Intersection numbers as mixed volumes of Newton-Okounkov bodies
Robert Wilms
TL;DR
The paper establishes a precise link between intersection theory and convex geometry by showing that for ample $\mathbb{Q}$-line bundles $L_1,\dots,L_d$ on an irreducible projective variety $X$, there exists an admissible flag $Y_\bullet$ such that $(L_1\cdots L_d)=d!\,V(\Delta_{Y_\bullet}(L_1),\dots,\Delta_{Y_\bullet}(L_d))$, where $\Delta_{Y_\bullet}(L_i)$ are Newton–Okounkov bodies. The construction uses Bertini to produce the flag and leverages the slice formula for Newton–Okounkov bodies together with a key mixed-volume lemma to prove the equality; the approach extends to a family of line bundles inside a convex cone and to big line bundles via positive intersection numbers. The work translates algebraic intersection inequalities into convex-geometry inequalities, allowing derivation of corollaries on positive intersection numbers and giving a framework to compare intersection products through valuations and Newton–Okounkov bodies. Overall, the paper provides a concrete, flag-dependent convex-geometric expression for intersection numbers and demonstrates how mixed volumes of NO-bodies encode rich intersection-theoretic information.
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 by the mixed volume $V(Δ_{Y_\bullet}(L_1),\dots,Δ_{Y_\bullet}(L_d))$ of their Newton--Okounkov bodies. The admissible flag $Y_\bullet$ of subvarieties is constructed by sections of the line bundles using Bertini's theorem, leaving a small possibility of changing 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.