Extension of the Topological Abel-Jacobi Map for Cubic Threefolds
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914627774840832 |
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| author | Zhang, Yilong |
| author_facet | Zhang, Yilong |
| contents | The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space $T_v$ of a Zariski open subspace of $(\mathbb P^4)^*$. Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of $T_v$ to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15751 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension of the Topological Abel-Jacobi Map for Cubic Threefolds Zhang, Yilong Algebraic Geometry 14J30, 32J05 primary, 32S55, 14C05 secondary The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space $T_v$ of a Zariski open subspace of $(\mathbb P^4)^*$. Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of $T_v$ to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective. |
| title | Extension of the Topological Abel-Jacobi Map for Cubic Threefolds |
| topic | Algebraic Geometry 14J30, 32J05 primary, 32S55, 14C05 secondary |
| url | https://arxiv.org/abs/2308.15751 |