On torus quotients of Schubert varieties in orthogonal Grassmannian-II

TL;DR

This paper proves the projective normality of GIT quotients of specific Schubert varieties in the orthogonal Grassmannian, advancing understanding of their geometric and algebraic properties.

Contribution

It establishes projective normality for GIT quotients of certain Schubert varieties in orthogonal Grassmannians, a novel result in geometric invariant theory.

Findings

GIT quotients of Schubert varieties are projectively normal

The results apply to varieties in the orthogonal Grassmannian

Provides new insights into the algebraic structure of these quotients

Abstract

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.