Volume of line bundles via valuation vectors (different from Okounkov bodies)

TL;DR

This paper introduces a new volume formula for big line bundles that combines valuation vectors from multiple flags, extending the classical Okounkov body approach.

Contribution

It presents a novel volume computation method using mixed valuation data from different flags, differing from traditional single-flag Okounkov body techniques.

Findings

New volume formula valid in the ample cone

Uses valuation vectors from multiple flags

Extends Okounkov body volume interpretation

Abstract

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a different volume formula, valid in the ample cone, also based on top rank valuation vectors, but mixing data along several different flags.