Quadratic Differentials and Equivariant Deformation Theory of Curves

Köck, Bernhard; Kontogeorgis, Aristides

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1015-1043.

Soit G un $p$-groupe fini agissant sur une courbe lisse projective $X$ sur un corps algébriquement clos $k$ de caractéristique $p$. Alors la dimension de l’espace tangent du foncteur de déformations équivariantes associé est égal à la dimension de l’espace de co-invariants de $G$ agissant sur l’espace $V$ de différentielles holomorphes quadratiques globales sur $X$. On applique des résultats connus sur la structure de module de Galois des espaces Riemann-Roch pour calculer cette dimension dans le cas où $G$ est cyclique ou dans le cas où l’action de $G$ sur $X$ est faiblement ramifiée. De plus, on détermine certaines sous-représentations de $V$, qui s’appellent rang-$p$ représentations.

Given a finite $p$-group $G$ acting on a smooth projective curve $X$ over an algebraically closed field $k$ of characteristic $p$, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of $G$ acting on the space $V$ of global holomorphic quadratic differentials on $X$. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when $G$ is cyclic or when the action of $G$ on $X$ is weakly ramified. Moreover we determine certain subrepresentations of $V$, called $p$-rank representations.

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