论文标题
在正交司法
On torus quotients of Schubert varieties in Orthogonal Grassmannian
论文作者
论文摘要
令$ g = spin（8n，\ mathbb {c}）（n \ ge 1）$和$ t_ {g} $是$g。$g。$ p^{α__{4n}}（\ supset t_ {g}）$的最大圆环。 $ x $是$ g/p^{α_{4n}} $在$ g/p^{α_{4n}} $相对于$ t $ linearized的半稳定点的非常丰富的线束$ \ MATHCAL {l}（2Ω__{4n}）。 0}} r_k，$ where $ r_k = h^{0}（x，x，\ mathcal {l}^{\ otimes k}（2Ω__{4n}））^{t_ {g}}}。在本文中，我们证明了$ n = 1 $和$ x = g/p = g/p^g/p^= g/p^= g/p^ran $ \ mathbb {c} $ - 代数$ r $由$ r_1。$。因此，我们证明$ g/p^{α_{4}} $的git商与$ t_ {g} $ linearized $ hilearearized $ noupe able ummate $ $ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r \ math cancal companty的git商（α_{4}} $ a射影空间$（\ Mathbb {p}^{2}，\ Mathcal {o} _ {\ Mathbb {p}^{2}}（1））$作为偏光品种。此外，我们证明$ r $由$ r_1 $和$ r_2 $生成，对于某些Schubert品种，$ g/p^{α_{4n}} $（对于$ n \ geq 2 $）。结果，我们证明这些舒伯特品种的git商相对于$ t_g $ linearized的下降非常正常，非常丰富的线束$ \ mathcal $ \ mathcal {l}（4Ω__{4N}）。$ $ 此外，对于$ g = spin（2n，\ mathbb {c}）（n \ ge 4）$（分别为$ g = sp = sp（2n，\ mathbb {c}）（n \ ge 2）$）和最大$ t_g $ g of $ g，$ g，$我们证明了$ g/p g/p randistient at $ g/p^randist is juppligity is justplion uppjectiment在$ t_g $中固定非常丰富的线条捆绑$ \ MATHCAL {l}（2Ω_{1}）$，并且对投射空间$（\ Mathbb {p}^{n-2}^{n-2}，\ Mathcal {O} $（\ mathbb {p}^{n-1}，\ Mathcal {o} _ {\ Mathbb {p}^{n-1}}}（1））$作为两极化的品种。
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.
