Projective normality of torus quotients of flag varieties

Abstract

Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus in $G$. We show that the quotient $T \backslash \backslash G/{P_{α_1}\cap P_{α_2}}$ is projectively normal with respect to the descent of a suitable line bundle, where $P_{α_i}$ is the maximal parabolic subgroup in $G$ associated to the simple root $α_i$, $i=1,2$. 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 $T \backslash \backslash (G/P_{α_2})^{ss}_T(\mathcal{L}_{2\varpi_2})$ whereas we prove that the quotient $T\backslash\backslash (G/P_{α_3})^{ss}_T(\mathcal{L}_{4\varpi_3})$ is projectively normal with respect to the descent of the line bundles $\mathcal{L}_{4\varpi_3}$.