𝜇_{𝑝}- and 𝛼_{𝑝}-actions on K3 surfaces in characteristic 𝑝

Abstract

We consider μ p \mu _p - and α p \alpha _p -actions on RDP K3 surfaces (K3 surfaces with rational double point (RDP) singularities allowed) in characteristic p > 0 p > 0 . We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of μ p \mu _p - and α p \alpha _p -actions are analogous to those of Z / l Z \mathbb {Z}/l\mathbb {Z} -actions (for primes l ≠ p l \neq p ) and Z / p Z \mathbb {Z}/p\mathbb {Z} -quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a μ p \mu _p - or α p \alpha _p - or Z / p Z \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 2 .