On the GIT quotient of Grassmannians by one dimensional torus

TL;DR

This paper investigates the GIT quotients of Grassmannians under a specific one-parameter subgroup, providing explicit descriptions of semistable loci and geometric properties using combinatorics and algebraic geometry techniques.

Contribution

It offers a detailed combinatorial and geometric analysis of GIT quotients of Grassmannians by a one-dimensional torus, including descriptions of semistable loci and geometric structures.

Findings

Explicit description of semistable loci using Weyl group combinatorics

Identification of cases with parabolic induction of projective spaces

Analysis of geometric properties like Fano status and Picard groups

Abstract

We consider the action of the one-parameter subgroup of the special linear group corresponding to a simple root on Grassmannians and describe the structure of the associated Geometric Invariant Theory (GIT) quotients with respect to Pl\"ucker line bundle. Using the combinatorics of Weyl group elements, we explicitly describe the semistable loci and identify cases where the resulting quotient admits the structure of a parabolic induction of a projective space. We further analyze the orbit structure under the Levi subgroup, compute the Picard group, connected component of the automorphism group and examine key geometric features such as Fano properties, cohomology of line bundles, and projective normality with respect to the descended linearization.