Convex Bodies associated to Linear series of Adelic Divisors on Quasi-Projective Varieties

TL;DR

This paper introduces convex bodies associated with linear series of adelic divisors on quasi-projective varieties, extending geometric methods to study volumes and base loci in arithmetic geometry.

Contribution

It develops a convex geometric framework for adelic divisors, analyzing volumes and base loci using methods inspired by Lazarsfeld and Mustata.

Findings

Properties of volumes of adelic divisors are established.

Convex bodies provide a new approach to restricted volumes.

Analogous properties for base loci are derived.

Abstract

In this article we define and study convex bodies associated to linear series of adelic divisors over quasi-projective varieties that have been introduced recently by Xinyi Yuan and Shou-Wu Zhang. Restricting our attention to big adelic divisors, we deduce properties of volumes obtained by Yuan and Zhang using different convex geometric arguments. We go on to define augmented base loci and restricted volumes of adelic divisors following the works of Michael Nakamaye and develop a similar study using convex bodies to obtain analogous properties for restricted volumes. We closely follow methods developed originally by Robert Lazarsfeld and Mircea Mustata.