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 ⊂ P m of multi-degree d ̲ = ( d 1 , … , d s ) . It is known that, if t : = ∑ i = 1 s d i + k k - ( k + 1 ) ( m - k ) ⩽ 0 and ∏ i = 1 s 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 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 ] ∈ W d ̲ , k the scheme F k ( Y ) is zero-dimensional of length one. This implies that W d ̲ , k is rational.