On the group of automorphisms of Horikawa surfaces

TL;DR

This paper investigates the automorphism groups of Horikawa surfaces, showing that within their moduli space, most surfaces have automorphism groups isomorphic to , highlighting a common symmetry structure.

Contribution

It proves that for admissible pairs , most Horikawa surfaces in the moduli space have automorphism groups isomorphic to , revealing a typical symmetry property.

Findings

Most Horikawa surfaces have automorphism group .

Every component of the moduli space contains surfaces with automorphism group .

Automorphism groups are generically  in the moduli space.

Abstract

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ are called Horikawa surfaces. In this note the group of automorphisms of Horikawa surfaces is studied. The main result states that given an admissible pair $(K^2, \chi)$ such that $K^2=2\chi-6$, every irreducible component of Gieseker's moduli space $\mathfrak{M}_{K^2,\chi}$ contains an open subset consisting of surfaces with group of automorphisms isomorphic to $\mathbb{Z}_2$.