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

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.

28 pages