The Geometry of Rings of Components of Hurwitz Spaces

TL;DR

This paper studies a commutative ring of components associated with Hurwitz spaces of covers of the projective line, analyzing its geometric structure and relating it to asymptotic counting invariants.

Contribution

It introduces a stratification of the prime spectrum of the ring of components and computes the dimensions and degrees of its strata, linking algebraic geometry with counting problems.

Findings

Stratification of the prime spectrum of the ring of components.

Explicit computation of dimensions and degrees of strata.

Complete description of the spectrum in specific cases.

Abstract

We consider a variant of the ring of components of Hurwitz spaces introduced by Ellenberg, Venkatesh and Westerland. By focusing on Hurwitz spaces classifying covers of the projective line, the resulting ring of components is commutative, which lets us study it from the point of view of algebraic geometry and relate its geometric properties to numerical invariants involved in our previously obtained asymptotic counts. Specifically, we describe a stratification of the prime spectrum of the ring of components, and we compute the dimensions and degrees of the strata. Using the stratification, we give a complete description of the spectrum in some cases.