Computing invariants of semi-log-canonical surfaces

TL;DR

This paper introduces methods for computing invariants of semi-log-canonical surfaces and applies them to classify certain Gorenstein stable surfaces with specific invariants.

Contribution

It provides new computational techniques for fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces, and classifies irregular Gorenstein stable surfaces with $K^2=1$.

Findings

Exactly two irregular Gorenstein stable surfaces with $K^2=1$

Both have $ ext{chi}(X)=0$ and $ ext{Pic}^0(X)= ext{C}^*$

They differ in homotopy type

Abstract

We describe some methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with $K^2=1$, both of which have $\chi(X) = 0$ and $\mathrm{Pic}^0(X)=\mathbb{C}^*$ but different homotopy type.