Dickenstein, Alicia; Di Rocco, Sandra; Piene, Ragni
Higher order duality and toric embeddings  [ Dualité d’ordre supérieur et immersions toriques ]
Annales de l'institut Fourier, Tome 64 (2014) no. 1 , p. 375-400
MR 3330552 | Zbl 06387278
doi : 10.5802/aif.2851
URL stable : http://www.numdam.org/item?id=AIF_2014__64_1_375_0

Classification:  14M25,  14T05
Mots clés: variété torique, dualité projective d’ordre supérieur, tropicalisation
La notion de variété duale d’ordre supérieur d’une variété projective, introduite par Piene en 1983, est une généralisation naturelle de la notion classique de dualité projective. Dans cet article, nous étudions les variétés duales d’ordre supérieur d’une immersion torique projective. Nous exprimons le degré de la variété duale d’ordre 2 d’une immersion 2-jet régulière, lisse et de dimension 3 en termes géometriques et combinatoires, et nous donnons une classification des variétés ayant une variété duale d’ordre 2 de dimension plus petite que celle attendue. Nous décrivons aussi la tropicalisation des variétés duales de tout ordre d’une variété torique immergée de façon équivariante (pas nécessairement normale). Dedicated to the memory of our friend Mikael Passare (1959–2011)
The notion of higher order dual varieties of a projective variety, introduced by Piene in 1983, is a natural generalization of the classical notion of projective duality. In this paper we study higher order dual varieties of projective toric embeddings. We express the degree of the second dual variety of a 2-jet spanned embedding of a smooth toric threefold in geometric and combinatorial terms, and we classify those whose second dual variety has dimension less than expected. We also describe the tropicalization of all higher order dual varieties of an equivariantly embedded (not necessarily normal) toric variety.

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