Torus quotient of the Grassmannian $G_{n,2n}$
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 $\mathrm{SL}(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1)$....
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.501/ |
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| author | Nayek, Arpita Saha, Pinakinath |
| author_facet | Nayek, Arpita Saha, Pinakinath |
| author_sort | Nayek, Arpita |
| collection | DOAJ |
| description | 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 $\mathrm{SL}(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1)$. By [10, Theorem 3.10, p. 764], $2$ 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 $\mathcal{O}(2)$. |
| format | Article |
| id | doaj-art-50f56942386e4d6a84241c81adc1092e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-50f56942386e4d6a84241c81adc1092e2025-02-07T11:10:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G91499150910.5802/crmath.50110.5802/crmath.501Torus quotient of the Grassmannian $G_{n,2n}$Nayek, Arpita0Saha, Pinakinath1Department of Mathematics, IIT Bombay, Powai, Mumbai 400076, IndiaDepartment of Mathematics, IIT Bombay, Powai, Mumbai 400076, IndiaLet $G_{n,2n}$ 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 $\mathrm{SL}(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1)$. By [10, Theorem 3.10, p. 764], $2$ 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 $\mathcal{O}(2)$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.501/GrassmannianLine bundleSemi-stable pointGIT-quotientProjective normality |
| spellingShingle | Nayek, Arpita Saha, Pinakinath Torus quotient of the Grassmannian $G_{n,2n}$ Comptes Rendus. Mathématique Grassmannian Line bundle Semi-stable point GIT-quotient Projective normality |
| title | Torus quotient of the Grassmannian $G_{n,2n}$ |
| title_full | Torus quotient of the Grassmannian $G_{n,2n}$ |
| title_fullStr | Torus quotient of the Grassmannian $G_{n,2n}$ |
| title_full_unstemmed | Torus quotient of the Grassmannian $G_{n,2n}$ |
| title_short | Torus quotient of the Grassmannian $G_{n,2n}$ |
| title_sort | torus quotient of the grassmannian g n 2n |
| topic | Grassmannian Line bundle Semi-stable point GIT-quotient Projective normality |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.501/ |
| work_keys_str_mv | AT nayekarpita torusquotientofthegrassmanniangn2n AT sahapinakinath torusquotientofthegrassmanniangn2n |