Projective normality of torus quotients of flag varieties

Abstract

Let $G=S{L}_n(\mathbb{C})$ 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 $T\backslash \backslash (G_{3,6}{)}_T^{ss}({\mathcal{L}}_{2{\varpi }_3})$. If $G=Spi{n}_7$, we give a degree bound of the generators of the homogeneous coordinate ring of $T\backslash \backslash (G/P_{{\alpha }_2}{)}_T^{ss}({\mathcal{L}}_{2{\varpi }_2})$ whereas we prove that the quotient $T\backslash \backslash (G/P_{{\alpha }_3}{)}_T^{ss}({\mathcal{L}}_{4{\varpi }_3})$ is projectively normal with respect to the descent of the line bundles ${\mathcal{L}}_{4{\varpi }_3}$.