K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups

TL;DR

This paper classifies certain K3 surfaces arising from quotients of product of curves by cyclic groups, focusing on those with non-symplectic automorphisms of prime order, and explores their moduli and Hodge structures.

Contribution

It provides a classification of K3 surfaces obtained from product-quotient constructions with cyclic groups and analyzes their automorphisms and moduli spaces.

Findings

Most K3 surfaces with a non-symplectic automorphism of order p are obtained via product-quotient constructions.

A subset of these K3 surfaces can be realized with one curve being a rigid hyperelliptic curve with an automorphism of order p.

Explicit equations and Hodge structure variations are provided for some constructed surfaces.

Abstract

We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that one can obtain the same set of K3 surfaces under more restrictive assumptions namely one of the two curves, say $C_2$, is isomorphic to a rigid hyperelliptic curve with an automorphism $\delta_p$ of order $p$ and the automorphism of the K3 surface is induced by $\delta_p$.\\ Finally, we describe the variation of the Hodge structures of the surfaces constructed and we give an equation for some of them.