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