On the Variety of Paths on Complete Intersections in Grassmannians

Abstract

In this article we study the Fano variety of lines on the complete intersection of the grassmannian G(n, 2n) with hypersurfaces of degrees d₁ ..., 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 nonempty and connected if ∑d_j < 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 ×F_n of spaces of full flags. After this we construct on Fn ×Fn a globally generated vector bundle 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 d₁,...,d_k. Using a presentation of 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 × 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.