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

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.