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

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.