Joins of Projective Varieties and Multisecant Spaces

Let X1, . . . ,Xs ⊂ PN , s ≥ 1, be integral varieties. For any integers ki > 0, 1 ≤ i ≤ s, and t ≥ 0 set ~k := (k1, . . . , ks) and ~ X := (X1, . . . ,Xs). Let Sec( ~ X; t,~k) be the set of all linear t-spaces contained in a linear (k1 + · · · + ks − 1)-space spanned by k1 points of X1, k2 points of X2, . . . , ks points of Xs. Here we study some cases where Sec( ~ X ; t,~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 Xi is a surface, more particularly: s = 3, k1 = k2 = k3 = 1 and t = 1