$\mathbb{Z}_2^2$-actions on Horikawa surfaces

TL;DR

This paper investigates $ ext{Z}_2^2$-group actions on Horikawa surfaces, showing that such symmetries are present in all connected components of the moduli space for these surfaces, enriching understanding of their symmetry properties.

Contribution

It demonstrates that every connected component of the moduli space of certain Horikawa surfaces includes surfaces admitting $ ext{Z}_2^2$-actions, revealing pervasive symmetry structures.

Findings

All components of the moduli space contain surfaces with $ ext{Z}_2^2$-actions.

The study extends understanding of symmetries in algebraic surfaces of general type.

Provides new insights into the structure of the moduli space of Horikawa surfaces.

Abstract

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ or $K^2_X=2\chi(\mathcal{O}_X)-5$ are called Horikawa surfaces. In this note we study $\mathbb{Z}^2_2$-actions on Horikawa surfaces. The main result is that all the connected components of Gieseker's moduli space of canonical models of surfaces of general type with invariants satisfying these relations contain surfaces with $\mathbb{Z}^2_2$-actions.