Effective model of a finite group action

Matthieu Romagny (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a flat $R$-scheme of finite type and $G$ a finite flat group scheme acting on $X$ so that $G_K$ is faithful on the generic fibre $X_K$. We prove that there is an effective model of $G$ i.e. a finite flat group scheme dominated by $G$, isomorphic to it on the generic fibre, and extending the action of $G_K$ on $X_K$ to an action on all of $X$ that is faithful also on the special fibre. It is unique with these properties. We give examples and applications to degenerations of coverings of curves.

Mots clés: cover of stable curves; reduction mod $p$; group scheme; moduli.; moduli

Domaines: Géométrie algébrique [math.AG]

Matthieu Romagny. Effective model of a finite group action. 2006. ⟨hal-00017906⟩