Extended Okounkov bodies and multi-point Seshadri constants

TL;DR

This paper introduces extended convex bodies for big divisors on projective varieties to analyze multi-point local positivity, providing new geometric tools and insights into Seshadri constants.

Contribution

It develops a new convex body framework for multi-point positivity, extending Okounkov bodies to handle multiple points simultaneously.

Findings

Extended convex bodies effectively describe multi-point local positivity.

The shape of these bodies encodes positivity data at multiple points.

The paper demonstrates the irrationality of Seshadri constants using these tools.

Abstract

Based on the work of Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata independently associated a convex body, called the Okounkov body, to a big divisor on a normal projective variety with respect to an admissible flag. Although the Okounkov bodies carry rich positivity data of big divisors, they only provide information near a single point. The purpose of this paper is to introduce a convex body of a big divisor that is effective in handling the positivity theory associated with multi-point settings. These convex bodies open the door to approach the local positivity theory at multiple points from a convex-geometric perspective. We study their properties and shapes, and describe local positivity data via them. Finally, we observe the irrationality of Seshadri constants with the help of a relation between Nakayama constants and Seshadri constants.