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Towards Lang--Vojta via Degeneration

Ryan C. Chen, Natalia Garcia-Fritz, Siddharth Mathur, Hector Pasten

TL;DR

This work advances the Lang--Vojta program by developing a degeneration framework that transfers finiteness and non-density results for $(D,S)$-integral points from arithmeticly hyperbolic moduli spaces to ambient varieties via degenerating classifying maps. A central theme is constructing geometrically irreducible divisors $D$, often as duals of curves, so that $(X,D)$ is arithmetically hyperbolic or $(D,S)$-integral points are finite; this includes explicit families such as duals of smooth plane curves (e.g., the Fermat plane cubic) and shows that every projective variety of dimension at least $2$ admits such a divisor. The strategy weaves together moduli spaces, Hilbert schemes, and GIT quotients with analytic degeneration arguments for marked curves, producing a flexible mechanism (Theorem MainCriterion) to obtain Zariski degeneracy or arithmetic hyperbolicity for a wide class of complements. Overall, the results provide concrete, explicit geometric constructions of arithmetically hyperbolic pairs and offer a positive answer to questions about the existence of geometrically irreducible divisors with limited integral points, thus contributing significantly to the Lang--Vojta landscape and its applications to Diophantine geometry.

Abstract

Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of $S$-integral points of varieties over number fields $k$, including many cases with geometrically irreducible boundary divisors. Our approach builds on the study of arithmetic and geometric properties of moduli spaces of curves with extra structure. As an application, we provide families of explicit examples of geometrically irreducible divisors on the projective plane (such as the dual of any smooth curve of degree at least $3$), with respect to which the sets of $S$-integral points are finite. Answering a question of Achenjang and Morrow, we show that, other than the case of curves, every normal projective variety admits a geometrically irreducible divisor $D$ for which finiteness of $(D,S)$-integral points holds over every finite extension of $k$.

Towards Lang--Vojta via Degeneration

TL;DR

This work advances the Lang--Vojta program by developing a degeneration framework that transfers finiteness and non-density results for -integral points from arithmeticly hyperbolic moduli spaces to ambient varieties via degenerating classifying maps. A central theme is constructing geometrically irreducible divisors , often as duals of curves, so that is arithmetically hyperbolic or -integral points are finite; this includes explicit families such as duals of smooth plane curves (e.g., the Fermat plane cubic) and shows that every projective variety of dimension at least admits such a divisor. The strategy weaves together moduli spaces, Hilbert schemes, and GIT quotients with analytic degeneration arguments for marked curves, producing a flexible mechanism (Theorem MainCriterion) to obtain Zariski degeneracy or arithmetic hyperbolicity for a wide class of complements. Overall, the results provide concrete, explicit geometric constructions of arithmetically hyperbolic pairs and offer a positive answer to questions about the existence of geometrically irreducible divisors with limited integral points, thus contributing significantly to the Lang--Vojta landscape and its applications to Diophantine geometry.

Abstract

Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of -integral points of varieties over number fields , including many cases with geometrically irreducible boundary divisors. Our approach builds on the study of arithmetic and geometric properties of moduli spaces of curves with extra structure. As an application, we provide families of explicit examples of geometrically irreducible divisors on the projective plane (such as the dual of any smooth curve of degree at least ), with respect to which the sets of -integral points are finite. Answering a question of Achenjang and Morrow, we show that, other than the case of curves, every normal projective variety admits a geometrically irreducible divisor for which finiteness of -integral points holds over every finite extension of .
Paper Structure (18 sections, 24 theorems, 29 equations)

This paper contains 18 sections, 24 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. The Lang--Vojta conjecture
  3. Arithmetic results
  4. Degenerating classifying maps
  5. Previous work
  6. Conventions
  7. Preliminaries
  8. Integral points
  9. Moduli spaces and their arithmetic
  10. The Hilbert scheme
  11. Moduli stacks of curves with a multi-section
  12. GIT quotients
  13. Arithmetic hyperbolicity of moduli
  14. Duals of smoothly branched curves
  15. Analytic degenerations of marked points on isotrivial families of curves
  16. ...and 3 more sections

Key Result

Theorem 1.2

Fix a number field $k$ and let $C \subset \mathbb{P}^n$ be a geometrically integral smoothly branched projective curve of geometric genus $g \geq 1$, defined over $k$, which is not contained in any hyperplane. Then the pair $(\mathbb{P}^n, C^*)$ is arithmetically hyperbolic.

Theorems & Definitions (88)

  • Conjecture 1.1: The Lang--Vojta conjecture
  • Theorem 1.2: see Theorem \ref{['thm:deg4']}
  • Theorem 1.3: see Proposition \ref{['lemma:dualrational']}
  • Example 1.4
  • Theorem 1.5: see Corollary \ref{['cor:morrowachen']}
  • Example 1.7
  • Example 1.8
  • Theorem 1.9: see Theorem \ref{['ThmMainCriterion']}
  • Definition 2.1
  • Remark 2.2
  • ...and 78 more