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

Arpita Nayek, Pinakinath Saha

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 $\mathcal{O}(2).$

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

Abstract

Let be the Grassmannian parameterizing the -dimensional subspaces of The Picard group of is generated by a unique ample line bundle Let be a maximal torus of which acts on and By \cite[Theorem 3.10, p.764]{Kum}, is the minimal integer such that descends to the GIT quotient. In this article, we prove that the GIT quotient of () by with respect to is not projectively normal when polarized with the descent of

Paper Structure

This paper contains 4 sections, 10 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Main Theorem
  4. Appendix

Key Result

Theorem 1.1

The GIT quotient of $G_{n,2n}$$(n \geq 3)$ by a maximal torus $T$ of $SL(2n,\mathbb{ C})$ with respect to the descent of $\mathcal{O}(2)$ is not projectively normal $($for more precise see corollary3.5$).$

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 12 more