Scanning the moduli of smooth hypersurfaces
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908876674170880 |
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| author | Aumonier, Alexis |
| author_facet | Aumonier, Alexis |
| contents | We study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth projective complex variety. In the spirit of scanning, we construct a map to a continuous section space of a projective bundle, and show that it induces an isomorphism in integral homology in a range of degrees growing with the ampleness of the hypersurfaces. When the ambient variety is a curve, this recovers a result of McDuff about configuration spaces. We compute the rational cohomology of the section space and exhibit a phenomenon of homological stability for hypersurfaces with first Chern class going to infinity. For simply connected varieties, the rational cohomology is shown to agree with the stable cohomology of a moduli space of hypersurfaces, with a peculiar tangential structure, as studied by Galatius and Randal-Williams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07560 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scanning the moduli of smooth hypersurfaces Aumonier, Alexis Algebraic Geometry Algebraic Topology 55R80, 14J70 (Primary), 14C05, 14M12, 57R15 (Secondary) We study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth projective complex variety. In the spirit of scanning, we construct a map to a continuous section space of a projective bundle, and show that it induces an isomorphism in integral homology in a range of degrees growing with the ampleness of the hypersurfaces. When the ambient variety is a curve, this recovers a result of McDuff about configuration spaces. We compute the rational cohomology of the section space and exhibit a phenomenon of homological stability for hypersurfaces with first Chern class going to infinity. For simply connected varieties, the rational cohomology is shown to agree with the stable cohomology of a moduli space of hypersurfaces, with a peculiar tangential structure, as studied by Galatius and Randal-Williams. |
| title | Scanning the moduli of smooth hypersurfaces |
| topic | Algebraic Geometry Algebraic Topology 55R80, 14J70 (Primary), 14C05, 14M12, 57R15 (Secondary) |
| url | https://arxiv.org/abs/2311.07560 |