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Volume 40, Issue 2
Symplectic Conditions on Grassmannian, Flag, and Schubert Varieties

Jiajun Xu & Guanglian Zhang

DOI: 10.4208/cmr.2023-0034

Commun. Math. Res., 40 (2024), pp. 154-190.

Published online: 2024-05

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  • Abstract

In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Plücker coordinates, and it is proved that such equations generate the defining ideal of variety of type C in those of type A. As applications of this result, the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed, and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections. Finally, the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed, filling gaps in the study of algebraic varieties of the same type.

  • Keywords

Grassmannian variety, generalized flag variety, Schubert variety, Plücker embedding, complete intersection.

  • AMS Subject Headings

14M15, 14L30, 15A15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-40-154, author = {Xu , Jiajun and Zhang , Guanglian}, title = {Symplectic Conditions on Grassmannian, Flag, and Schubert Varieties}, journal = {Communications in Mathematical Research }, year = {2024}, volume = {40}, number = {2}, pages = {154--190}, abstract = {

In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Plücker coordinates, and it is proved that such equations generate the defining ideal of variety of type C in those of type A. As applications of this result, the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed, and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections. Finally, the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed, filling gaps in the study of algebraic varieties of the same type.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2023-0034}, url = {http://global-sci.org/intro/article_detail/cmr/23086.html} }
TY - JOUR T1 - Symplectic Conditions on Grassmannian, Flag, and Schubert Varieties AU - Xu , Jiajun AU - Zhang , Guanglian JO - Communications in Mathematical Research VL - 2 SP - 154 EP - 190 PY - 2024 DA - 2024/05 SN - 40 DO - http://doi.org/10.4208/cmr.2023-0034 UR - https://global-sci.org/intro/article_detail/cmr/23086.html KW - Grassmannian variety, generalized flag variety, Schubert variety, Plücker embedding, complete intersection. AB -

In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Plücker coordinates, and it is proved that such equations generate the defining ideal of variety of type C in those of type A. As applications of this result, the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed, and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections. Finally, the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed, filling gaps in the study of algebraic varieties of the same type.

Xu , Jiajun and Zhang , Guanglian. (2024). Symplectic Conditions on Grassmannian, Flag, and Schubert Varieties. Communications in Mathematical Research . 40 (2). 154-190. doi:10.4208/cmr.2023-0034
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