. Fantechi, 
% L. G\"{o}ttsche, ... who else?
\oremind{\bibitem[Ge1]{m14} E. Getzler, {\em Intersection theory on
$\cmbar_{1,4}$ and elliptic Gromov-Witten invariants}, J. Amer. Math.
Soc. {\bf 10} (1997), no. 4, 973--998.  
 \oremind{Extracted, from mathreview too.}
 \lremind{m14}  \cited}
\oremind{\bibitem[Ge2]{gconj} E. Getzler, {\em Topological recursion relations in 
genus 2}, in {\em Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)},
73--106, World Sci. Publishing, River Edge, NJ, 1998.  
\lremind{gconj} \cited }
\oremind{Not in library; nothing in teresting in mathscinet.}
%\bibitem[Ge4]{m1n}  E. Getzler, {\em The \Poincare polynomial of 
%$\cmbar_{1,n}$}, preprint, 1996.  \lremind{m1n} \cited 
%\oremind{This was cited by 
%Pikaart; take a look for $\cmbar_{1,11}$ fact!}
% It hasn't appeared!
\oremind{\bibitem[Ge3]{ezranew} E. Getzler, manuscript in preparation.
\cited \lremind{ezranew}}
\oremind{\bibitem[GeP]{fabvir} E. Getzler and R. Pandharipande,
{\em Virasoro constraints and the Chern classes of the Hodge bundle},
Nuclear Phys. B {\bf 530} (1998), 701--714.
\cited \lremind{fabvir} \oremind{Info extracted.}}
\oremind{\bibitem[Gi]{givental} A. Givental, manuscript in preparation.
\cited \lremind{givental}}
\oremind{\bibitem[GJ]{gj} I. P. Goulden and D. M. Jackson, {\em The
number of ramified coverings of the sphere by the double
torus, and a general form for higher genera}, J. Combin. Theory
A {\bf 88} (1999) 259--275. \cited\lremind{gj}}
\oremind{\bibitem[GJVn]{gjvn} 
I. P. Goulden, D. M. Jackson, and A. Vainshtein, {\em 
The number of ramified coverings of the sphere by the torus and surfaces of
higher genera}, Ann. Combin. {\bf 4} (2000), 27--46.  \lremind{gjvn}}
\bibitem[GJV]{gjvk} I. P. Goulden, D. M. Jackson, and R. Vakil,
{\em The Gromov-Witten potential of a point, Hurwitz numbers,
and Hodge integrals}, Proc. London Math. Soc. (3), {\bf 83} (2001), no. 3, 563--581.  \lremind{gjvk} \oremind {Extracted.}  \oremind{Currently keeping.}
\oremind{\bibitem[GrP]{nontaut}  T. Graber and R. Pandharipande, 
{\em A non-tautological algebraic class on $\overline{M}_{2,22}$},
math.AG/0104057.  \cited
\lremind{nontaut} \oremind{Info extracted.}}
\oremind{\bibitem[GrV1]{gvelsv} T. Graber and R. Vakil, {\em Hodge
integrals and Hurwitz numbers via virtual localization},
Compositio Math., to appear. \lremind{gvelsv}}
\bibitem[GrV1]{socle} T. Graber and R. Vakil, {\em On the tautological
ring of $\cmbar_{g,n}$}, in {\em Proceedings of the Seventh
G\"{o}kova Geometry-Topology Conference}, 
International Press, 2000. \lremind{socle}
\bibitem[GrV2]{theoremstar} T. Graber and R. Vakil, {\em Relative
virtual localization, and vanishing
of tautological classes on moduli spaces of curves}, manuscript in preparation. \lremind{theoremstar}
\bibitem[HL]{hl} R. Hain and E. Looijenga, {\em Mapping class groups and moduli spaces of curves}, Proc. Sympos. Pure Math. {\bf 62}, pp. 97--142,
Amer. Math. Soc., Providence, RI, 1997.
\lremind{hl} \cited  \oremind{Keep!}
\oremind{Quickly skimmed.}
\oremind{\bibitem[Har1]{harer1} J. Harer, {\em The second homology group
of the mapping class group of an orientable surface}, Invent.
Math. {\bf 72} (1983), 221--239. \cited \lremind{harer1}
\bibitem[Har2]{harer2} J. Harer, {\em The cohomology of the
moduli space of curves}, in {\em Theory of moduli}, 138--221,
LNM 1337, Springer, Berlin-New York 1988. \cited \lremind{harer2}}
\oremind{\bibitem[HarZ]{hz}  J. Harer and D. Zagier, {\em The Euler
characteristic of the moduli space of curves}, Invent. Math. {\bf 85} (1986),
no. 3, 457--485.  \cited
\lremind{hz} \oremind{Looked through mathreview.}}
\bibitem[HM]{hmo}
 J. Harris and I. Morrison, {\em Moduli 
of Curves}, Graduate Texts in Mathematics {\bf 187}, Springer-Verlag, New 
York, 1998. \lremind{hmo} \cited \oremind{Keep!}
%\bibitem[HMu]{hmum} J. Harris and D. Mumford, {\em On the
%Kodaira dimension of the moduli space of curves}, Invent. Math. {\bf 67}
%(1982), 23--86.  \lremind{hmum}
%\oremind{Should I put Eisenbud there too?  
% Likely so.  I've printed this out.  I've actually deleted this ref!
% Note that Joe also proved another case.}  \cited  
\oremind{\bibitem[Hu]{hurwitz}  A. Hurwitz, {\em \"{U}ber Riemann'sche
Fl\"{a}chen mit gegeben Verzweigungspunkten}, Math. Ann. {\bf 39} (1891),
1--60.
\lremind{hurwitz} \cited}
\oremind{\bibitem[Io1]{igetz} E. Ionel, {\em Topological recursive relations in
    $H^{2g}(M_{g,n})$}, Invent. Math. {\bf 148} (2002), no. 3, 627--658.
  \lremind{igetz} \cited }
\oremind{\bibitem[Io2]{ifaber} E. Ionel, lecture, University of Michigan, March 2002.
\oremind{Vet this with her.} \cited  \lremind{ifaber}}
\oremind{\bibitem[ItZ]{iz} C. Itzykson and J.-B. Zuber, 
{\em Combinatorics of the modular group II:  The Kontsevich integrals},
Internat. J. Modern Phys. A {\bf 7} (1992) no. 23, 5661--5705.
\lremind{iz} \cited}
\oremind{\bibitem[Iz]{izadi} E. Izadi, {\em The Chow ring of the moduli space
of curves of genus 5}, in {\em The Moduli Space of Curves}, Progress in Math. {\bf 129},
Birkh\"{a}user, Boston, 1995. \cited \lremind{izadi}
\remind{ See HL p. 107 for completion of Izadi's calculation.}}
\oremind{\bibitem[Ke]{keel} S. Keel, {\em Intersection theory of moduli space
of stable $n$-pointed curves of genus zero}, Trans. Amer. Math. Soc.
{\bf 330} (1992), no. 2, 545--574. \lremind{keel} \cited}
\oremind{\bibitem[Ko]{kont} M. Kontsevich, {\em Intersection theory on 
the moduli space of curves and the matrix Airy function}, Comm. Math. Phys. {\bf 147} (1992), 1--23. \cited}
\bibitem[L]{lkont} E. Looijenga, {\em Intersection theory
on Deligne-Mumford compactifications (after Witten and Kontsevich)},
S\'{e}minaire Bourbaki, Vol. 1992/93, Ast\'{e}risque No. 216 (1993), Exp. No. 768, 4,
187--212.  \lremind{lkont} \cited
\oremind{\bibitem[Lo]{lthm}
E. Looijenga, {\em On the tautological ring of $\cm_g$}, Invent. Math. {\bf 121} (1995), no. 2, 411--419.
\lremind{lthm} \cited}
%\bibitem[LM]{lm} Laumon and Moret-Bailly
\oremind{\bibitem[MaW]{mw} I. Madsen and M. Weiss, manuscript in preparation. 
\cited \oremind{Perhaps excise this ref.}}
\oremind{\bibitem[Mi]{miller} E. Miller, {\em The homology of the mapping class group}, J. Diff. Geom,
{\bf 24} (1986), 1--14.   \cited \lremind{miller}}
\bibitem[MirSym]{mirror} K. Hori, S. Katz, A. Klemm, R. Pandharipande,
  R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, {\em Gromov-Witten
    Theory and Mirror Symmetry}, Clay Math. Inst., Amer.  Math. Soc., 2002, to appear.
  \cited \oremind{Extracted.} \oremind{Keep?}
\oremind{\bibitem[Mo]{morita} S. Morita, {\em Characteristic classes of surface 
bundles}, Invent. Math. {\bf 90} (1987), 551--577. \cited \lremind{morita}}
\bibitem[M]{m} D. Mumford, {\em Toward an enumerative geometry of the moduli
space of curves}, in {\em Arithmetic and Geometry}, Vol. II, M. Artin
and J. Tate ed., 271--328, Prog.
Math. {\bf 36}, Birk. Boston, Boston, MA, 1983.  \lremind{m}  \oremind{Looked
over.} \oremind{Keep!}
\oremind{\bibitem[OP1]{op} A. Okounkov and R. Pandharipande, {\em Gromov-Witten theory,
Hurwitz numbers, and matrix models, I}, math.AG/0101147.
\cited}
\oremind{\bibitem[OP2]{op2} A. Okounkov and R. Pandharipande, {\em Gromov-Witten 
theory, Hurwitz theory, and completed cycles}, math.AG/0204305.
\cited \lremind{op2}
\bibitem[OP3]{op3} A. Okounkov and R. Pandharipande,
{\em The equivariant Gromov-Witten theory of $\proj^1$}, math.AG/0204305. 
\cited \lremind{op3}
\bibitem[OP4]{op4} A. Okounkov and R. Pandharipande,
{\em Virasoro constraints and descendents of odd classes in
the Gromov-Witten theory of target curves}, in preparation.
\cited  \lremind{op4}}
\bibitem[P]{icm} R. Pandharipande, {\em Three questions
in Gromov-Witten theory}, in {\em Proceedings of the 2002 ICM}
Vol. II, 503--512. \lremind{icm}
% NOT cited.
%\bibitem[Pi1]{p1} M. Pikaart, {\em An orbifold partition of $\cmbar_g^n$},
%in {\em The Moduli Space of Curves}, Progress in Math. {\bf 129},
%Birkh\"{a}user, Boston, 1995. \cited \lremind{p1}
% This was included in \cite{p}, so I've excluded it here!
\oremind{\bibitem[Pi]{p} M. Pikaart, {\em Moduli spaces of curves:  
Stable cohomology and Galois covers}, Ph.D. thesis, Universiteit Utrecht,
1997.  \oremind{Look over this!} \cited \lremind{p}}
\oremind{\bibitem[R]{riemann} B. Riemann, {\em Theorie der Abel'schen Funktionen}, J.
reine angew. Math. {\bf 54} (1857), 115--155.  \cited
 This was from Pikaart p. 4 and 60.}
\bibitem[V]{quartic} R. Vakil, {\em Characteristic numbers of quartic
plane curves}, Can. J. Math. {\bf 51} (1999), no. 5, 1089--1120.
\cited \lremind{quartic}
%\bibitem[Vi]{v} A. Vistoli, his appendix.
\oremind{\bibitem[W]{w}  E. Witten, {\em Two dimensional gravity and
intersection theory on moduli space}, Surveys in Diff. Geom. {\bf 1}
(1991), 243--310.  \cited
\lremind{w}}
\end{thebibliography}
\end{document}
