Extended Okounkov bodies and multi-point Seshadri constants

Jaesun Shin

Published 2017-10-12Version 1

Based on the work of Okounkov, Lazarsfeld-Mustata and Kaveh-Khovanskii independently associated a convex body, called the Okounkov body, to a big divisor on a smooth 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 various properties and their shapes, and observe local positivity data via them. Finally, we give some applications, especially for the Nagata conjecture.