Gorenstein stable surfaces with $K_X^2 = 1$ and $p_g>0$

TL;DR

This paper classifies Gorenstein stable surfaces with minimal canonical class and positive geometric genus, describing their structure as weighted complete intersections and exploring their moduli space and degenerations.

Contribution

It extends classical results by providing a simple description of such surfaces and analyzing their moduli space stratification, including non-Gorenstein examples.

Findings

Surfaces admit a weighted complete intersection description.

Moduli space stratification for p_g=2 surfaces.

Existence of non-Gorenstein degenerations.

Abstract

In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with $K_X^2=1$; for $p_g=2$ this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.