Vojta's conjecture on weighted projective varieties

TL;DR

This paper extends Vojta's conjecture to weighted projective varieties, establishing their equivalence, and explores implications for height bounds and weighted gcds under the conjecture.

Contribution

It formulates Vojta's conjecture for weighted projective varieties and proves their equivalence, introducing weighted divisors and height bounds.

Findings

All three versions of Vojta's conjecture are equivalent.

Assuming the conjecture, bounds on weighted gcd heights are established.

Analogous results are obtained for weighted homogeneous polynomials.

Abstract

We formulate Vojta's conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. In the process, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up relative to an exceptional divisor. Furthermore, we prove that assuming Vojta's conjecture for weighted projective varieties one can bound the $\log {\rm h_{wgcd}}$ for any subvariety of codimension $\geq 2$ and a finite set of places $S$. An analogue result is proved for weighted homogeneous polynomials with integer coefficients.