Hyperbolicity of cyclic covers and complements

Abstract

Abstract. We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the hyperbolicity of complements of those branch divisors. As an application, we find new examples of Brody hyperbolic hypersurfaces in P n+1 that are cyclic covers of P n .

Construction of hyperbolic Horikawa surfaces

“…As an application of Theorem 1.2, we give new examples of Brody hyperbolic surfaces in P 3 of minimal degree 10 that are cyclic covers of P 2 under linear projections (see Theorem 5.3). This also improves [Liu16,Theorem 25]. We mention that a Brody hyperbolic Horikawa surface of even c 2 1 has to be a double cover of P 2 branched along a degree 10 curve (see Remark 5.5).…”

“…We remark that here that some Brody hyperbolic double covers of P 2 have been constructed in [Liu16,Theorem 5] with branch loci of minimal degree 30.…”

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 .…”