Construction of hyperbolic Horikawa surfaces

Abstract

We construct a Brody hyperbolic Horikawa surface that is a double cover of 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.

A remark on the intersection of plane curves

“…Unfortunately, the genus g works against us in (0.2); however, for g = 0, 1 and d even, (0.2) is better than (0.1). Further related problems have been recently considered in [19,64,65]. As a final remark, note that (0.2) is more useful than (0.1) if one looks, as we do in this paper, at the geometric genera of curves contained in a double plane X d , that is, a cyclic double cover of P 2 branched along a very general plane curve D of even degree d. For instance, letting g = 0, δ(D, Γ) = 0, 2 and d even, we are looking actually for rational curves on X d .…”