On complete intersections containing a linear subspace

Consider the Fano scheme F-k(Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y subset of P-m of multi-degree d = (d(1),..., d(s)). It is known that, if t := Sigma(s)(i=1) ((k) (di+k)) - (k + 1)(m - k) <= 0 and Pi (s)(i=1) d(i) > 2, for Y a general complete intersection as above, then F-k (Y) has dimension -t. In this paper we consider the case t > 0. Then the locus Wd, k of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general [Y]. Wd, k the scheme Fk (Y) is zero-dimensional of length one. This implies that Wd, k is rational.