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

TL;DR

This paper studies the deformation spaces of certain singular product-quotient surfaces and introduces a new method to construct Todorov surfaces with specific invariants using $ ext{Q}$-Gorenstein smoothings.

Contribution

It provides a novel construction of Todorov surfaces via $ ext{Q}$-Gorenstein smoothings and analyzes the deformation spaces of specific singular product-quotient surfaces.

Findings

New construction of Todorov surfaces with $p_g=1$, $q=0$, and $2 \\le K^2 \\le 8$.

Analysis of deformation spaces of singular product-quotient surfaces.

Application of $ ext{Q}$-Gorenstein smoothings to surface reconstruction.

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.