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: ballico@science.unitn.it

Abstract.Let $V_{n,d} \subseteq {\bf {P}}^N$ , $N:= \binom{n+d}{n}-1$ , be the order 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 -secant variety of $X_{n,d}$ , i.e. the closure in ${\bf {P}}^N$ of the union of all -linear spaces spanned by 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 , $n\ge 2s$ or and . In this paper we prove their conjecture when and , $d \ge 4$ , and an asympotic case of the conjecture for all $n \ge 6$ .