Higher Secant Varieties of ℙⁿ × ℙ¹ Embedded in Bi-Degree (a, b)

Abstract

In this article, we compute the dimension of all the higher secant varieties to the Segre–Veronese embedding of ℙ ⁿ × ℙ¹ via the section of the sheaf 𝒪(a, b) for any n, a, b ∈ ℤ⁺. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann (1, s − 1)-defective Veronese varieties.

Key Words:

• Dimensions
• Secant Varieties
• Segre–Veronese

2010 Mathematics Subject Classification:

• 15A69
• 14Q15
• 14N05

ACKNOWLEDGMENTS

All authors supported by MIUR funds. The second author was partially supported by CIRM–FBK (Centro Internazionale Ricerca Matematica, Fondazione Bruno Kessler), Trento, Italy; by Project Galaad, INRIA (Institut national de recherche en informatique et en automatique), Sophia Antipolis Méditerranée, France; Institut Mittag-Leffler (Sweden); and by International Individual Marie Curie Fellowship (IFP7 Program).

Notes

Communicated by R. Piene.