It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is and that if such a group has order , then it is isomorphic to the Mathieu group . Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is and this group contains with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface . In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order , both groups contains as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.
Mukai a montré que l’ordre maximal d’un groupe fini agissant fidèlement et symplectiquement sur une surface K3 est et que, si un tel groupe a pour ordre , alors il est isomorphe au groupe de Mathieu . Kondo a ensuite montré que l’ordre maximal d’un groupe fini agissant fidèlement sur une K3 surface est et qu’un tel groupe contient comme sous-groupe d’indice . Kondo a aussi montré qu’il existe une unique surface K3 sur laquelle ce groupe agit fidèlement : c’est la surface de Kummer . Dans cet article, nous décrivons deux autres surfaces K3 admettant un groupe fini d’automorphismes d’ordre , ces deux groupes et ces deux surfaces K3 étant uniques. Ce résultat a été obtenu indépendamment par S. Brandhorst and K. Hashimoto dans un article à venir, dont le but est de classifier les groupes finis agissant fidèlement sur des K3 surfaces et dont la partie symplectique est maximale.
Revised:
Accepted:
Published online:
Keywords: K3 surfaces, automorphisms
Mots-clés : Surfaces K3, automorphismes
Bonnafé, Cédric 1; Sarti, Alessandra 2
CC-BY-ND 4.0
@article{AIF_2021__71_2_711_0,
author = {Bonnaf\'e, C\'edric and Sarti, Alessandra},
title = {K3 surfaces with maximal finite automorphism groups containing $M_{20}$},
journal = {Annales de l'Institut Fourier},
pages = {711--730},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {71},
number = {2},
year = {2021},
doi = {10.5802/aif.3411},
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url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3411/}
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Bonnafé, Cédric; Sarti, Alessandra. K3 surfaces with maximal finite automorphism groups containing $M_{20}$. Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 711-730. doi : 10.5802/aif.3411. https://aif.centre-mersenne.org/articles/10.5802/aif.3411/
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