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

### Alice Garbagnati

Università degli Studi di Milano, Italy### Matteo Penegini

Università degli Studi di Milano, Italy

## 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 $\mathbb Z/p\mathbb Z$ or the group $\mathbb Z/2p\mathbb Z$ where $p$ is an odd prime. 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.

## Cite this article

Alice Garbagnati, Matteo Penegini, K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1277–1310

DOI 10.4171/RMI/869