On torus quotients of Schubert varieties in Orthogonal Grassmannian

TL;DR

This paper studies the geometric invariant theory quotients of Schubert varieties in orthogonal Grassmannians, proving projective normality and explicit isomorphisms to projective spaces for certain cases, and describing the generators of associated graded algebras.

Contribution

It establishes projective normality and explicit geometric descriptions of GIT quotients for Schubert varieties in orthogonal Grassmannians, including generator degrees of coordinate rings.

Findings

GIT quotient of G/P^{α_4} is projectively normal and isomorphic to P^2.

For certain Schubert varieties, the coordinate ring is generated by R_1 and R_2.

GIT quotients of specific Schubert varieties are projectively normal and isomorphic to projective spaces.

Abstract

Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{\alpha_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n}.$ Let $X$ be a Schubert variety in $G/P^{\alpha_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2\omega_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2\omega_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{\alpha_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence, we prove that the GIT quotient of $G/P^{\alpha_{4}}$ is projectively normal with respect to the descent of the $T_{G}$-linearized very ample line bundle $\mathcal{L}(2\omega_{4})$ and is isomorphic to the projective space $(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$ as a polarized variety. Further, we prove that $R$ is generated by $R_1$ and $R_2$ for some Schubert varieties in $G/P^{\alpha_{4n}}$ (for $n \geq 2$). As a consequence, we prove that the GIT quotient of those Schubert varieties are projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(4\omega_{4n}).$ Moreover, for $G = Spin(2n,\mathbb{C})(n \ge 4)$ (respectively, $G=Sp(2n, \mathbb{C}) (n\ge 2)$) and a maximal torus $T_G$ of $G,$ we prove that the GIT quotient of $G/P^{\alpha_{1}}$ is projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(2\omega_{1})$ and is isomorphic to the projective space $(\mathbb{P}^{n-2},\mathcal{O}_{\mathbb{P}^{n-2}}(1))$ (respectively, $(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1))$ as a polarized variety.