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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.501/ |
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| Summary: | 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 $\mathcal{O}(2)$. |
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| ISSN: | 1778-3569 |