Construction of hyperbolic Horikawa surfaces

Abstract

We construct a Brody hyperbolic Horikawa surface that is a double cover of $\mathbb{P}^2$ branched along a smooth curve of degree $10$. We also construct Brody hyperbolic double covers of Hirzebruch surfaces with branch loci of the lowest possible bidegree.Comment: 16 pages; comments are very welcome. Final version, to appear in Annales de l'Institut Fourie