Divisibility of polynomials and degeneracy of integral points

TL;DR

This paper explores the arithmetic hyperbolicity of blow-up varieties, extending previous results to higher dimensions, and applies advanced strategies to connect polynomial divisibility with degeneracy of integral points.

Contribution

It generalizes known results on hyperbolicity from dimension 2 to arbitrary dimensions using Ru-Vojta's strategy.

Findings

Multiple examples of pseudo-arithmetically hyperbolic varieties.

Extension of hyperbolicity results to higher dimensions.

Analogues established in function fields and Nevanlinna theory.

Abstract

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.