Deligne-Mumford compactification

The Deligne-Mumford compactification is obtained as a moduli space for stable genus g curves with marked points, (instead of considering just smooth curves, as in $ \mathcal{M}_{g,n}$). A stable genus $ g$ curve is a connected, projective curve with at worst nodal singularities and finite automorphism group. This translates to: every genus 0 irreducible component has at least three marked or nodal points and every genus 1 component has at least one marked or nodal point. This space is important because it is a smooth compactification (as a stack, at least) with easy-to-understand boundary components that give an inductive structure to all the moduli spaces of curves. For instance, Deligne and Mumford used this compactification to prove that $ \mathcal{M}_{g}$ is irreducible for any characteristic.

Jeffrey Herschel Giansiracusa 2005-06-27