Given a finite -group acting on a smooth projective curve over an algebraically closed field of characteristic , the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of acting on the space of global holomorphic quadratic differentials on . We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when is cyclic or when the action of on is weakly ramified. Moreover we determine certain subrepresentations of , called -rank representations.
Soit G un -groupe fini agissant sur une courbe lisse projective sur un corps algébriquement clos de caractéristique . 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 agissant sur l’espace de différentielles holomorphes quadratiques globales sur . On applique des résultats connus sur la structure de module de Galois des espaces Riemann-Roch pour calculer cette dimension dans le cas où est cyclique ou dans le cas où l’action de sur est faiblement ramifiée. De plus, on détermine certaines sous-représentations de , qui s’appellent rang- représentations.
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
@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}, pages = {1015--1043}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {3}, year = {2012}, doi = {10.5802/aif.2715}, mrnumber = {3013815}, zbl = {1256.14026}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2715/} }
TY - JOUR AU - Köck, Bernhard AU - Kontogeorgis, Aristides TI - Quadratic Differentials and Equivariant Deformation Theory of Curves JO - Annales de l'Institut Fourier PY - 2012 SP - 1015 EP - 1043 VL - 62 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2715/ DO - 10.5802/aif.2715 LA - en ID - AIF_2012__62_3_1015_0 ER -
%0 Journal Article %A Köck, Bernhard %A Kontogeorgis, Aristides %T Quadratic Differentials and Equivariant Deformation Theory of Curves %J Annales de l'Institut Fourier %D 2012 %P 1015-1043 %V 62 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2715/ %R 10.5802/aif.2715 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2715/
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