On the torus quotients of Schubert varieties

In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space $SL(n,\mathbb{C})\mathrm{/}P$ is projectively normal with respect to the line bundle ${\mathcal{L}}_{{\alpha }_0}$ and the quotient space is a projective space, where the line bundle ${\mathcal{L}}_{{\alpha }_0}$ and the parabolic subgroup $P$ of $SL(n,\mathbb{C})$ are associated to the highest root ${\alpha }_0$.