K3 surfaces with maximal finite automorphism groups containing $M_{20}$

Cédric Bonnafé (1) , Alessandra Sarti (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

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.

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Cédric Bonnafé, Alessandra Sarti. K3 surfaces with maximal finite automorphism groups containing $M_{20}$. Annales de l'Institut Fourier, 2021, 71 (2), pp.711-730. ⟨10.5802/aif.3411⟩. ⟨hal-02317043v2⟩