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arXiv · 2204.05589

Symplectic conditions on Grassmannian, flag, and Schubert varieties

Jiajun Xu·Guanglian Zhang
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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 Pl$\ddot{u}$cker 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.

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Jiajun Xu, Guanglian Zhang. 2022-04-12. Symplectic conditions on Grassmannian, flag, and Schubert varieties. https://arxiv.org/abs/2204.05589

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