On complete intesections containing a linear subspace

Consider the Fano scheme F_k(Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y in IP^m of multi-degree d = (d_1,....,d_s). It is known that, if t:= t(m,d,k) <= 0 and d_1....d_s >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 W(d,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] in W(d,k) the scheme F_k(Y) is zero-dimensional of length one. This implies that W(d,k) is rational.