Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic bigness and a uniform Bogomolov-type result

Xinyi Yuan

Abstract

In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a different approach to the uniform Mordell-Lang type of result of Dimitrov-Gao-Habegger and Kuhne. The treatment is based on the recent theory of adelic line bundles of Yuan-Zhang.

Arithmetic bigness and a uniform Bogomolov-type result

Abstract

In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a different approach to the uniform Mordell-Lang type of result of Dimitrov-Gao-Habegger and Kuhne. The treatment is based on the recent theory of adelic line bundles of Yuan-Zhang.

Paper Structure

This paper contains 61 sections, 40 theorems, 412 equations.

Table of Contents

  1. Introduction
  2. Uniform Bogomolov-type result
  3. Potential bigness
  4. Admissible canonical bundle
  5. Bigness of admissible canonical bundle
  6. Uniform fiberwise bigness
  7. Notations and terminology
  8. Admissible adelic line bundles
  9. Review on adelic line bundles
  10. Admissible adelic line bundles
  11. Basics on Jacobian schemes
  12. Admissible extensions
  13. Construction of canonical admissible extensions
  14. Intersection and pull-back
  15. Adjunction formula and Hodge index theorem
  16. ...and 46 more sections

Key Result

Theorem 1.1

Let $g>1$ be an integer. Then there are constants $c_1,c_2>0$ depending only on $g$ satisfying the following properties. Let $K$ be either a number field or a function field of one variable over a field $k$. Then for any geometrically integral, smooth and projective curve $C$ of genus $g$ over $K$,

Theorems & Definitions (79)

  • Theorem 1.1: Theorem \ref{['small points2']}
  • Theorem 1.2: Theorem \ref{['bigness7']}
  • Theorem 1.3: Theorem \ref{['bigness5']}
  • Theorem 1.4: Theorem \ref{['fiberwise2']}, Theorem \ref{['fiberwise1']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Definition 2.4
  • ...and 69 more