On the Variety of Paths on Complete Intersections in Grassmannians

In this article we study the Fano variety of lines on the complete intersection of the grassmannian $G(n,2n)$ with hypersurfaces of degrees $d_1,...,d_i$. A length $l$ path on such a variety is a connected curve composed of $l$ lines. The main result of this article states that the space of length $l$ paths connecting any two given points on the variety is non-empty and connected if $\sum d_j<\frac{n}{4}$. To prove this result we first show that the space of length $n$ paths on the grassmannian $G(n,2n)$ that join two generic points is isomorphic to the direct product $F_n\times F_n$ of spaces of full flags. After this we construct on $F_n\times F_n$ a globally generated vector bundle $\mathcal E$ with a distinguished section $s$ such that the zeros of $s$ coincide with the space of length $n$ paths that join $x$ and $y$ and lie in the intersection of hypersurfaces of degrees $d_1$,...,$d_k$. Using a presentation of $\mathcal E$ as a sum of linear bundles we show that zeros of its generic and, hence, any section form a non empty connected subvariety of $F_n\times F_n$. Apart from its immediate geometric interest, this result will be used in our future work on generalisation of splitting theorems for finite rank vector bundles on ind-manifolds.