On Fano schemes of linear spaces of general complete intersections

We consider the Fano scheme F_k(X) of k-dimensional linear subspaces contained in a complete intersection X⊂P^n of multi-degree d̲=(d1,…,ds). Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when X is a very general complete intersection and Π_{i=1}^s d_i>2 and we find conditions on n, d̲, and k under which F_k(X) does not contain either rational or elliptic curves. At the end of the paper, we study the case Π_{i=1}^s d_i=2