Rational curves in a quadric threefold via an $\text{SL}(2,\mathbb{C})$-representation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913928024424448 |
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| author | Chung, Kiryong Huh, Sukmoon Yoo, Sang-Bum |
| author_facet | Chung, Kiryong Huh, Sukmoon Yoo, Sang-Bum |
| contents | In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very well-known facts. If the degree of a rational curve is $3$, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in $Q_3$ has a specific projective bundle structure. From this, we can immediately obtain the cohomology ring of the moduli space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11539 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational curves in a quadric threefold via an $\text{SL}(2,\mathbb{C})$-representation Chung, Kiryong Huh, Sukmoon Yoo, Sang-Bum Algebraic Geometry 14E05, 14E15, 14M15 In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very well-known facts. If the degree of a rational curve is $3$, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in $Q_3$ has a specific projective bundle structure. From this, we can immediately obtain the cohomology ring of the moduli space. |
| title | Rational curves in a quadric threefold via an $\text{SL}(2,\mathbb{C})$-representation |
| topic | Algebraic Geometry 14E05, 14E15, 14M15 |
| url | https://arxiv.org/abs/2301.11539 |