Alexis Aumonier: Moduli of embedded hypersurfaces

Abstract

What is the cohomology of the space of smooth hypersurfaces embedded inside a given projective complex variety? As a first step, one can restrict the family of hypersurfaces in two ways: either fixing the linear equivalence class, or fixing the Chern class of the hypersurfaces. I will explain how to partially answer the question in the first case using tools from algebraic topology. The second case is part of ongoing work which I will also outline. A main idea is to "scan" the parameter space of hypersurfaces and compare it to an associated continuous section space. In the case of curves, where a hypersurface is a configuration of points, this recovers a result of McDuff.