On the torus quotients of Schubert varieties

TL;DR

This paper investigates the geometric invariant theory (GIT) quotients of Schubert varieties under torus actions, revealing conditions for smoothness, projective normality, and the structure of the quotients as projective spaces.

Contribution

It characterizes minuscule Schubert varieties with contained semistable loci, studies smoothness of torus quotients in Grassmannians, and proves projective normality of quotients in homogeneous spaces.

Findings

Semistable locus is contained in the smooth locus for certain minuscule Schubert varieties.

Torus quotients of Schubert varieties in Grassmannians can be smooth under specific conditions.

Quotients in $SL(n, \\mathbb C)/P$ are projectively normal and isomorphic to projective spaces.

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 $SL(n, \mathbb C)/P$ 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 $\alpha_0$.