A family of surfaces with $p_g=q=2, \, K^2=7$ and Albanese map of degree $3$

TL;DR

This paper presents a new, computer-free construction of a specific family of algebraic surfaces with particular invariants, clarifying their Albanese map and moduli space, building on previous computer-assisted methods.

Contribution

It introduces an alternative, computer-free method to construct and analyze these surfaces, expanding understanding of their geometric properties.

Findings

Explicit description of the Albanese map.

Characterization of the moduli space of these surfaces.

Confirmation of the surfaces' properties without computer algebra.

Abstract

We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA. We provide an alternative, computer-free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.