Intersection numbers as mixed volumes of Newton-Okounkov bodies

TL;DR

This paper establishes a geometric interpretation of intersection numbers of ample line bundles on projective varieties as mixed volumes of their associated Newton-Okounkov bodies, linking algebraic geometry with convex geometry.

Contribution

It introduces a method to express intersection numbers as mixed volumes of Newton-Okounkov bodies using a flexible flag construction from line bundle sections.

Findings

Intersection numbers equal mixed volumes of Newton-Okounkov bodies.

Construction of flags from line bundle sections via Bertini's theorem.

Application of slice formula and convex geometry in proof.

Abstract

In this paper we express any intersection number $(L_1\cdot\ldots\cdot L_d)$ of ample line bundles on an irreducible projective variety as the mixed volume $V(\Delta_{Y_\bullet}(L_1),\dots,\Delta_{Y_\bullet}(L_d))$ of their Newton-Okounkov bodies. The admissible flag $Y_\bullet$ of subvarieties is constructed from sections of the line bundles using Bertini's theorem, allowing some flexibility to vary the line bundles after the flag is fixed. The proof relies on the slice formula for Newton-Okounkov bodies and on mixed-volume calculations in convex geometry.