An h-principle for complements of discriminants

2021/12/01 by Alexis Aumonier, Aumonier, Alexis · 1 citation
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #14J10 #14J70 (Secondary) #55R80 (Primary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2112.00326

openalex publication_date 2021/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compare spaces of non-singular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a range of degrees growing with the jet ampleness. As an application, when L is a very ample line bundle on a smooth projective complex variety, we prove that the rational cohomology of the space of non-singular algebraic sections of L^{⊗ d} stabilises as d → ∞ and compute the stable cohomology. We also prove that the integral homology does not stabilise using tools from stable homotopy theory.

Cited by

• Homological stability and weak approximation

Related

• Stable Cohomology of Discriminant Complements for an algebraic curve
• Scanning the moduli of smooth hypersurfaces
• Homological stability for the space of hypersurfaces with marked points
• A generalization of Vassiliev's h-principle
• Hartogs and open embeddings, proper maps, compactifications, cohomologies