K3 surfaces with maximal finite automorphism groups containing M 20
K3 surfaces with maximal finite automorphism groups containing M 20
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具有包含 M 20 的最大有限自同构群的 K3 曲面
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
A. Sarti
中科院分区：
文献类型：
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作者：
C'edric Bonnaf'e;A. Sarti
Mukai证明了在K3曲面上忠实地辛作用的有限群的最大阶为960 $，并且该群同构于群M_{20}$。Kondo证明了忠实作用在K3曲面上的有限群的最大阶为3\,840，且此群包含指数为4的Mathieu群M_{20}。Kondo还证明了存在一个唯一的K3曲面，这个曲面是库默曲面Km$（E_i\times E_i）$。本文又刻画了两个K3曲面，它们都含有一个阶为1\,920的大的有限自同构群，这两个群都含有M_{20}$作为指数为2的子群.我们还证明了这两个群和两个K3曲面是唯一的。这一结果由S. Brandhorst和K. Hashimoto在即将发表的论文中，其目的是对所有忠实地作用于具有最大辛部分的K3曲面的有限群进行分类。
It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface Km$(E_i\times E_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M_{20}$ 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.