Surfaces with p_g=q=2, K^2=6 and Albanese map of degree 2

TL;DR

This paper classifies certain minimal surfaces of general type with specific invariants, describing their moduli space structure and geometric properties, including the presence of a special elliptic curve contracted by the Albanese map.

Contribution

It provides a detailed classification of surfaces with p_g=q=2, K^2=6, and Albanese map degree 2, identifying moduli space components and geometric features.

Findings

Moduli space has three irreducible components of dimensions 4, 4, and 3.

General surfaces contain a smooth elliptic curve with self-intersection -2.

The elliptic curve is contracted by the Albanese map and stable under first-order deformations.

Abstract

We classify minimal surfaces $S$ of general type with $p_g=q=2$ and $K_S^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth, irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension 4, 4, 3, respectively. The general surface $S$ contains a smooth elliptic curve $Z$ such that $Z^2=-2$, which is contracted by the Albanese map and which is preserved by any first-order deformation.