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 $\underline{d}=(d_1,\dots ,d_s)$. 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 ${\Pi }_{i=1}^s{d}_i=2$.

Referee's suggestions added. To appear on Archiv der Matematik