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

Arpita Nayek; Pinakinath Saha

doi:10.5802/crmath.501

Algebra, Combinatorics

Torus quotient of the Grassmannian

G_{n, 2 n}

Arpita Nayek ¹; Pinakinath Saha ¹

¹ Department of Mathematics, IIT Bombay, Powai, Mumbai 400076, India

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1499-1509.

Abstract
Résumé

Let $G_{n, 2 n}$ be the Grassmannian parameterizing the $n$ -dimensional subspaces of $ℂ^{2 n}$ . The Picard group of $G_{n, 2 n}$ is generated by a unique ample line bundle $𝒪 (1)$ . Let $T$ be a maximal torus of $SL (2 n, ℂ)$ which acts on $G_{n, 2 n}$ and $𝒪 (1)$ . By [10, Theorem 3.10, p. 764], $2$ is the minimal integer $k$ such that $𝒪 (k)$ descends to the GIT quotient. In this article, we prove that the GIT quotient of $G_{n, 2 n}$ ( $n \geq 3$ ) by $T$ with respect to $𝒪 (2) = 𝒪 {(1)}^{\otimes 2}$ is not projectively normal when polarized with the descent of $𝒪 (2)$ .

Soit $G_{n, 2 n}$ la Grassmannienne des sous-espaces de dimension $n$ de $ℂ^{2 n}$ . Le groupe de Picard de $G_{n, 2 n}$ est engendré par un unique fibré en droites ample $𝒪 (1)$ . Fixons un tore maximal $T$ du groupe $SL (2 n, ℂ)$ qui agit sur $G_{n, 2 n}$ et $𝒪 (1)$ . D’après [10, Theorem 3.10, p. 764], $2$ est l’entier minimal $k$ tel que $𝒪 (k)$ descende au quotient GIT. Dans cet article, nous prouvons que le quotient GIT de $G_{n, 2 n}$ ( $n \geq 3$ ) par $T$ par rapport à $𝒪 (2) = 𝒪 {(1)}^{\otimes 2}$ n’est pas projectivement normal lorsqu’il est polarisé avec la descente de $𝒪 (2)$ .

Received: 2022-09-14
Revised: 2023-03-29
Accepted: 2023-03-29
Published online: 2023-11-10

DOI: 10.5802/crmath.501

Classification: 14M15, 05E10
Keywords: Grassmannian, Line bundle, Semi-stable point, GIT-quotient, Projective normality

Author's affiliations:

Arpita Nayek ¹; Pinakinath Saha ¹

¹ Department of Mathematics, IIT Bombay, Powai, Mumbai 400076, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Arpita Nayek and Pinakinath Saha},
     title = {Torus quotient of the {Grassmannian} $G_{n,2n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1499--1509},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.501},
     language = {en},
}

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Arpita Nayek; Pinakinath Saha. Torus quotient of the Grassmannian $G_{n,2n}$. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1499-1509. doi : 10.5802/crmath.501. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.501/

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