# K3 surfaces with 9 cusps in characteristic p

## TL;DR

This paper investigates K3 surfaces with nine cusps over fields of characteristic p not equal to 3, establishing their relation to abelian surfaces and exploring their position in the supersingular locus.

## Contribution

It proves that such K3 surfaces admit a triple cover by an abelian surface and characterizes which abelian surfaces produce these K3 surfaces.

## Key findings

- Each K3 surface with 9 cusps admits a triple cover by an abelian surface.
- Characterization of abelian surfaces with automorphisms leading to K3 surfaces.
- Analysis of the occurrence of these K3 surfaces in the supersingular locus.

## Abstract

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.