IJPAM: Volume 50, No. 3 (2009)

ON THE SECANT VARIETIES TO THE TANGENT
DEVELOPABLE OF VERONESE VARIETIES

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]


Abstract.Let $V_{n,d} \subseteq {\bf {P}}^N$, $N:= \binom{n+d}{n}-1$, be the order $d$ Veronese embedding of ${\bf {P}}^n$, $X_{n,d}:= T(V_{n,d}) \subseteq {\bf {P}}^N$ the tangent developable of $V_{n,d}$ and $S^{s-1}(X_{n,d}) \subseteq {\bf {P}}^N$ the $s$-secant variety of $X_{n,d}$, i.e. the closure in ${\bf {P}}^N$ of the union of all $(s-1)$-linear spaces spanned by $s$ points of $X_{n,d}$. $S^{s-1}(X_{n,d})$ has expected dimension $\min \{N,(2n+1)s-1\}$. Catalisano, Geramita and Gimigliano conjectured that $S^{s-1}(X_{n,d})$ has always the expected dimension, except when $d=2$, $n\ge 2s$ or $d=3$ and $n=2,3,4$. In this paper we prove their conjecture when $n=4$ and $n=5$, $d \ge 4$, and an asympotic case of the conjecture for all $n \ge 6$.

Received: September 16, 2008

AMS Subject Classification: 14N05

Key Words and Phrases: tangent developable, secant variety, Veronese variety, fat point, zero-dimensional scheme, postulation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 3