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 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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2014-08-01
|
| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/96 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849338812326477824 |
|---|---|
| author | S. M. Yermakova |
| author_facet | S. M. Yermakova |
| author_sort | S. M. Yermakova |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-c5a1a50c432e4ac8a0640ae7670ff748 |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2014-08-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-c5a1a50c432e4ac8a0640ae7670ff7482025-08-20T03:44:18ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172014-08-01214354610.18255/1818-1015-2014-4-35-4690On the Variety of Paths on Complete Intersections in GrassmanniansS. M. Yermakova0P.G. Demidov Yaroslavl State UniversityIn 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.https://www.mais-journal.ru/jour/article/view/96grassmannianvector bundlefano variety of lines |
| spellingShingle | S. M. Yermakova On the Variety of Paths on Complete Intersections in Grassmannians Моделирование и анализ информационных систем grassmannian vector bundle fano variety of lines |
| title | On the Variety of Paths on Complete Intersections in Grassmannians |
| title_full | On the Variety of Paths on Complete Intersections in Grassmannians |
| title_fullStr | On the Variety of Paths on Complete Intersections in Grassmannians |
| title_full_unstemmed | On the Variety of Paths on Complete Intersections in Grassmannians |
| title_short | On the Variety of Paths on Complete Intersections in Grassmannians |
| title_sort | on the variety of paths on complete intersections in grassmannians |
| topic | grassmannian vector bundle fano variety of lines |
| url | https://www.mais-journal.ru/jour/article/view/96 |
| work_keys_str_mv | AT smyermakova onthevarietyofpathsoncompleteintersectionsingrassmannians |