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 Model. Anal. Inform. Sist., 2014, Volume 21, Number 4, Pages 35–46 (Mi mais385)

On the Variety of Paths on Complete Intersections in Grassmannians

S. M. Yermakova

P. G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

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 $d_1,...,d_i$. 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.

Keywords: grassmannian, vector bundle, Fano variety of lines.

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Citation: S. M. Yermakova, “On the Variety of Paths on Complete Intersections in Grassmannians”, Model. Anal. Inform. Sist., 21:4 (2014), 35–46

Citation in format AMSBIB
\Bibitem{Erm14} \by S.~M.~Yermakova \paper On the Variety of Paths on Complete Intersections in Grassmannians \jour Model. Anal. Inform. Sist. \yr 2014 \vol 21 \issue 4 \pages 35--46 \mathnet{http://mi.mathnet.ru/mais385} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. M. Ermakova, “Ravnomernost vektornykh rassloenii konechnogo ranga na polnykh peresecheniyakh konechnoi korazmernosti v lineinykh ind-grassmanianakh”, Model. i analiz inform. sistem, 22:2 (2015), 209–218
2. S. M. Ermakova, “Finite-Rank Vector Bundles on Complete Intersections of Finite Codimension in the Linear Ind-Grassmannian”, Math. Notes, 98:5 (2015), 852–856
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