On the Variety of Paths on Complete Intersections in Grassmannians - S. M. Yermakova (Моделирование и анализ информационных систем)
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 ..., di . 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 ∑dj < 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 Fn ×Fn 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,...,dk. 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 Fn × Fn. 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.