Symplectic conditions on Grassmannian, flag, and Schubert varieties

TL;DR

This paper characterizes symplectic Grassmannian, flag, and Schubert varieties within type A varieties using linear equations in Plücker coordinates, and explores their geometric properties and smoothness.

Contribution

It provides explicit defining equations for symplectic varieties in type A, and analyzes their geometric and smoothness properties, filling gaps in algebraic geometry literature.

Findings

Equations generate the ideal of type C varieties in type A

Number of local equations for Schubert varieties is computed

Conditions for smoothness in non-minuscule cases are discussed

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.