Gorenstein stable surfaces with $K_X^2 = 2$ and $\chi(\mathcal O_X) = 4$

TL;DR

This paper classifies Gorenstein stable surfaces with specific invariants by stratifying their moduli space and relating it to plane octics with singularities, revealing 47 strata and 78 components.

Contribution

It provides a detailed stratification of the moduli space of these surfaces and establishes an isomorphism with a moduli space of plane octics, facilitating concrete analysis.

Findings

47 inhabited strata identified

78 total components in the moduli space

Isomorphism with moduli space of plane octics

Abstract

We define and study a concrete stratification of the moduli space of Gorenstein stable surfaces $X$ satisfying $K_{X}^2 = 2$ and $\chi(\mathcal{O}_{X}) = 4$, by first establishing an isomorphism with the moduli space of plane octics with certain singularities, which is then easier to handle concretely. In total, there are 47 inhabited strata with altogether 78 components.