# Reference on Deligne-Mumford compactness for Riemann surfaces

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.

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.