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Weakly-special threefolds and non-density of rational points

Finn Bartsch, Ariyan Javanpeykar, Erwan Rousseau

TL;DR

This paper advances Campana’s conjectures by proving a function-field non-density statement for rational points on Bogomolov–Tschinkel BTCP-threefolds. It develops a comprehensive orbifold framework, defining Campana orbifolds, their Chern classes, and the moduli space of orbifold maps, and proves a finite-type, low-dimensional moduli result using an orbifold version of Kobayashi–Ochiai and Bogomolov-type hyperbolicity. The key strategy reduces to examining curves on a Kodaira dimension one base (B,Δ) with a non-isotrivial elliptic fibration, establishes finiteness of non-constant orbifold maps, and then lifts this finiteness to higher-dimensional settings via cutting arguments. Consequently, for BT threefolds X over finitely generated fields, the union of graphs of non-constant maps V→X is not dense in X×V, highlighting a robust disparity between weakly-special and special varieties and contributing to the broader understanding of rational points in the Campana program.

Abstract

We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish fundamental properties of moduli spaces of orbifold maps, and prove a dimension bound for such moduli spaces by using the recent extension of Kobayashi-Ochiai's finiteness theorem for Campana's orbifold pairs.

Weakly-special threefolds and non-density of rational points

TL;DR

This paper advances Campana’s conjectures by proving a function-field non-density statement for rational points on Bogomolov–Tschinkel BTCP-threefolds. It develops a comprehensive orbifold framework, defining Campana orbifolds, their Chern classes, and the moduli space of orbifold maps, and proves a finite-type, low-dimensional moduli result using an orbifold version of Kobayashi–Ochiai and Bogomolov-type hyperbolicity. The key strategy reduces to examining curves on a Kodaira dimension one base (B,Δ) with a non-isotrivial elliptic fibration, establishes finiteness of non-constant orbifold maps, and then lifts this finiteness to higher-dimensional settings via cutting arguments. Consequently, for BT threefolds X over finitely generated fields, the union of graphs of non-constant maps V→X is not dense in X×V, highlighting a robust disparity between weakly-special and special varieties and contributing to the broader understanding of rational points in the Campana program.

Abstract

We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish fundamental properties of moduli spaces of orbifold maps, and prove a dimension bound for such moduli spaces by using the recent extension of Kobayashi-Ochiai's finiteness theorem for Campana's orbifold pairs.
Paper Structure (17 sections, 38 theorems, 33 equations)

This paper contains 17 sections, 38 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. Campana's orbifold pairs
  4. Chern classes
  5. Main result
  6. The moduli space of orbifold maps
  7. Vanishing of sections of line bundles
  8. Families of orbifold maps
  9. The Hom-scheme of orbifold maps
  10. The dimension of the moduli space of orbifold maps
  11. Bend-and-break and orbifold Kobayashi--Ochiai
  12. Pseudo-bounded orbifold pairs
  13. Kodaira dimension one and rational points on moduli spaces of orbifold maps
  14. Bogomolov's theorem in the orbifold setting
  15. A cutting argument and the proof of Theorem \ref{['thm:main2']}
  16. ...and 2 more sections

Key Result

Theorem 1.4

The following statements hold.

Theorems & Definitions (89)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Theorem 1.4: Bogomolov--Tschinkel, Campana--Păun, Campana--Winkelmann
  • Theorem A
  • Theorem B: Bogomolov's theorem for orbifold surfaces
  • Definition 2.1
  • Definition 2.2: Orbifold base
  • Definition 2.3: Morphisms
  • Definition 2.4
  • ...and 79 more