Estimation of arithmetic linear series

TL;DR

This paper introduces a method to estimate arithmetic linear series using Okounkov's idea, extending classical results to the arithmetic setting and providing tools for arithmetic volume analysis.

Contribution

It develops a general approach to estimate arithmetic linear series based on Yuan's ideas, extending Fujita's approximation theorem to the arithmetic context.

Findings

Established an arithmetic analogue of Fujita's approximation theorem

Introduced the concept of arithmetic linear series

Provided a method to estimate arithmetic restricted volumes

Abstract

Lazarsfeld and Mustata propose general and systematic usage of Okounkov's idea in order to study asymptotic behavior of linear series on an algebraic variety. It is a very simple way, but it yields a lot of consequences, like Fujita's approximation theorem. Yuan generalized this way to the arithmetic situation, and he established the arithmetic Fujita's approximation theorem, which was also proved by Chen independently. In this paper, we introduce arithmetic linear series and give a general way to estimate them based on Yuan's idea. As an application, we consider an arithmetic analogue of the algebraic restricted volumes.