Explicit computation of symmetric differentials and its application to quasi-hyperbolicity

TL;DR

This paper develops explicit methods using symmetric differentials to study algebraic quasi-hyperbolicity of singular surfaces, applying them to specific surfaces to derive constraints on rational and genus 1 curves.

Contribution

It introduces explicit techniques for analyzing algebraic quasi-hyperbolicity via symmetric differentials and applies these to several complex surfaces, establishing new bounds and properties.

Findings

Rational curves on Barth's sextic pass through at least four singularities.

Genus 1 curves on Barth's sextic pass through at least two singularities.

Barth's decic, Sarti's surface, and the magic square surface are algebraically quasi-hyperbolic.

Abstract

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with $A_1$-singularities, and use our work to show that Barth's decic, Sarti's surface, and the surface parametrizing $3\times 3$ magic squares of squares are all algebraically quasi-hyperbolic.