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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.25645 |
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| _version_ | 1866918471841873920 |
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| author | Ghosh, Arkadev Kannan, S. S. |
| author_facet | Ghosh, Arkadev Kannan, S. S. |
| contents | Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $ω_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(nω_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $nω_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(nω_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | GIT quotient of minimal dimensional Schubert variety modulo a subtorus Ghosh, Arkadev Kannan, S. S. Algebraic Geometry 14M15, 14L35, 14F25 Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $ω_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(nω_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $nω_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(nω_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$. |
| title | GIT quotient of minimal dimensional Schubert variety modulo a subtorus |
| topic | Algebraic Geometry 14M15, 14L35, 14F25 |
| url | https://arxiv.org/abs/2604.25645 |