Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

TL;DR

This paper explores the connection between the SHGH Conjecture and the rationality of one-point Seshadri constants on blow-ups of the projective plane, using Newton-Okounkov bodies to test rationality.

Contribution

It introduces a novel approach linking the SHGH Conjecture to Seshadri constants via Newton-Okounkov bodies, providing a new perspective for testing rationality.

Findings

Established a relationship between the SHGH Conjecture and Seshadri constant rationality.

Proposed a method to test rationality using functions on Newton-Okounkov bodies.

Clarified the role of Okounkov functions in understanding Seshadri constants.

Abstract

In this note we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane, and explain how rationality of Seshadri constants can be tested with the help of functions on Newton--Okounkov bodies.