# 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 ). A stable genus 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 is irreducible for any characteristic.

Jeffrey Herschel Giansiracusa 2005-06-27