Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3

TL;DR

This work proves the vanishing of $H^1$ for the Hurwitz spaces of fully-marked admissible covers of degree $3$, and deduces the same for the classical simply-branched Hurwitz spaces, while showing nonvanishing in degree $4$ via explicit 1-dimensional base cases. The authors develop an inductive framework that leverages the boundary stratification and a monodromy-centric encoding of covers, using a natural target map to $\overline{\mathcal{M}}_{0,m}$ to reduce genus computations to monodromy data and the Riemann–Hurwitz formula, aided by a Python tool for base-case calculations. This approach mirrors Arbarello–Cornalba’s method for moduli of curves, yet adapts it to Hurwitz spaces by analyzing boundary strata that involve generalized admissible covers. The paper provides concrete base-case data up to degree $5$, establishing vanishing for $d=3$ and demonstrating nonvanishing phenomena for $d\ge 4$, which has implications for the Picard Rank Conjecture and the odd-cohomology program in Hurwitz spaces.

Abstract

We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ vanishes for covers of degree $ d = 3$ and deduce the same result for the classical Hurwitz space of simply-branched covers. In degree 4, we compute examples where $H^1(\overline{\mathcal{H}}_{\underline{4},\underline{g}}(\underlineμ))$ is nonzero, which implies that $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ is nonvanishing for $d \geq 4$. We describe the stratification of the boundary of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$ by lower-dimensional $\overline{\mathcal{H}}_{\underline{d'},\underline{g'}}(\underline{μ'})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$.

Figures (7)

• Figure 1: A $1$-dimensional picture of an admissible cover with two labelings $\{L_k\}$ and $\{L'_k\}$ of $f^{-1} (q)$. In terms of $\{L_k\}$, the fiber over $b_i$ is associated to the permutation $(12)(3)(45)$, with markings $p_i^1, p_i^2, p_i^3$ respectively.
• Figure 2: The dual graphs of two boundary divisors of $\overline{\mathcal{H}}_{3,2}((2,1)^8)$ lying above a single component of $\Delta_j$ for $j=2$ (with marking data suppressed to minimize cluttering the picture). The two branch points which have bubbled off to the left vertex have associated transpositions in $S_3$. If these transpositions are identical, then the resulting boundary divisor is $\Delta_\Gamma$. If these transpositions have one element in common, then the resulting boundary divisor is $\Delta_{\Gamma'}.$
• Figure 3: The standard generating set for $\pi_1(\mathbb{P}^1 \setminus B, q)$.
• Figure 4: A loop in $\overline{\mathcal{M}}_{0,4}$ based at $\lambda$ and winding around $\infty$, which perturbs the genrators of $\pi_1(D\setminus B, q)$ into a new generating set on the right. This is a two-dimensional picture.
• Figure 5: The target map locally near $\lambda$. Two sheets of the Hurwitz space are given by fully-marked monodromy representations. This is a one-dimensional picture.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1: Fully-marked admissible covers
• Definition 2.2
• Definition 2.3: Morphisms of fully-marked admissible covers
• Definition 2.4: Hurwitz space of fully-marked admissible covers
• Remark 2.5
• Definition 3.1
• Remark 4.1
• Lemma 5.1: Injectivity lemma