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Vojta's conjecture on weighted projective varieties

Sajad Salami, Tony Shaska

TL;DR

This work extends Vojta's conjecture to the setting of weighted projective varieties, introducing weighted heights, weighted multiplier ideals, and weighted log pairs, and proves the three formulations are equivalent. It develops a coherent theory of weighted divisors and heights, and shows that weighted analogues of Vojta’s inequalities hold and are compatible with the standard framework via appropriate resolutions and multiplier ideals. A central contribution is the identification of generalized weighted gcds with heights on weighted blow-ups, leading to explicit bounds on log h_{wgcd} for subvarieties of codimension at least 2 under Vojta's conjecture, with an analogue for weighted homogeneous polynomials. The results unify arithmetic properties of weighted and classical projective varieties and provide a natural mechanism to bound the size of points on weighted spaces, potentially impacting questions about rational points in weighted geometries.

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.

Vojta's conjecture on weighted projective varieties

TL;DR

This work extends Vojta's conjecture to the setting of weighted projective varieties, introducing weighted heights, weighted multiplier ideals, and weighted log pairs, and proves the three formulations are equivalent. It develops a coherent theory of weighted divisors and heights, and shows that weighted analogues of Vojta’s inequalities hold and are compatible with the standard framework via appropriate resolutions and multiplier ideals. A central contribution is the identification of generalized weighted gcds with heights on weighted blow-ups, leading to explicit bounds on log h_{wgcd} for subvarieties of codimension at least 2 under Vojta's conjecture, with an analogue for weighted homogeneous polynomials. The results unify arithmetic properties of weighted and classical projective varieties and provide a natural mechanism to bound the size of points on weighted spaces, potentially impacting questions about rational points in weighted geometries.

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 for any subvariety of codimension and a finite set of places . An analogue result is proved for weighted homogeneous polynomials with integer coefficients.
Paper Structure (18 sections, 10 theorems, 152 equations)

This paper contains 18 sections, 10 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on weighted projective varieties
  3. Weighted projective varieties
  4. Rational points on weighted varieties in $\mathbb P_{\mathfrak q}^n$
  5. Weighted heights on $\mathbb P_\mathfrak q^n$
  6. Vojta's conjecture for projective varieties
  7. Vojta's conjecture for smooth varieties
  8. Vojta's conjecture using a multiplier ideal sheaf
  9. Vojta's conjecture for log pairs over $\mathbb Q$
  10. Vojta's conjecture for weighted projective varieties
  11. Local and global weighted heights
  12. Weil and Cartier divisors on weighted varieties
  13. Vojta's conjecture for smooth weighted varieties
  14. Vojta's conjecture using a weighted multiplier ideal sheaf
  15. Vojta's conjecture for weighted log pairs
  16. ...and 3 more sections

Key Result

Lemma 1

The definitions of $\mathcal{I}(\mathcal{X}, \mathcal{Y})$, $\mathcal{I}^-(\mathcal{X}, \mathcal{Y})$, and $\mathcal{H}(\mathcal{X}, \mathcal{Y})$ are independent of the choice of a weighted log resolution.

Theorems & Definitions (19)

  • Example 1
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • Lemma 1
  • Proposition 1
  • Conjecture 6
  • Corollary 1
  • ...and 9 more