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Transactions of the American Mathematical Society

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On non-secant defectivity of Segre-Veronese varieties

Authors: Carolina Araujo, Alex Massarenti and Rick Rischter
Journal: Trans. Amer. Math. Soc. 371 (2019), 2255-2278
MSC (2010): Primary 14N05, 14N15; Secondary 14E05, 15A69
Published electronically: August 9, 2018
MathSciNet review: 3896080
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Abstract: Let $ SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese variety given as the image of $ \mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_r}$ under the embedding induced by the complete linear system $ \big \vert\mathcal {O}_{\mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_r}}(d_1,\dots , d_r)\big \vert$. We prove that asymptotically $ SV^{\pmb n}_{\pmb d}$ is not $ h$-defective for $ h\leq (\min \{n_i\})^{\lfloor \log _2(d-1)\rfloor }$, where $ d = d_1+\dots +d_r$.

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Additional Information

Carolina Araujo
Affiliation: IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil

Alex Massarenti
Affiliation: Universidade Federal Fluminense, Rua Mário Santos Braga, 24020-140, Niterói, Rio de Janeiro, Brazil
Address at time of publication: Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy

Rick Rischter
Affiliation: Universidade Federal de Itajubá, Av. BPS 1303, Bairro Pinheirinho, Itajubá, Minas Gerais, Brazil

Keywords: Segre-Veronese varieties, secant varieties, osculating spaces, secant defect, degenerations of rational maps
Received by editor(s): February 20, 2017
Received by editor(s) in revised form: June 7, 2017
Published electronically: August 9, 2018
Additional Notes: The first named author was partially supported by CNPq and Faperj Research Fellowships.
The second named author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica “F. Severi” (GNSAGA-INDAM)
The third named author would like to thank CNPq for the financial support.
Article copyright: © Copyright 2018 American Mathematical Society