2-Gorenstein stable surfaces with $K_X^2 = 1$ and $\chi(X) = 3$

TL;DR

This paper classifies 2-Gorenstein stable surfaces with specific invariants, revealing new components and divisors in the moduli space of surfaces of general type.

Contribution

It provides a complete classification of 2-Gorenstein stable surfaces with $K_X^2=1$ and $\chi(X)=3$, discovering new moduli space components and divisors.

Findings

Classification into four types of surfaces

Identification of a new divisor in the moduli space

Discovery of a new irreducible component

Abstract

The compactification $\overline M_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $\chi(X)=3$ in the moduli space of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.