The moduli space of stable curves

Abstract

In this chapter we construct the moduli space $\overline{M}_{g,n}$ of stable n-pointed curves of genus g and look at its structure from various points of view. First we construct $\overline{M}_{g,n}$ as an analytic space, and then we show that this analytic space has a natural structure of algebraic space. After a utilitarian introduction to orbifolds and stacks, in particular to Deligne–Mumford stacks, we then show that $\overline{M}_{g,n}$ is just a coarse reflection of a more fundamental object, the moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable n-pointed curves of genus g, and we show, using in an essential way the results of Chapter X, that $\overline{\mathcal{M}}_{g,n}$ is a Deligne–Mumford stack. After discussing some basic constructions, such as normalization and quotient by a finite group, in the context of Deligne–Mumford stacks, we close by interpreting the fundamental constructions of projection and clutching as morphisms of moduli spaces, and by observing that contraction and stabilization give an isomorphism of stacks between $\overline{\mathcal{M}}_{g,n+1}$ and the “universal curve” $\overline{\mathcal{C}}_{g,n}$ over $\overline{\mathcal{M}}_{g,n}$.