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arXiv 2207.01477math.AGmath.COmath.RT
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正交Grassmannian中Schubert簇的环面

On torus quotients of Schubert varieties in Orthogonal Grassmannian

Arpita Nayek, Pinakinath Saha

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41 pages
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摘要：

设$G=Spin(8n, \mathbb{C})(n\ge 1)$和$T_{G}$是$G.$的极大环面，$P^{α_{4n}}(\supset T_{G})$是$G$的极大抛物子群，对应于单根$α_{4n}.$，$X$是$G/P^{α_{4n}}$中的Schubert簇，允许关于$T$ -线性化的非常充分线丛$\mathcal{L}(2ω_{4n}).$的半稳定点，$R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$其中$R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$在本文中，证明了对于$n=1$和$X=G/P^{α_4},$，分次$\mathbb{C}$ -代数$R$是由$R_1.$生成的。我们证明了$G/P^{α_{4}}$的GIT商关于$T_{G}$的下降是射影正规的。线性化的非常丰富的线丛$\mathcal{L}(2ω_{4})$，并且同构于作为极化簇的射影空间$(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$。进一步，我们证明了$R$是由$R_1$和$R_2$生成的一些舒伯特品种在$G/P^{α_{4n}}$（为$n \geq 2$）。因此，我们证明了这些Schubert簇的GIT商关于$T_G$ -线性化的非常充裕线丛$\mathcal{L}(4ω_{4n}).$的下降是射影正规的。此外，对于$G = Spin(2n,\mathbb{C})(n \ge 4)$，（分别为，$G=Sp(2n, \mathbb{C}) (n\ge 2)$）和$G,$的极大环面$T_G$，我们证明了$G/P^{α_{1}}$的GIT商关于$T_G$的下降是射影正规的。线性化的非常丰富的线丛$\mathcal{L}(2ω_{1})$，并且同构于射影空间$(\mathbb{P}^{n-2},\mathcal{O}_{\mathbb{P}^{n-2}}(1))$（分别地，$(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1))$作为极化簇。

英文摘要：

Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{α_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n}.$ Let $X$ be a Schubert variety in $G/P^{α_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2ω_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{α_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence, we prove that the GIT quotient of $G/P^{α_{4}}$ is projectively normal with respect to the descent of the $T_{G}$-linearized very ample line bundle $\mathcal{L}(2ω_{4})$ 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 $G/P^{α_{4n}}$ (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 $\mathcal{L}(4ω_{4n}).$ 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 $G/P^{α_{1}}$ is projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(2ω_{1})$ 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.

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