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arXiv · 2311.07560

Scanning the moduli of smooth hypersurfaces

Alexis Aumonier
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Abstract

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.

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Alexis Aumonier. 2023-11-13. Scanning the moduli of smooth hypersurfaces. https://doi.org/10.1017/fms.2026.10184

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