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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.17721 |
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| _version_ | 1866913623589257216 |
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| author | Chung, Kiryong Kim, Jaehyun Kim, Jeong-Seop |
| author_facet | Chung, Kiryong Kim, Jaehyun Kim, Jeong-Seop |
| contents | Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian variety $\mathrm{Gr}(3,V)$, $\dim V=7$ with the universal subbundle $\mathcal{F}$. When the section $\mathbf{s}$ is given by the net of the $\mathrm{SL}_2$-invariant skew forms, we call it by the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth and compute its Poincaré polynomial by applying the Białynicki-Birula's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rational quartic curves in the Mukai-Umemura variety Chung, Kiryong Kim, Jaehyun Kim, Jeong-Seop Algebraic Geometry Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian variety $\mathrm{Gr}(3,V)$, $\dim V=7$ with the universal subbundle $\mathcal{F}$. When the section $\mathbf{s}$ is given by the net of the $\mathrm{SL}_2$-invariant skew forms, we call it by the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth and compute its Poincaré polynomial by applying the Białynicki-Birula's theorem. |
| title | Rational quartic curves in the Mukai-Umemura variety |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2412.17721 |