G\'eom\'etrie, points entiers et courbes enti\`eres

TL;DR

This paper proves that on certain algebraic varieties over number fields or complex numbers, the set of quasi-integral points or integral curves outside a large divisor is not Zariski dense, revealing geometric restrictions on such points.

Contribution

It establishes non-density results for quasi-integral points and integral curves on projective varieties with respect to divisors composed of many positive components.

Findings

Quasi-integral points outside the divisor are not Zariski dense over number fields.

Integral curves outside the divisor are not Zariski dense over complex numbers.

The results connect the geometry of divisors with arithmetic and complex-analytic properties.

Abstract

Let $X$ be a projective variety over a number field $K$ (resp. over $\mathbb{C}$). Let $H$ be the sum of ``sufficiently many positive divisors'' on $X$. We show that any set of quasi-integral points (resp. any integral curve) in $X-H$ is not Zariski dense.