[Surfaces K3 avec un groupe fini d’automorphismes maximal contenant ]
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.
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.
Révisé le :
Accepté le :
Publié le :
Keywords: K3 surfaces, automorphisms
Mot 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},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3411/}
}
TY - JOUR
AU - Bonnafé, Cédric
AU - Sarti, Alessandra
TI - K3 surfaces with maximal finite automorphism groups containing $M_{20}$
JO - Annales de l'Institut Fourier
PY - 2021
SP - 711
EP - 730
VL - 71
IS - 2
PB - Association des Annales de l’institut Fourier
UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3411/
DO - 10.5802/aif.3411
LA - en
ID - AIF_2021__71_2_711_0
ER -
%0 Journal Article
%A Bonnafé, Cédric
%A Sarti, Alessandra
%T K3 surfaces with maximal finite automorphism groups containing $M_{20}$
%J Annales de l'Institut Fourier
%D 2021
%P 711-730
%V 71
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3411/
%R 10.5802/aif.3411
%G en
%F AIF_2021__71_2_711_0
Bonnafé, Cédric; Sarti, Alessandra. K3 surfaces with maximal finite automorphism groups containing $M_{20}$. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 711-730. doi : 10.5802/aif.3411. https://aif.centre-mersenne.org/articles/10.5802/aif.3411/
[1] Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer, 1984, 304 pages | DOI | Zbl
[2] The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl
[3] On K3 surfaces with maximal symplectic action (2019) (https://arxiv.org/abs/1910.05952, to appear in Ann. Henri Lebesgue)
[4] Towards spetses. I, Transform. Groups, Volume 4 (1999) no. 2-3, pp. 157-218 | DOI | MR | Zbl
[5] Quartic surfaces with icosahedral symmetry, Adv. Geom., Volume 18 (2018) no. 1, pp. 119-132 | DOI | MR | Zbl
[6] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | DOI | MR
[7] Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, 158, Cambridge University Press, 2016, xi+485 pages | DOI | Zbl
[8] On certain Kummer surfaces which can be realized as non-singular quartic surfaces in , J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 23 (1976) no. 3, pp. 545-560 | MR | Zbl
[9] The maximum order of finite groups of automorphisms of surfaces, Am. J. Math., Volume 121 (1999) no. 6, pp. 1245-1252 | DOI | MR | Zbl
[10] On the automorphisms of hypersurfaces, J. Math. Kyoto Univ., Volume 3 (1964), pp. 347-361 | DOI | MR | Zbl
[11] Finite groups of automorphisms of surfaces and the Mathieu group, Invent. Math., Volume 94 (1988) no. 1, pp. 183-221 | DOI | MR | Zbl
[12] Finite groups of automorphisms of Kählerian surfaces of type , Usp. Mat. Nauk, Volume 31 (1976) no. 2(188), pp. 223-224
[13] Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 43 (1979) no. 1, pp. 111-177 | MR
[14] Singularities. II. Automorphisms of forms, Math. Ann., Volume 231 (1978) no. 3, pp. 229-240 | DOI | MR | Zbl
[15] Finite unitary reflection groups, Can. J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl
[16] On singular surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, 1977, pp. 119-136 | DOI | Zbl
[17] Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds (Lecture Notes in Mathematics), Volume 412, Springer, 1974, pp. 259-287 | DOI | MR | Zbl
[18] Galois covers between surfaces, Ann. Inst. Fourier, Volume 46 (1996) no. 1, pp. 73-88 | DOI | Numdam | MR | Zbl
Cité par Sources :
