Intersection numbers as mixed volumes of Newton-Okounkov bodies

Fuente: arXiv
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Autore principale: Wilms, Robert
Natura: Preprint
Pubblicazione: 2025
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author Wilms, Robert
author_facet Wilms, Robert
contents 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 $V(Δ_{Y_\bullet}(L_1),\dots,Δ_{Y_\bullet}(L_d))$ of their Newton-Okounkov bodies. The admissible flag $Y_\bullet$ of subvarieties is constructed 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.
format Preprint
id arxiv_https___arxiv_org_abs_2502_18441
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle Intersection numbers as mixed volumes of Newton-Okounkov bodies
Wilms, Robert
Algebraic Geometry
14M25, 14C17, 52A39
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 $V(Δ_{Y_\bullet}(L_1),\dots,Δ_{Y_\bullet}(L_d))$ of their Newton-Okounkov bodies. The admissible flag $Y_\bullet$ of subvarieties is constructed 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.
title Intersection numbers as mixed volumes of Newton-Okounkov bodies
topic Algebraic Geometry
14M25, 14C17, 52A39
url https://arxiv.org/abs/2502.18441