GIT quotient of minimal dimensional Schubert variety modulo a subtorus

TL;DR

This paper investigates the GIT quotient of a minimal Schubert variety in a Grassmannian under a subtorus action, revealing it as an iterated projective space bundle over projective space.

Contribution

It explicitly describes the GIT quotient of a specific minimal Schubert variety modulo a subtorus as an iterated projective bundle over projective space.

Findings

GIT quotient is isomorphic to an iterated projective space bundle.

The structure of the quotient is explicitly determined for the case n=rq+1.

Provides a geometric description of the quotient space.

Abstract

Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $\omega_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(n\omega_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $n\omega_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(n\omega_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$.