# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.04317