Vojta's conjecture on weighted projective varieties and an application on GCD's

Author:

Sajad Salami, Tony Shaska

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-09-18 16:00:00

Abstract

We state 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. Moreover, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up at a point, relative to an exceptional divisor. Furthermore, we also prove that assuming Vojta's conjecture for weighted projective varieties one can bound the $\log {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. As an application of our results we obtain a bound on greatest common divisors, which restricted to projective space is the same as bounds obtained by Corvaja, Zannier, et al.