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Joins of projective varieties and multisecant spaces

Author: E. Ballico
Journal: Proc. Amer. Math. Soc. 133 (2005), 1-10
MSC (2000): Primary 14N05, 14M15
Published electronically: August 24, 2004
MathSciNet review: 2085145
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Abstract: Let $X_1,\dots ,X_s \subset {\mathbf {P}}^N$, $s \ge 1$, be integral varieties. For any integers $k_i>0$, $1 \le i \le s$, and $t \ge 0$ set $\vec{k}:= (k_1,\dots ,k_s)$ and $\vec{X}:= (X_1,\dots ,X_s)$. Let $\mbox{Sec}(\vec{X} ;t,\vec{k} )$be the set of all linear $t$-spaces contained in a linear $(k_1+\cdots +k_s-1)$-space spanned by $k_1$ points of $X_1$, $k_2$ points of $X_2$, ..., $k_s$ points of $X_s$. Here we study some cases where $\mbox{Sec}(\vec{X} ;t,\vec{k} )$ has the expected dimension. The case $s=1$ was recently considered by Chiantini and Coppens and we follow their ideas. The two main results of the paper consider cases where each $X_i$ is a surface, more particularly:

\begin{displaymath}s=3, k_1=k_2=k_3=1 \mbox{and} \ t=1 \end{displaymath}


\begin{displaymath}s=2, k_1=2, k_2 = 1 \ \mbox{and} t=1 . \end{displaymath}

References [Enhancements On Off] (What's this?)

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Additional Information

E. Ballico
Affiliation: Department of Mathematics, University of Trento, 38050 Povo, Trento, Italy

Keywords: Joins, multisecant spaces, secant variety, Grassmannian
Received by editor(s): August 16, 2002
Published electronically: August 24, 2004
Additional Notes: The author was partially supported by MIUR and GNSAGA of INdAM (Italy).
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society

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