Geography of minimal surfaces of general type with $\mathbb{Z}_2^2$-actions and the locus of Gorenstein stable surfaces

TL;DR

This paper investigates the distribution of minimal surfaces of general type with specific group actions within moduli spaces, revealing that the Gorenstein stable surfaces locus is not closed in the compactification for certain invariants.

Contribution

It demonstrates the existence of surfaces with $bZ_2^2$-actions across a range of invariants and shows the non-closure of the Gorenstein stable surfaces locus in the moduli space compactification.

Findings

Surfaces with $bZ_2^2$-actions exist for all admissible pairs within specified bounds.

The locus of Gorenstein stable surfaces is not closed in the KSBA-compactification.

The results cover a broad range of invariants $(K^2, \chi)$.

Abstract

In this note the geography of minimal surfaces of general type admitting $\mathbb{Z}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space $\mathfrak{M}_{K^2,\chi}$ contains surfaces admitting a $\mathbb{Z}_2^2$-action for every admissible pair $(K^2, \chi)$ such that $2\chi-6\leq K^2\leq 8\chi-8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification $\overline{\mathfrak{M}}_{K^2,\chi}$ of Gieseker's moduli space $\mathfrak{M}_{K^2,\chi}$ for every admissible pair $(K^2, \chi)$ such that $2\chi-6\leq K^2\leq 8\chi-8$.