Deformations of product-quotient surfaces and reconstruction of Todorov surfaces via $\mathbb{Q}$-Gorenstein smoothing

Abstract

We consider the deformation spaces of some singular product-quotient surfaces $X= (C_1 \times C_2)/G$, where the curves $C_i$ have genus 3 and the group $G$ is isomorphic to $\mathbb{Z}_4$. As a by-product, we give a new construction of Todorov surfaces with $p_g = 1$, $q = 0$ and $2 \leq K^2 \leq 8$ by using $\mathbb{Q}$-Gorenstein smoothings.