The exceptional set in Vojta's conjecture for algebraic points of bounded degree

TL;DR

This paper investigates the exceptional set in Vojta's conjecture for algebraic points of bounded degree, demonstrating that the conjecture does not extend to all algebraic points without degree restrictions.

Contribution

It provides a negative answer to whether Vojta's conjecture holds universally for all algebraic points, using elliptic surfaces to illustrate the dependence on parameters.

Findings

Vojta's conjecture does not hold for all algebraic points without degree bounds

Elliptic surfaces are used to demonstrate the failure of the conjecture in this setting

The exceptional set's dependence on parameters is clarified

Abstract

We study the dependence on various parameters of the exceptional set in Vojta's conjecture. In particular, by making use of certain elliptic surfaces, we answer in the negative the often-raised question of whether Vojta's conjecture holds when extended to all algebraic points (that is, if the conjecture holds without fixing a bound on the degree of the algebraic points).