Torus quotient of the Grassmannian $G_{n,2n}$

TL;DR

This paper investigates the geometric properties of the GIT quotient of the Grassmannian $G_{n,2n}$ by a maximal torus, showing it is not projectively normal with respect to a specific polarization.

Contribution

It proves that the GIT quotient of $G_{n,2n}$ by a maximal torus with respect to $ ext{O}(2)$ is not projectively normal for $n geq 2$, providing new insights into its geometric structure.

Findings

The GIT quotient is not projectively normal when polarized with the descent of $ ext{O}(2)$.

The minimal integer for descent of the line bundle is 2.

The result applies for all $n geq 2$.

Abstract

Let $G_{n,2n}$ be the Grassmannian parameterizing the $n$-dimensional subspaces of $\mathbb{C}^{2n}.$ The Picard group of $G_{n,2n}$ is generated by a unique ample line bundle $\mathcal{O}(1).$ Let $T$ be a maximal torus of $SL(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1).$ By \cite[Theorem 3.10, p.764]{Kum}, $2$ is the minimal integer $k$ such that $\mathcal{O}(k)$ descends to the GIT quotient. In this article, we prove that the GIT quotient of $G_{n,2n}$ ($n\ge 3$) by $T$ with respect to $\mathcal{O}(2)=\mathcal{O}(1)^{\otimes 2}$ is not projectively normal when polarized with the descent of $\mathcal{O}(2).$