On Fano schemes of linear spaces of general complete intersections

Abstract

We consider the Fano scheme $$F_k(X)$$ of k-dimensional linear subspaces contained in a complete intersection $$X \subset {\mathbb {P}}^n$$ of multi-degree $${\underline{d}} = (d_1, \ldots , d_s)$$ . 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 $$\Pi _{i=1}^s d_i > 2$$ and we find conditions on n, $${\underline{d}}$$ , 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 $$\Pi _{i=1}^s d_i = 2$$ .