Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus in $G$. We show that the quotient $T \backslash \backslash G/{P_{\alpha_1}\cap P_{\alpha_2}}$ is projectively normal with respect to the descent of a suitable line bundle, where $P_{\alpha_i}$ is the maximal parabolic subgroup in $G$ associated to the simple root $\alpha_i$, $i=1,2$. We give a degree bound of the generators of the homogeneous coordinate ring of $T \backslash \backslash (G_{3,6})^{ss}_T(\mathcal{L}_{2\varpi_3})$. If $G =Spin_7$, we give a degree bound of the generators of the homogeneous coordinate ring of $T \backslash \backslash (G/P_{\alpha_2})^{ss}_T(\mathcal{L}_{2\varpi_2})$ whereas we prove that the quotient $T\backslash\backslash (G/P_{\alpha_3})^{ss}_T(\mathcal{L}_{4\varpi_3})$ is projectively normal with respect to the descent of the line bundles $\mathcal{L}_{4\varpi_3}$.

中文翻译：

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旗品种环面商的投影正态性

令 $G=SL_n(\mathbb C)$ 和 $T$ 是 $G$ 中的最大圆环。我们证明商 $T \backslash \backslash G/{P_{\alpha_1}\cap P_{\alpha_2}}$ 对于合适的线丛的下降是射影正态的，其中 $P_{\alpha_i}$是与简单根 $\alpha_i$ 相关联的 $G$ 中的最大抛物线子群，$i=1,2$。我们给出了 $T \backslash \backslash (G_{3,6})^{ss}_T(\mathcal{L}_{2\varpi_3})$ 齐次坐标环的生成元的度界。如果$G =Spin_7$，我们给出$T \backslash \backslash (G/P_{\alpha_2})^{ss}_T(\mathcal{L}_{2 \varpi_2})$ 而我们证明商 $T\backslash\backslash (G/P_{\alpha_3})^{ss}_T(\mathcal{L}_{4\varpi_3})$ 是投影正常的到线束 $\mathcal{L}_{4\varpi_3}$ 的下降。