Motivic random variables and representation stability II: Hypersurface sections

TL;DR

This paper establishes geometric and cohomological stabilization results for smooth hypersurface sections of fixed varieties as degree increases, revealing new examples of representation stability in algebraic geometry.

Contribution

It introduces new stabilization theorems for hypersurface sections and configuration spaces, with explicit formulas and probabilistic interpretations, extending representation stability concepts.

Findings

Stabilization of the Hodge Euler characteristic for hypersurface families

Explicit formulas for stable values using probabilistic models

New geometric examples of weak representation stability

Abstract

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of varieties, and deduce from this the stabilization of the Hodge Euler characteristic of natural families of local systems constructed from the vanishing cohomology. We prove explicit formulas for the stable values using a probabilistic interpretation, along with the natural analogs in point counting over finite fields. We explain how these results provide new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. This interpretation of representation stability was studied in the prequel for configuration spaces.