# Projective normality of torus quotients of flag varieties

## TL;DR

This paper proves projective normality of certain torus quotients of flag varieties and provides degree bounds for generators of their coordinate rings, advancing understanding of their algebraic and geometric properties.

## Contribution

It establishes projective normality for specific torus quotients of flag varieties and derives degree bounds for generators of their coordinate rings, including new results for $SL_n$ and $Spin_7$ cases.

## Key findings

- Proved projective normality of $T ackslash ackslash G/P$ for certain parabolics.
- Provided degree bounds for generators of coordinate rings.
- Extended results to $Spin_7$ and specific flag varieties.

## Abstract

Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus in $G$. We show that the quotient $T \backslash \backslash G/{P_{\alpha_1}\cap P_{\alpha_2}}$ 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 $\mathcal{L}_{4\varpi_3}$.