# Construction of hyperbolic Horikawa surfaces

## TL;DR

This paper constructs new examples of Brody hyperbolic surfaces, specifically Horikawa surfaces and double covers of Hirzebruch surfaces, expanding the known classes of hyperbolic algebraic surfaces.

## Contribution

It introduces explicit constructions of Brody hyperbolic surfaces as double covers with specific branch loci, including the first such examples of Horikawa surfaces.

## Key findings

- Constructed a Brody hyperbolic Horikawa surface as a double cover of P^2.
- Built Brody hyperbolic double covers of Hirzebruch surfaces with minimal branch degrees.
- Expanded the catalog of known hyperbolic algebraic 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.