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Campana conjecture for coverings of toric varieties over function fields

Carlo Gasbarri, Ji Guo, Julie Tzu-Yueh Wang

TL;DR

The paper advances the function-field analogue of Campana’s orbifold conjecture by proving $abc$-type bounds for Campana points on projective toric varieties with large boundary multiplicities, then extending these methods to show Campana’s conjecture for finite coverings of such toric varieties. The core strategy combines a function-field version of Vojta’s philosophy with the main technical GNSW $S$-unit framework, plus local Weil functions for Campana integral points. The results yield finiteness-type (non-density) statements for Campana integral points on orbifolds of general type arising as toric covers and their finite ramified extensions, with effective control of exceptional sets. This significantly broadens the classes of varieties and orbifold structures for which Campana–Lang–Vojta-type predictions hold over function fields, with potential explicit diophantine applications.

Abstract

We first prove Vojta's abc conjecture over function fields for Campana points on projective toric varieties with high multiplicity along the boundary. As a consequence, we obtain a version of Campana's conjecture on finite coverings of projective toric varieties over function fields.

Campana conjecture for coverings of toric varieties over function fields

TL;DR

The paper advances the function-field analogue of Campana’s orbifold conjecture by proving -type bounds for Campana points on projective toric varieties with large boundary multiplicities, then extending these methods to show Campana’s conjecture for finite coverings of such toric varieties. The core strategy combines a function-field version of Vojta’s philosophy with the main technical GNSW -unit framework, plus local Weil functions for Campana integral points. The results yield finiteness-type (non-density) statements for Campana integral points on orbifolds of general type arising as toric covers and their finite ramified extensions, with effective control of exceptional sets. This significantly broadens the classes of varieties and orbifold structures for which Campana–Lang–Vojta-type predictions hold over function fields, with potential explicit diophantine applications.

Abstract

We first prove Vojta's abc conjecture over function fields for Campana points on projective toric varieties with high multiplicity along the boundary. As a consequence, we obtain a version of Campana's conjecture on finite coverings of projective toric varieties over function fields.
Paper Structure (7 sections, 8 theorems, 67 equations)

This paper contains 7 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and heights
  4. Local Weil Functions and Campana integral points
  5. Proof of Theorem \ref{['main_thm']}
  6. Proof of Theorem \ref{['toric']}
  7. Proof of Theorem \ref{['toricGG']} and Theorem \ref{['vojtaconj']}

Key Result

Theorem 1.9

Let $X$ be an admissible toric variety of dimension $n\ge 2$. Let $D$ be an effective reduced divisor on $X$ whose support is in general position with the boundary $X\setminus \mathbb G_m^n$. Let $A$ be a big divisor on $X$. Then, for every $\epsilon >0$, there exists a positive integer $\ell$ such for any $P\in \mathcal{R}\setminus Z$.

Theorems & Definitions (27)

  • Conjecture 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 17 more