  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 $t :=
\sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ 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.
