Vojta's conjecture for singular varieties

TL;DR

This paper generalizes Vojta's conjecture to include singular varieties using log pairs and multiplier ideals, revealing that rational points tend to cluster near singularities and extending related conjectures to these cases.

Contribution

It introduces a new formulation of Vojta's conjecture for singular varieties via log pairs, establishing equivalence with the original conjecture on log resolutions.

Findings

Rational points are more abundant near singularities.

The generalized conjecture aligns with the original when applied to log resolutions.

Extension of Lang's conjecture to singular varieties and log pairs.

Abstract

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is equivalent to the original conjecture applied to a log resolution of the pair. A special case of the generalized conjecture can be interpreted as representing a general phenomenon that there tend to exist more rational points near singular points than near smooth points. The same phenomenon is also observed in relation between greatest common divisors of integer pairs satisfying an algebraic equation and plane curve singularities, which is discussed in Appendix. As an application of the generalization of Vojta's conjecture, we also derive a generalization of a geometric conjecture of Lang concering varieties of general type to singular varieties and log pairs.