The $\theta$-density in Arakelov geometry

TL;DR

This paper introduces a new $ heta$-density concept for global sections of Hermitian line bundles on arithmetic varieties, extending classical density results and proving analogues of Bertini theorems in Arakelov geometry.

Contribution

It constructs a $ heta$-density in Arakelov geometry and establishes its properties, including analogues of Bertini theorems for irreducibility and regularity.

Findings

$ heta$-density behaves similarly to classical density in Arakelov setting

Proves $ heta$-density analogues of Bertini theorems

Extends understanding of section distributions in arithmetic geometry

Abstract

In this article, we construct a $\theta$-density for the global sections of ample Hermitian line bundles on a projective arithmetic variety. We show that this density has similar behaviour to the usual density in the Arakelov geometric setting, where only global sections of norm smaller than $1$ are considered. In particular, we prove the analogue by $\theta$-density of two Bertini kind theorems, on irreducibility and regularity respectively.