On the Torus quotients of Schubert varieties

Abstract

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 [Formula: see text] is projectively normal with respect to the line bundle [Formula: see text] and the quotient space is a projective space, where the line bundle [Formula: see text] and the parabolic subgroup [Formula: see text] of [Formula: see text] are associated to the highest root [Formula: see text].