A new irreducible component of the moduli space of stable Godeaux surfaces

TL;DR

This paper constructs a new class of stable Godeaux surfaces from del Pezzo surfaces, revealing an irreducible component of their moduli space, and analyzes their canonical rings and pluricanonical maps.

Contribution

It introduces a novel construction of stable Godeaux surfaces with specific invariants, expanding understanding of their moduli space structure.

Findings

Constructed stable Godeaux surfaces with $K_X^2=1$ and $p_g=q=0$

Identified an irreducible component of the moduli space containing these surfaces

Explicitly computed the canonical ring and studied pluricanonical maps

Abstract

We construct from a general del Pezzo surface of degree 1 a Gorenstein stable surfaces $X$ with $K_X^2=1$ and $p_g(X)=q(X)=0$. These surfaces are not smoothable but give an open subset of an irreducible component of the moduli space of stable Godeaux surfaces. In a particular example we also compute the canonical ring explicitly and discuss the behaviour of pluricanonical maps.