${\mu }_p$- and ${\alpha }_p$-actions on K3 surfaces in characteristic $p$

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 $l\ne p$) 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 $2$.

43 pages, comments welcome / v2: Section 6 simplified. Construction of Theorem 7.3(2) changed accordingly. Non-existence of $D_8^1$ and $E_8^1$ moved to arXiv:1907.04686. / v3: Presentation improved. / v4: Added non-existence of ${\alpha }_p$-actions for $p=13,17,19$. Reorganized Lem 6.3-6.4 according to the change on arXiv:1907.04686. Strengthened Cor 3.5 and Prop 4.1 slightly / v5: Minor revision

References in corpus (1)

• A 1-dimensional family of Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1

Cited by in corpus (1)

• Torsors over the Rational Double Points in Characteristic $\mathbf{p}$