Appears in collection : Rational points on irrational varieties

We report on a joint project with Nils Bruin. A surface X is algebraically quasi- hyperbolic if it contains finitely many curves of genus 0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal sur- faces of high degree in projective three-space carry symmetric differentials, and they used this to prove quasi-hyperbolicity of these surfaces. We explain how a granu- lar analysis of their ideas, combined with computational tools and insights, yield explicit results for the existence of symmetric differentials, and we show how these results can be used to give constraints on the locus of rational curves on surfaces like the Barth Decic, Buechi’s surface, and certain complete intersections of general type.