Note on a family of surfaces with $p_g=q=2$ and $K^2=7$

TL;DR

This paper explores a specific family of algebraic surfaces with particular invariants, offering an alternative construction and detailed analysis of their moduli space, including its structure and components.

Contribution

It provides a new construction method for these surfaces and describes the structure and components of their moduli space in detail.

Findings

The moduli space has three connected components.

Two components are two-dimensional, irreducible, and generically smooth.

The paper characterizes the Albanese map for these surfaces.

Abstract

We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by C. Rito. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $\mathcal{M}$ in the moduli space of the surfaces of general type. In particular we prove that $\mathcal{M}$ is an open subset, and it has three connected components, two dimensional, irreducible and generically smooth.