Symplectic Conditions on Grassmannian, Flag, and Schubert Varieties (Communications in Mathematical Research, 2024)
In this paper, a description of the set-theoretical defining equationsof 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 ofvariety of type C in those of type A. As applications of this result, the numberof local equations required to obtain the Schubert variety of type C from theSchubert variety of type A is computed, and further geometric properties ofthe 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 algebraicvarieties of the same type.