IJPAM: Volume 70, No. 4 (2011)

ON THE DEFECTIVITY OF JOINS OF VERONESE
VARIETIES AND TANGENTIAL VARIETIES TO THEM

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract. Let $X_{n,k}\subset \mathbb {P}^r$, $r:= \binom{n+k}{n}-1$ be the order $k$ Veronese embedding of $\mathbb {P}^n$ and $\tau (X_{n,k}) \subset \mathbb {P}^r$ the tangential variety of $X_{n,k}$. Let $\tau (X_{n,k}(a,b)) \subseteq \mathbb {P}^r$ be the join of $a$ copies of $X_{n,k}$ and $b$ copies of $\tau (X_{n,k})$. Here we conjecture that $\tau (X_{n,k}(a,b))$ has always the expected dimension if $k\ge 5$. We prove this conjecture if $2\le n\le 4$. We use in an essential way a paper by Catalisano, Geramita and Gimigliano.

Received: April 1, 2011

AMS Subject Classification: 14N05, 14J99

Key Words and Phrases: secant variety, Veronese variety, tangential variety, join

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 4