Smooth torus quotients of Schubert varieties in the Grassmannian

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.

Categories: math.AG math.CO