IJPAM: Volume 16, No. 1 (2004)

GENERALIZED SECANT VARIETIES AND
JOINS OF PROJECTIVE CURVES

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


Abstract.Let $X\subset {\bf {P}}^r$ be an integral non-degenerate variety. For any integer $t\ge 2$ let $S^{\{t\}}(X)
\subseteq {\bf {P}}^r$ denote the closure in ${\bf {P}}^r$ of the union of all $(t-1)$-dimensional linear spaces spanned by a length $t$ zero-dimensional subscheme of $X$. $S^{\{t\}}(X)$ may be reducible. Here we show that when $\mbox{\rm dim}(X)=1$ and $r \ge 4$ the scheme $S^{\{2\}}(X)$ is irreducible if and only if the curve $X$ has only planar singularities. We extend the definition and the result to joins of different varieties.

Received: June 30, 2004

AMS Subject Classification: 14N05

Key Words and Phrases: secant variety, joins of projective varieties

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