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\author{Dan Abramovich}
\date{February 27, 2012}

\title{Logarithmic Stable Maps}

\begin{document}
\maketitle

Plan for the talk:
\begin{enumerate}
\item{Log structures--what are they?  when are they present?}
\item{Log smooth curves}
\item{Log stable maps}
\item{An example}
\item{What does ``log smoothness'' really mean?}
\end{enumerate}

Logarithmic smoothness allows us to treat certain singular objects
as if they were smooth.  

Jun Li is one pioneer in this; he has given applications to Gromow-Witten theory.

Kazuya Kato first introduced log structures, attributing the notion to Fontaine
and Illusie.  The objects are pairs $X=(\underline{X},\cM)$ and 
$\cM\stackrel{\alpha}{\lra} \cO_X$
where
\begin{itemize}
\item{$\underline{X}$ is a scheme}
\item{$\cM$ is a sheaf of monoids}
\item{$\alpha:\cM \ra \cO_X$ is a monoid homomorphisms, where $\cO_X$
is a monoid under multiplication}
\item{$\alpha^{-1}\cO_X^{\times} \stackrel{\sim}{\lra} \cO_X^{\times}$
an isomorphism}
\end{itemize}
(The first three conditions alone are the data of a pre-logarithmic structure.)

A morphism $(X,\cM_X) \stackrel{f}{\lra} (Y,\cM_Y)$ consists of a 
morphism $\underline{f}:\underline{X} \ra \underline{Y}$ of the underlying schemes
with a commutative diagram
$$\begin{array}{ccc}
{\underline f}^{-1} \cM_Y & \stackrel{f^{\flat}}{\ra} & \cM_X \\
\quad  \downarrow {\scriptstyle \alpha_Y}  &   & {\scriptstyle \alpha_X} \downarrow \quad \\
{\underline f}^{-1} \cO & \stackrel{f^{\sharp}}{\ra} & \cO_X 
\end{array}
$$

\noindent {\bf Basic Example:} ${\underline D} \subset {\underline X}$ a normal crossings divisor such that
$$ \cM_{{\underline D} \subset {\underline X}}(V) =\{f\in \cO_X(V):
f|_{V-D} \in \cO_X(V)^{\times} \},
$$
e.g., $X=\bA^n$, $D=Z(x_1\cdots x_k)$, $\cM$ generated by $\cO_X^{\times}$
and $x_1,\ldots,x_k$.


\noindent {\bf In this case:} $\cM$ can be thought of as ``partial 
log data for some functions on $X$''. 

\noindent {\bf In this case:} $\Omega^1_X(\log D)$ locally free \\
This remains true when $X$ is toric and $D$ is the complement of the
torus $T$.  Hence these are `log-smooth'.  

How about $\bA^1$ with boundary $0$?  How do we restrict the log structure
to the point $0$?  

\noindent {\bf Important case:}  $P$ a sharp monoid (i.e., $P^{\times}=\{0\}$),
${\underline X}=\Spec(\bC)$
$$\begin{array}{rcl}
\cM=P \oplus \bC & \stackrel{\alpha}{\lra} & \bC \\
(p,z) & \mapsto & \begin{cases}  z & p=0 \\
                                 0 & \text{otherwise}
		\end{cases} \\
(p,z) & \mapsto & 0^p\cdot z
\end{array}
$$
Notation: $\Spec(P \ra \bC)$; this is a log point with monoid $P$. \\
When $P=\bN$, we get the standard log point, or `pointed point' in French

\noindent {\bf Good cases:} \\
$P$ `integral', i.e., $P \hookrightarrow P^{\text{gp}}$  \\
$P$ `saturated', e.g., $\{0,3,4,5,\ldots \}$ is not saturated

One more example
$$\begin{array}{rcl}
\bA^2 & \ra & \bA^1 \\
  t & \mapsto & xy
\end{array}$$
where $xy$ and $t$ give the log structures.  Note that 
$$\Omega^1_{\bA^1}(\log \{0\}) \ra \Omega^1_{\bA^2}(\log D)$$
has locally-free cokernel, hence the morphism is `log smooth'.

K.~Kato introduced these for hard arithmetic problems, so geometers
were slow to adopt them.

F.~Kato (a student of Kenji Ueno) considered $C\ra S$, fine and saturated \\
$\underline C$ a curve and ${\underline S}=\Spec(\bC)$. \\
Assume that $C\ra S$ is log smooth, interpreted via the infinitesimal
lifting property.  \\
Points of $C$ can be one of three different types:
\begin{itemize}
\item{general point:  $C=\underline{C} \times_{\underline S} S$
and $\underline{C}$ smooth}
\item{special smooth point:  $C=S\times \Spec(\bN\ra \bC[x])$
where $1\mapsto x$, \'etale locally}
\item{singular point: necessarily nodes, given locally by the log structure
$$C=\Spec(P[m_x,m_y]/(m_x+m_y=p) \ra \bC[x,y]/\left<xy\right>)$$
for some $p \in P$.
}
\end{itemize}
Category:  objects are log smooth curves over log schemes $C\ra S$ and
arrows are diagrams
$$\begin{array}{rcl}
C_1 & \ra & C_2 \\
\downarrow &  & \downarrow \\
S_1 & \ra & S_2 
\end{array}
$$
This naturally fibered over the category of log schemes.

\begin{theo}[F.~Kato]
Consider $(\oM_{g,n},\cM)$ where $\cM$ arises from the modular boundary.
This represents the category of stable log smooth curves with $n$
markings.
\end{theo}

What about mappings?  
$$\begin{array}{rcl}
C & \ra & X \\
\downarrow &  & \\
S  &  & 
\end{array}
$$
where $C$ is nodal and $C\ra X$ satisfies a stability condition.


\begin{theo}[Q.Chen, Gross-Siebert, Abramovich-Chen]
$X$ `nice': fine + saturated, .... (missing technical assumptions
on global generation of monoid on $X$)

Fix $g,n,\beta \in H_2(\underline{X},\bZ)$.  Then 
$\oM_{g,n}(X,\beta)$ is represented by a log DM stack
$$(\underline{\oM_{g,n}}(X,\beta), \cM)$$
that is finite over $\oM_{g,n}(\underline{X},\beta)$.
\end{theo}

Furthermore, if $X$ is log smooth then this log DM stack carries
the virtual fundamental classes needed for Gromov-Witten analysis.

\noindent {\bf Example}:  \\
Let $X$ be a cubic surface.  Fix two points and consider the number
of rational hyperplane sections containing these; there are twelve
by an elementary argument.

Suppose $X$ specializes to a union of three planes meeting transversally;
resolve singularities in the total space, which entails blowing up 
three points on each component of the conductor, so altogether there
are nine
exceptional $\bP^1$'s.  

How do we see `twelve' rational hyperplane sections after
specialization?  Suppose the two points specialize
to distinct components.  We get one distinguished hyperplane section
for each exceptional $\bP^1$, for a total of nine.  What about the missing
three curves?  In fact, there is just one more curve with multiplicity
three.  The domain of the associated stable map will be a tree of four $\bP^1$'s,
three mapping
to lines on each of the components of the degenerate surfaces, and the remaining
one contracted to the point $p$ where the three planes meet.  The image
is the hyperplane spanned by $p$ and the two fixed points.    

Kov\'acs:  Is there an interpretation of this in terms of 
deformations of $\bQ$-Gorenstein singularities?
Abramovich:  Yes, there are many approaches to this question.  





\end{document}
