# $\mu_p$- and $\alpha_p$-actions on K3 surfaces in characteristic $p$

## TL;DR

This paper investigates $_p$- and $__p$-actions on rational double point K3 surfaces in positive characteristic, analyzing their properties, quotient surfaces, and singularities, revealing analogies with group actions in characteristic zero.

## Contribution

It establishes the behavior of $_p$- and $__p$-actions on RDP K3 surfaces and shows their correspondence with certain cyclic group actions, including existence of specific coverings.

## Key findings

- $_p$- and $__p$-actions mirror cyclic group actions in characteristic zero.
- Characterization of quotient surfaces and singularities under these actions.
- Existence of coverings by K3-like surfaces with specific singularity configurations.

## Abstract

We consider $\mu_p$- and $\alpha_p$-actions on RDP K3 surfaces (K3 surfaces with rational double point singularities allowed) in characteristic $p > 0$. We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of $\mu_p$- and $\alpha_p$-actions are analogous to those of $\mathbb{Z}/l\mathbb{Z}$-actions (for primes $l \neq p$) and $\mathbb{Z}/p\mathbb{Z}$-quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a $\mu_p$- or $\alpha_p$- or $\mathbb{Z}/p\mathbb{Z}$-covering by a "K3-like" surface, which is often an RDP K3 surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic $2$.