The defect of a cubic threefold and applications to intermediate Jacobian fibrations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913922303393792 |
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| author | Marquand, Lisa Viktorova, Sasha |
| author_facet | Marquand, Lisa Viktorova, Sasha |
| contents | The defect of a cubic threefold $X$ with isolated singularities is a global invariant that measures the failure of $\mathbb{Q}$-factoriality. We compute the defect for such cubics in terms of topological data about the curve of lines through a singular point. We express the mixed Hodge structure on the middle cohomology of $X$ in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of $X$: in particular, we show that a cubic threefold is not $\mathbb{Q}$-factorial if and only if it contains either a plane or a cubic scroll. We relate the defect to existence of compactified intermediate Jacobian fibrations with irreducible fibers associated to a cubic fourfold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05118 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The defect of a cubic threefold and applications to intermediate Jacobian fibrations Marquand, Lisa Viktorova, Sasha Algebraic Geometry 14J17, 14J30, 14J45 The defect of a cubic threefold $X$ with isolated singularities is a global invariant that measures the failure of $\mathbb{Q}$-factoriality. We compute the defect for such cubics in terms of topological data about the curve of lines through a singular point. We express the mixed Hodge structure on the middle cohomology of $X$ in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of $X$: in particular, we show that a cubic threefold is not $\mathbb{Q}$-factorial if and only if it contains either a plane or a cubic scroll. We relate the defect to existence of compactified intermediate Jacobian fibrations with irreducible fibers associated to a cubic fourfold. |
| title | The defect of a cubic threefold and applications to intermediate Jacobian fibrations |
| topic | Algebraic Geometry 14J17, 14J30, 14J45 |
| url | https://arxiv.org/abs/2312.05118 |