K3 surfaces with action of the group $M_{20}$

TL;DR

This paper explores K3 surfaces with a symplectic action of the Mathieu group M20, demonstrating the existence of infinitely many such surfaces and providing explicit models and examples.

Contribution

It establishes the existence of infinitely many K3 surfaces admitting a faithful symplectic M20 action and describes their projective models.

Findings

Existence of infinitely many K3 surfaces with M20 action

Explicit descriptions and models of these surfaces

Examples illustrating the symplectic actions

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 if such a group has order 960, then it is isomorphic to the Mathieu group $M_{20}$. In this paper, we are interested in projective K3 surfaces admitting a faithful symplectic action of the group $M_{20}$. We show that there are infinitely many K3 surfaces with this action and we describe them and their projective models, giving some explicit examples.