# Revista Matemática Iberoamericana

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**Volume 31, Issue 4, 2015, pp. 1277–1310**

**DOI: 10.4171/RMI/869**

Published online: 2015-12-23

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

Alice Garbagnati^{[1]}and Matteo Penegini

^{[2]}(1) Università degli Studi di Milano, Italy

(2) Università degli Studi di Milano, Italy

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.

*Keywords: *K3 surfaces, automorphisms of K3 surfaces, product-quotient surfaces

Garbagnati Alice, Penegini Matteo: K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups. *Rev. Mat. Iberoam.* 31 (2015), 1277-1310. doi: 10.4171/RMI/869