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

This article is cited in 2 scientific papers (total in 2 papers)

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.

Full text: PDF file (440 kB)
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UDC: 512.7
Received: 04.08.2014

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
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    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  mathnet  mathscinet  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
  • Моделирование и анализ информационных систем
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