Intersection numbers as mixed volumes of Newton-Okounkov bodies

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.