IJPAM: Volume 2, No. 2 (2002)

WEAKLY DEFECTIVE PROJECTIVE VARIETIES

E. Ballico
Dept. of Mathematics
University of Trento
38050 Povo (Trento) - Via Sommarive 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Fix positive integers $s$ and $a_i$, $1 \le i \le s$. Set $\alpha := (a_1,\dots ,a_s)$. Let $X_i \subset {\bf {P}}^N$, $1 \le i \le s$, be irreducible subvarieties. The $s$-ple $(X_1,\dots ,X_s)$ is called $\alpha$-defective if the join of $a_1$ copies of $X_i$, $a_2$ copies of $X_2$, and so on until we take the join with $a_s$ copies of $X_s$ has not the expected dimension. Fix general $a_i$-ples $(P_{i,1},\dots ,P_{i,a_{i}})\in X_i^{a_i}$ and let ${\bf {X}}(P_{1,1},\dots ,P_{s,a_{s}})$ be the linear span of all tangent spaces $T_{P_{i,x_{i}}}X_i$, $1 \le i \le s$, $1 \le x_i \le a_i$. $(X_1,\dots ,X_s)$ is called weakly $\alpha$-defective if there is a pair $(i,j)$ such that for a general hyperplane $H$ containing ${\bf {X}}(P_{1,1},\dots ,P_{s,a_{s}})$ there is a positive dimensional variety $\Phi$ such that $P_{i,j}\in \Phi \subseteq
X_i$ and $H$ is tangent to $X_i$ at a general point of $\Phi$. Here we prove that $\alpha$-defectivity implies weakly $\alpha$-defectivity and classify the weakly defective $s$-ples such that $\mbox{dim}(X_s) = 1$.

Received: June 6, 2002

AMS Subject Classification: 14N05

Key Words and Phrases: projective varieties, joins, contact locus, defective projective varieties, weakly defective projective varieties

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 2
Issue: 2