# Arithmetic ampleness and an arithmetic Bertini theorem

## TL;DR

This paper proves that for a projective arithmetic variety with an ample hermitian line bundle, the proportion of effective sections defining irreducible divisors approaches 1 as the tensor power increases, extending to related schemes.

## Contribution

It establishes an arithmetic Bertini theorem showing generic irreducibility of divisors from high tensor powers of ample hermitian line bundles on arithmetic varieties.

## Key findings

- Proportion of irreducible divisors tends to 1 as n increases.
- Provides bounds for effective sections of hermitian line bundles.
- Discusses properties and restriction theorems of arithmetic ampleness.

## Abstract

Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal L}^{\otimes n}$ that define an irreducible divisor on $\mathcal X$ tends to $1$ as $n$ tends to $\infty$. We prove variants of this statement for schemes mapping to such an $\mathcal X$. On the way to these results, we discuss some general properties of arithmetic ampleness, including restriction theorems, and upper bounds for the number of effective sections of hermitian line bundles on arithmetic varieties.