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 $2\le K^2\le 8$ by using $\mathbb{Q}$-Gorenstein smoothings.

arXiv comment21 pages, Minor changes are made. It will apper in Advanced Studies in Pure Mathematics (Proceeding of Algebraic Geometry in East Asia, Taipei)