A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes

Alicia Dickenstein; Benjamin Nill; Michèle Vergne

doi:10.1016/j.crma.2012.02.001

Combinatorics

A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes

Presented by: Claire Voisin

Alicia Dickenstein ¹; Benjamin Nill ²; Michèle Vergne ³

¹ Departamento de Matemática, FCEN, Universidad de Buenos Aires and IMAS, CONICET, Ciudad Universitaria, Pab I, (C1428EGA) Buenos Aires, Argentina
² Case Western Reserve University, Department of Mathematics, 10900, Euclid Avenue, Cleveland, OH 44106, USA
³ Institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 229-233.

Abstract
Résumé

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $⌈ \frac{\dim P}{2} ⌉$ .

Nous donnons une formule pour le degré du discriminant dʼune variété torique projective non singulière associée à un polytope entier P, en terme du nombre de points entiers des intérieurs de dilatations de faces de dimension supérieure ou égale à $⌈ \frac{\dim P}{2} ⌉$ .

Received: 2011-12-12
Accepted: 2012-02-02
Published online: 2012-02-23

DOI: 10.1016/j.crma.2012.02.001

Author's affiliations:

Alicia Dickenstein ¹; Benjamin Nill ²; Michèle Vergne ³

¹ Departamento de Matemática, FCEN, Universidad de Buenos Aires and IMAS, CONICET, Ciudad Universitaria, Pab I, (C1428EGA) Buenos Aires, Argentina
² Case Western Reserve University, Department of Mathematics, 10900, Euclid Avenue, Cleveland, OH 44106, USA
³ Institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France

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     author = {Alicia Dickenstein and Benjamin Nill and Mich\`ele Vergne},
     title = {A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--233},
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Alicia Dickenstein; Benjamin Nill; Michèle Vergne. A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 229-233. doi : 10.1016/j.crma.2012.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.02.001/

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