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Bernhard Köck; Aristides Kontogeorgis
Quadratic Differentials and Equivariant Deformation Theory of Curves
(Différentielles quadratiques et théorie des déformations équivariantes de courbes)
Annales de l'institut Fourier, 62 no. 3 (2012), p. 1015-1043, doi: 10.5802/aif.2715
Article PDF | Reviews MR 3013815 | Zbl 1256.14026
Class. Math.: 14H30, 14D15, 14F10, 11R32
Keywords: quadratic differentials, tangent space, equivariant deformation functor, Galois modules, Riemann-Roch spaces, weakly ramified, $p$-rank representation

Résumé - Abstract

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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