Vojta's conjecture on weighted projective varieties and an application on greatest common divisors

We study 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. Furthermore, 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. We show that a point $\mathbf{x} \in {\mathbb P}_{{\mathbf{w}}, k}^n $ is smooth if and only if its generalized logarithmic weighted greatest common divisor $\log {h_{wgcd} \,} ({\mathbf x}) >0$. 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. An application of our results is a bound on greatest common divisors, which seems to be a better bound than previous bounds obtained by Corvaja, Zannier, et al.