On the hyperbolicity of surfaces of general type with small c 21

Surfaces of general type with positive second Segre number s2 := c 2 1 − c2 > 0 are known by the results of Bogomolov to be algebraically quasi-hyperbolic, that is, with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green–Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal c1, known as Horikawa surfaces. In principle, these surfaces should be the most difficult case for the above conjecture as illustrated by the quintic surfaces in P. Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and quasi-hyperbolic orbifold Horikawa surfaces.