# Concave transforms of filtrations and rationality of Seshadri constants

## TL;DR

This paper links concave transforms of filtrations to Newton--Okounkov bodies, providing a rationality criterion for Seshadri constants and introducing new geometric tools for semigroup analysis.

## Contribution

It establishes a connection between concave transforms and Newton--Okounkov bodies, and offers a rationality criterion for Seshadri constants based on this framework.

## Key findings

- Subgraph of concave transform equals Newton--Okounkov body of a semigroup
- Associated graded algebra corresponds to sections of a line bundle in higher dimension
- Provides a rationality criterion for Seshadri constants

## Abstract

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.