LAUSR

Consider the Fano scheme $$F_k(Y)$$ F k ( Y ) parameterizing k -dimensional linear subspaces contained in a complete intersection $$Y \subset \mathbb {P}^m$$ Y ⊂ P m of multi-degree $$\underline{d} = (d_1, \ldots , d_s)$$ d ̲ = ( d 1 , … , d s ) . It is known that, if $$t := \sum _{i=1}^s \left( {\begin{array}{c}d_i +k\\ k\end{array}}\right) -(k+1) (m-k)\leqslant 0$$ t : = ∑ i = 1 s d i + k k - ( k + 1 ) ( m - k ) ⩽ 0 and $$\prod _{i=1}^sd_i >2$$ ∏ i = 1 s d i > 2 , for Y a general complete intersection as above, then $$F_k(Y)$$ F k ( Y ) has dimension $$-t$$ - t . In this paper we consider the case $$t> 0$$ t > 0 . Then the locus $$W_{\underline{d},k}$$ 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 $$[Y]\in W_{\underline{d},k}$$ [ Y ] ∈ W d ̲ , k the scheme $$F_k(Y)$$ F k ( Y ) is zero-dimensional of length one. This implies that $$W_{\underline{d},k}$$ W d ̲ , k is rational.