On the hyperbolicity of surfaces of general type with small $c_1 ^2$

TL;DR

This paper investigates the hyperbolic properties of minimal surfaces of general type, specifically Horikawa surfaces, demonstrating that many such surfaces are algebraically hyperbolic or quasi-hyperbolic, using orbifold techniques.

Contribution

It provides the first examples of Horikawa surfaces with proven algebraic hyperbolicity and quasi-hyperbolicity, advancing understanding of hyperbolic properties in minimal surfaces.

Findings

Existence of infinitely many moduli components with hyperbolic members

Construction of explicit algebraically hyperbolic Horikawa surfaces

Demonstration that minimal surfaces with small $c_1^2$ can be hyperbolic

Abstract

Surfaces of general type with positive second Segre number $s_2:=c_1^2-c_2>0$ are known by results of Bogomolov to be quasi-hyperbolic i.e. 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 $c_1^2$, known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in $\bP^3$. Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or even is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and (quasi-)hyperbolic orbifold Horikawa surfaces.