Stability and semi-stability of (2,2)-type surfaces

TL;DR

This paper studies the geometric invariant theory (GIT) compactification of the moduli space of (2,2)-type surfaces in a product of projective spaces, characterizing stability conditions and describing the boundary of the moduli space.

Contribution

It provides a detailed characterization of stable and semi-stable (2,2)-type surfaces and determines the boundary components of their moduli space.

Findings

Classification of stable and semi-stable (2,2)-type surfaces

Description of the boundary of the moduli space

Identification of strictly semi-stable surface classes

Abstract

We describe the GIT compactification of the moduli of (2,2)-type effective divisors of $\mathbb{P}^1\times\mathbb{P}^2$ (i.e., surfaces of the linear system $\vert \pi_1^*\mathcal{O}_{\mathbb{P}^1}(2)\otimes \pi_2^*\mathcal{O}_{\mathbb{P}^2}(2)\vert$ ) which are generically Del Pezzo surfaces of degree two. In order to get the compactification, we characterize stable and semi-stable (2,2)-type surfaces, and also determine the equivalence classes of strictly semi-stable (2,2)-type surfaces. Moreover, we describe the boundary of the moduli of (2,2)-type surfaces.