Torus quotients of Schubert varieties in the Grassmannian $G_{2,n}$

TL;DR

This paper investigates the geometric invariant theory (GIT) quotients of Schubert varieties in the Grassmannian G_{2,n}, revealing their structure as projective spaces, toric varieties, and analyzing their singularities.

Contribution

It provides a detailed description of the GIT quotients of Schubert varieties in G_{2,n}, including their projectivity, toric nature, and singularity structure, which was previously not fully understood.

Findings

GIT quotients of Richardson varieties are projective spaces.

Certain Richardson varieties' quotients are projective toric varieties.

GIT quotients have finitely many singular points, with exact counts computed.

Abstract

Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular points. Further, we have computed the exact number of singular points of the GIT quotient of $G_{2,n}.$