# On the degeneracy of integral points and entire curves in the complement of nef effective divisors

## TL;DR

This paper proves a strong degeneracy result for integral points and entire curves in the complement of nef effective divisors, using a generalized Schmidt's subspace theorem and weak positivity assumptions.

## Contribution

It introduces a novel degeneracy theorem under weak positivity conditions and explores connections with hyperbolicity and divisor bounds.

## Key findings

- Degeneracy of integral points under weak positivity assumptions
- Analogous results for entire curves in complex geometry
- Conjecture on optimal divisor bounds for hyperbolicity

## Abstract

As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a Conjecture regarding (optimal) bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.