Sections and Unirulings of Families over ℙ1
We consider morphisms $\pi : X \to \mathbb{P}^{1}$ of smooth projective varieties over $\mathbb{C}$ . We show that if π has at most one singu...
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2024
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| Online Access: | https://hdl.handle.net/1721.1/154260 |
| _version_ | 1811078698655285248 |
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| author | Pieloch, Alex |
| author_facet | Pieloch, Alex |
| author_sort | Pieloch, Alex |
| collection | MIT |
| description | We consider morphisms
$\pi : X \to \mathbb{P}^{1}$
of smooth projective varieties over
$\mathbb{C}$
. We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π.
To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections. |
| first_indexed | 2024-09-23T11:04:03Z |
| format | Article |
| id | mit-1721.1/154260 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:04:03Z |
| publishDate | 2024 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1542602024-04-23T03:43:30Z Sections and Unirulings of Families over ℙ1 Pieloch, Alex Geometry and Topology Analysis We consider morphisms $\pi : X \to \mathbb{P}^{1}$ of smooth projective varieties over $\mathbb{C}$ . We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections. 2024-04-22T14:57:07Z 2024-04-22T14:57:07Z 2024-04-18 2024-04-21T03:10:37Z Article http://purl.org/eprint/type/JournalArticle 1016-443X 1420-8970 https://hdl.handle.net/1721.1/154260 Pieloch, A. Sections and Unirulings of Families over P1. Geom. Funct. Anal. (2024). PUBLISHER_CC en 10.1007/s00039-024-00679-6 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Science and Business Media LLC Springer International Publishing |
| spellingShingle | Geometry and Topology Analysis Pieloch, Alex Sections and Unirulings of Families over ℙ1 |
| title | Sections and Unirulings of Families over ℙ1 |
| title_full | Sections and Unirulings of Families over ℙ1 |
| title_fullStr | Sections and Unirulings of Families over ℙ1 |
| title_full_unstemmed | Sections and Unirulings of Families over ℙ1 |
| title_short | Sections and Unirulings of Families over ℙ1 |
| title_sort | sections and unirulings of families over p1 |
| topic | Geometry and Topology Analysis |
| url | https://hdl.handle.net/1721.1/154260 |
| work_keys_str_mv | AT pielochalex sectionsandunirulingsoffamiliesoverp1 |