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 $X\subset {\mathbb{P}}^n$ of multi–degree $\underset{‾}{d}=(d_1,\dots ,d_s)$. 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 ${\mathrm{\Pi }}_{i=1}^s{d}_i>2$ and we find conditions on $n$, $\underset{‾}{d}$ 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 $\Pi_{i=1}^s d_i = 2$.