Joins of projective varieties and multisecant spaces

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:

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