Higher order duality and toric embeddings

Dickenstein, Alicia; Di Rocco, Sandra; Piene, Ragni

doi:10.5802/aif.2851

Higher order duality and toric embeddings
[Dualité d’ordre supérieur et immersions toriques]

Dickenstein, Alicia ¹ ; Di Rocco, Sandra ² ; Piene, Ragni ³

¹ Department of Mathematics, FCEN Universidad de Buenos Aires and IMAS - CONICET Ciudad Universitaria - Pab. I C1428EGA Buenos Aires Argentina
² Department of Mathematics Royal Institute of Technology (KTH) 10044 Stockholm Sweden
³ CMA/Department of Mathematics University of Oslo P.O.Box 1053 Blindern NO-0316 Oslo Norway

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 375-400.

Résumé
Abstract

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.

MR Zbl

DOI : 10.5802/aif.2851

Classification : 14M25, 14T05
Keywords: toric variety, higher order projective duality, tropicalization
Mot clés : variété torique, dualité projective d’ordre supérieur, tropicalisation

Affiliations des auteurs :

Dickenstein, Alicia ¹ ; Di Rocco, Sandra ² ; Piene, Ragni ³

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     title = {Higher order duality and toric embeddings},
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     pages = {375--400},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Dickenstein, Alicia; Di Rocco, Sandra; Piene, Ragni. Higher order duality and toric embeddings. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 375-400. doi : 10.5802/aif.2851. http://www.numdam.org/articles/10.5802/aif.2851/

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