IJPAM: Volume 16, No. 1 (2004)

TRIPLE POINTS AND THE WEAK NON-DEFECTIVITY
OF VERONESE EMBEDDINGS OF
PROJECTIVE SPACES

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Fix integers $n \ge 2$, $d \ge 4$ and $k \ge 0$ such that \begin{equation*}
k(n+1) + (n+2)(n+1)/2 \le \binom{n+d}{n}
\end{equation*} and $k+1$ general points $P_1,\dots ,P_{k+1}\in {\bf {P}}^n$. Let $\Sigma (P_1,\dots ,P_{k+1})$ (resp. $\Sigma (P_1,\dots ,P_{k+1})'$) denote the projective space of all degree $d$ hypersurfaces of ${\bf {P}}^n$ singular at each point $P_i$, $1 \le i \le k+1$, (resp. singular at each $P_i$ and with a triple point at $P_{k+1}$). Here we use Horace Method to prove that $\mbox{\rm dim}(\Sigma (P_1,\dots ,P_{k+1})') = \binom{n+d}{n}
-(n+1)k - (n+2)(n+1)/2-1$, $\mbox{\rm dim}(\Sigma (P_1,\dots ,P_{k+1})) = \binom{n+d}{n} -(k+1)(n+1)-1$ and (in characteristic zero) a general $F\in \Sigma (P_1,\dots ,P_{k+1})$ satisfies $\mbox{\rm Sing}(F)
= \{P_1,\dots ,P_{k+1}\}$ and it has an ordinary node at each $P_i$. This result implies that the order $d$ Veronese embedding of ${\bf {P}}^n$ is not weakly $k$-defective in the sense of Ciliberto and Chiantini.

Received: July 5, 2004

AMS Subject Classification: 14N05

Key Words and Phrases: Veronese variety, weakly defective variety, zero-dimensional scheme, double point, fat point, Veronese embedding

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 16
Issue: 1