Quadratic Differentials and Equivariant Deformation Theory of Curves
Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 1015-1043.

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.

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.

DOI: 10.5802/aif.2715
Classification: 14H30, 14D15, 14F10, 11R32
Keywords: quadratic differentials, tangent space, equivariant deformation functor, Galois modules, Riemann-Roch spaces, weakly ramified, $p$-rank representation
Mot clés : différentiels quadratiques, espace tangent, foncteur de déformations équivariantes, modules de Galois, espaces Riemann-Roch, faiblement ramifiée, rang-$p$ réprésentations
Köck, Bernhard 1; Kontogeorgis, Aristides 2

1 University of Southampton School of Mathematics Highfield Southampton SO17 1BJ (United Kingdom)
2 National and Kapodistrian University of Athens Department of Mathematics Panepistimioupolis GR-157 84 Athens (Greece)
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Köck, Bernhard; Kontogeorgis, Aristides. Quadratic Differentials and Equivariant Deformation Theory of Curves. Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 1015-1043. doi : 10.5802/aif.2715. https://aif.centre-mersenne.org/articles/10.5802/aif.2715/

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