# Le th\'eor\`eme de r\'eduction stable de Deligne et Mumford

## TL;DR

The paper explains the stable reduction theorem of Deligne and Mumford, which is key to understanding the compactification of the moduli space of algebraic curves by including stable curves with controlled singularities.

## Contribution

It introduces the concepts of stable curves and the stable reduction theorem, providing foundational understanding of the compactification of moduli spaces of curves.

## Key findings

- Construction of the projective compactification of the moduli space.
- Proof of the properness of the compactification.
- Relation between stable reduction and the geometry of moduli spaces.

## Abstract

The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. This text is based on the oral presentation and aims at introducing these objects.