Smooth torus quotients of Schubert varieties in the Grassmannian

TL;DR

This paper investigates the geometric invariant theory (GIT) quotients of Schubert varieties in Grassmannians, providing combinatorial criteria for when these quotients are smooth, under specific coprimality conditions.

Contribution

It introduces necessary and sufficient combinatorial conditions for the smoothness of GIT quotients of Schubert varieties in Grassmannians when the acting torus has a specific linearization.

Findings

Identifies conditions for smooth GIT quotients

Connects combinatorial data with geometric properties

Focuses on coprime integers r and n

Abstract

Let $r < n$ be positive integers and further suppose $r$ and $n$ are coprime. We study the GIT quotient of Schubert varieties $X(w)$ in the Grassmannian $G_{r,n}$, admitting semistable points for the action of $T$ with respect to the $T$-linearized line bundle ${\cal L}(n\omega_r)$. We give necessary and sufficient combinatorial conditions for the GIT quotient $T\backslash\mkern-6mu\backslash X(w)^{ss}_{T}({\cal L}(n\omega_r))$ to be smooth.