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

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 $|{\pi }_1^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(2)\otimes {\pi }_2^{\ast }{\mathcal{O}}_{{\mathbb{P}}^2}(2)|$ ) 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.

Comments are welcome! 28 pages