Quelques remarques sur le champ des courbes hyperelliptiques en caracteristique deux

Abstract

Dans cette note on decrit le champ des courbes hyperelliptiques lisses defines sur un corps algebriquement clos de caracteristique deux comme un champ quotient. On en deduit le groupe de Picard. ----- In this note we describe the stack ${\cal H}_g$ of smooth hyperelliptic curves of genus $g$ over an algebraically closed field of characteristic two, as a quotient stack of a smooth variety of dimension $3g+5$ by a non reductive algebraic group, extending a well known result of Vistoli. As an application, we show the Mumford-Vistoli description of the Picard group of the stack ${\cal H}_g$ is valid in this characteristic. We also describe the natural stratification of ${\cal H}_g$ by means of higher ramification data. We point out that after stable compactification $\bar {\cal H}_g$ is not smooth in codimension at least two.