On the group of automorphisms of Horikawa surfaces

Abstract

Minimal algebraic surfaces of general type $X$ such that $K_X^2=2\chi ({\mathcal{O}}_X)-6$ 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 ${\mathbb{Z}}_2$.

Some changes following referee's report. Final version to appear in Comptes Rendus Mathématique. 12 pages