Joins of projective varieties and multisecant spaces

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{equation*} s=3, k_1=k_2=k_3=1 \ \mbox {and} \ t=1 \end{equation*} or \begin{equation*} s=2, k_1=2, k_2 = 1 \ \mbox {and} \ t=1 . \end{equation*}