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Scanning the moduli of smooth hypersurfaces

Alexis Aumonier

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.

Scanning the moduli of smooth hypersurfaces

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.
Paper Structure (39 sections, 62 theorems, 213 equations)

This paper contains 39 sections, 62 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Rational computations and stability
  3. Configuration spaces on curves
  4. Characteristic classes and moduli spaces of manifolds
  5. Outline
  6. The proof strategy for the main theorem
  7. The Picard scheme, jet bundles, and hypersurfaces
  8. The Picard scheme
  9. Jet bundles
  10. Jet ampleness
  11. The topology of hypersurfaces
  12. The moduli of hypersurfaces
  13. Moduli functors and the Hilbert scheme
  14. A convenient point-set model
  15. Statement of the main theorem
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth projective complex variety and let $\alpha \in \mathrm{NS}(X)$ be ample enough. Taking the first jet yields a map which induces an isomorphism in integral homology onto the path component that it hits, in degrees $* < \frac{d(\alpha) - 3}{2}$.

Theorems & Definitions (145)

  • Theorem 1.1: See \ref{['theorem:main-theorem']} for a precise version
  • Remark 1.2
  • Theorem 1.3: See \ref{['theorem:rational-cohomology-section-space-projective-bundle']} for a precise version
  • Theorem 1.4: See \ref{['corollary:rational-stability']}
  • Remark 1.5
  • Theorem 1.6: See \ref{['theorem:scanning-for-curves']}
  • Theorem 1.7: See \ref{['theorem:comparison-of-characteristic-classes']}, \ref{['corollary:simply-connected-gammans-bdiff']} and \ref{['corollary:without-mu-quotient']} for precise versions
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 135 more