Quadratic types and the dynamic Euler number of lines on a quintic threefold
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910378691133440 |
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| author | Pauli, Sabrina |
| author_facet | Pauli, Sabrina |
| contents | We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym^5 S^* -> Gr(2, 5) defined by the threefold. Furthermore, we define the dynamic Euler number which allows us to compute the A^1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym^5 S^* -> Gr(2, 5). Combining those two results we get that the sum of the types of lines on a general quintic threefold is 1445<1> + 1430<-1> in GW(k) when k is a field of characteristic not equal to 2 or 5. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_12089 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Quadratic types and the dynamic Euler number of lines on a quintic threefold Pauli, Sabrina Algebraic Geometry 14N15 (Primary) 14F42, 14G27 (Secondary) We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym^5 S^* -> Gr(2, 5) defined by the threefold. Furthermore, we define the dynamic Euler number which allows us to compute the A^1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym^5 S^* -> Gr(2, 5). Combining those two results we get that the sum of the types of lines on a general quintic threefold is 1445<1> + 1430<-1> in GW(k) when k is a field of characteristic not equal to 2 or 5. |
| title | Quadratic types and the dynamic Euler number of lines on a quintic threefold |
| topic | Algebraic Geometry 14N15 (Primary) 14F42, 14G27 (Secondary) |
| url | https://arxiv.org/abs/2006.12089 |