On complete intersections containing a linear subspace

Consider the Fano scheme $F_k(Y)$ parameterizing $k$--dimensional linear subspaces contained in a complete intersection $Y subset mathbb{P}^m$ of multi--degree $\underline{d} = (d_1, ldots, d_s)$. It is known that, if $t := sum_{i=1}^s inom{d_i +k}{k}-(k+1) (m-k)leqslant 0$ and $Pi_{i=1}^sd_i >2$, 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 $W_{\underline{d},k}$ 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 $[Y]in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero--dimensional of length one. This implies that $W_{\underline{d},k}$ is rational.