Low degree Hurwitz stacks in the Grothendieck ring

TL;DR

This work establishes stabilization in the Grothendieck ring of stacks for Hurwitz spaces of low-degree covers (degrees $2$–$5$) as genus grows, after normalizing by $\mathbb{L}^{\dim\mathrm{Hur}_{d,g,k}}$. The authors develop low-degree Casnati–Ekedahl parametrizations, express the Hurwitz stacks as quotient stacks, and compute local contributions via motivic Euler products and Tamagawa-number–type mass formulas, enabling explicit limit expressions such as $\lim_{g\to\infty}\{\mathrm{Hur}_{d,g,k}^{s}\}/\mathbb{L}^{\dim}\mathrm{Hur}_{d,g,k}=1-\mathbb{L}^{-2}$ for $d\le5$. They provide codimension bounds to control error terms, derive special simplifications for simply branched covers, and connect to arithmetic/topological motivations, including analogies with Bhargava’s discriminant density, Ellenberg–Venkatesh–Westerland stability, and vector-bundle methods on $\mathbb P^1$. The results form a motivic analogue of classical density theorems in number fields and offer a toolkit (CE-strata, motivic Euler products, and local-class computations) for probing stabilization phenomena in low-degree covers. The paper also discusses higher-degree conjectures, possible second-order terms, and broader questions about extending the stabilization pattern beyond $d=5$, and about other Galois groups and ramification structures.

Abstract

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb P^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

Theorems & Definitions (145)

• Theorem 1.1: Theorem A
• Theorem 1.2: Theorem B
• Remark 1
• Corollary 2
• Corollary 5
• Conjecture 6
• Remark 7
• Remark 8
• Corollary 9
• Definition 10