Reference on Deligne-Mumford compactness for Riemann surfaces

Question

I am working with closed degenerating hyperbolic Riemann surfaces, and I try to understand the compactification of the moduli space. Looking in different books, notably the one of Hummel, I now get a good intuition of what happens, but i have still not found a reference where this compactification is made precise in this setting. And I also look for a combinatorial description of what limit surface are possible? Even if I presume that all configuration are possible. Thank You.

Answer 1

I'm assuming you are not primarily looking for references to papers in algebraic geometry. Maybe http://arxiv.org/abs/1301.0062 is what you are looking for. They construct rigorously the DM-compactification as an analytic space and explain how to interpret it in terms of Teichmüller theory.

The combinatorial structure of the Deligne-Mumford boundary is easiest explained in terms of dual graphs.

To define the dual graph of a nodal Riemann surface with some marked smooth points, draw a vertex for each irreducible component of the Riemann surface. Usually you decorate each vertex $v$ with the genus of the compnent, $g(v)$. For each marked point you draw a half-edge at the vertex where it is attached. For each node you draw an edge (two half-edges) connecting the components given by the branches at the node (so you might have loops in the dual graph). Define $n(v)$ as the number of half-edges incident to a vertex $v$. Finally define the genus of the dual graph as $b_1 + \sum_v g(v)$, where $b_1$ is the first Betti number of the graph.

Then the strata of $\overline{M}_{g,n}$ correspond bijectively to isomorphism classes of dual graphs of genus $g$ with exactly $n$ "legs" (half-edges which are not part of an edge). If $\Gamma$ is a dual graph, then the stratum is given by the quotient $$ \left(\prod_{v \in \mathrm{Vert}(\Gamma)} M_{g(v),n(v)}\right)\big/\mathrm{Aut}(\Gamma),$$ where $ \mathrm{Aut}(\Gamma)$ is the finite group of automorphisms of the graph. This means that understanding the strata amounts to the same as understanding the moduli spaces of smooth curves.

A lot of the combinatorial structure of the boundary becomes transparent in this language. For instance the gluing maps between different moduli spaces become here just the operation of fusing two half-edges into an edge.