K3 surfaces with maximal finite automorphism groups containing $M\_20$

Abstract

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 $3840$ 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 $1920$, 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.

15 pages

References in corpus (1)

• Extensions of maximal symplectic actions on K3 surfaces

Cited by in corpus (4)

• 800 conics in a smooth quartic surface
• Extensions of maximal symplectic actions on K3 surfaces
• Conway Subgroup Symmetric Compactifications Redux
• Conics in Kummer quartics