Bounding heights uniformly in families of hyperbolic varieties

TL;DR

Under the assumption of Vojta's height conjecture, the paper establishes uniform bounds on the heights of rational points across families of hyperbolic varieties, extending previous results from curves to higher dimensions.

Contribution

It generalizes the uniform height bounds from curves to higher-dimensional hyperbolic varieties assuming Vojta's conjecture.

Findings

Height of rational points on hyperbolic varieties can be bounded uniformly in families.

Results extend to higher-dimensional varieties beyond curves.

Application to bounding heights on curves of general type and certain hyperbolic surfaces.

Abstract

We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$.