Towards Lang--Vojta via Degeneration

TL;DR

This paper advances the Lang--Vojta conjecture by proving finiteness and degeneracy results for $S$-integral points on varieties, using moduli space techniques, and provides explicit examples of divisors with finite integral points.

Contribution

It introduces new finiteness results for $S$-integral points on varieties with irreducible boundary divisors, expanding the understanding of the Lang--Vojta conjecture.

Findings

Finiteness of $S$-integral points on certain varieties.

Construction of explicit divisors with finite integral points.

Every normal projective variety admits a divisor with finite $S$-integral points.

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 $k$.