Quadratic Differentials and Equivariant Deformation Theory of Curves
Annales de l'Institut Fourier, Volume 62 (2012) no. 3, p. 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 : https://doi.org/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
@article{AIF_2012__62_3_1015_0,
     author = {K\"ock, Bernhard and Kontogeorgis, Aristides},
     title = {Quadratic Differentials and Equivariant Deformation Theory of Curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {3},
     year = {2012},
     pages = {1015-1043},
     doi = {10.5802/aif.2715},
     mrnumber = {3013815},
     zbl = {1256.14026},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_3_1015_0}
}
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. http://www.numdam.org/item/AIF_2012__62_3_1015_0/

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