On torus quotients of Schubert varieties in Orthogonal Grassmannian

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 $R={⨁}_{k\in {\mathbb{Z}}_{\ge 0}}{R}_k,$ 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 $n\ge 2$). 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.

41 pages