REVIEW 2 minor 2 cited by
The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves
T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read The integral Chow ring of the stack of n-pointed hyperelliptic curves is computed in full for n=1 and 2, and up to one degree-2 class for 3 up to 2g+2.
desk verdict This paper computes the integral Chow ring of H_{g,n} completely for n=1,2 and up to one unknown order in degree 2 for 3≤n≤2g+2, with the g=2 case giving new data on M_{2,n}. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The stack H_{g,n} of n-pointed smooth hyperelliptic curves of genus g, equipped with its universal curve and the tautological classes it carries.
What would settle it
An explicit cycle on H_{g,3} for small g whose self-intersection or order in the Chow group differs from the predicted value in codimension 2.
Extended reading notes
Core claim
The integral Chow ring of H_{g,n} admits an explicit presentation generated by classes pulled back from the hyperelliptic stack together with classes coming from the universal curve; the relations hold integrally for n=1 and 2 in all degrees, and for 3≤n≤2g+2 they determine the ring except for the order of one codimension-2 class.
Load-bearing premise
The usual generators and relations coming from the geometry of the hyperelliptic stack and its universal curve generate the full integral Chow ring without hidden torsion or extra relations that would change the additive order of the degree-2 class.
Editorial extensions
If this is right
- All intersection numbers on H_{g,1} and H_{g,2} can be computed from the given presentation.
- The same holds for M_{2,n} when n≤7.
- For 3≤n≤2g+2 the ring structure is known except for possible 2-torsion or higher torsion in one class.
- Partial results for n=2g+3 give bounds on further relations.
Reading between the lines
- The computations suggest that integral torsion in these Chow rings appears only in low degree and is controlled by the geometry of the universal curve.
- One could test the result by computing the order of the degree-2 class directly via localization or via the geometry of the hyperelliptic involution for small g.
- The same method may extend to other moduli stacks whose Chow rings are generated by similar tautological classes.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript computes the integral Chow ring of the moduli stack H_{g,n} of n-pointed smooth hyperelliptic curves of genus g. Complete presentations (generators and relations over Z) are obtained for n=1 and n=2; for 3 ≤ n ≤ 2g+2 the ring is determined up to the additive order of a single class in degree 2; partial results are given for n=2g+3. Specializing to g=2 recovers results for CH^*(M_{2,n}) when 1 ≤ n ≤ 7.
Significance. If the stated presentations are correct, the work supplies explicit integral generators and relations for these Chow rings, a concrete advance in the study of moduli stacks of curves. The precise tracking of the additive order of the remaining degree-2 class, together with the direct integral approach, is a notable strength. The reduction to the case of M_{2,n} for small n connects the results to an intensively studied low-genus moduli space.
minor comments (2)
- The abstract states the ranges of n for which complete or partial results are obtained, but does not indicate the method (e.g., pullbacks from the universal curve or explicit relation-checking) used to establish completeness of the presentations; a one-sentence outline would help readers locate the key arguments.
- [Introduction] Notation for the generators coming from the hyperelliptic involution and the marked points is introduced gradually; collecting the full set of generators and their degrees in a single early table or proposition would improve readability.
Simulated Author's Rebuttal
We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.
Circularity Check
No significant circularity
full rationale
The derivation consists of an explicit presentation of generators (from the hyperelliptic involution and marked points) and relations (pulled back from the universal curve), followed by direct verification that this presentation is complete over Z in the stated degrees. No equation reduces to a fitted parameter renamed as a prediction, no load-bearing premise rests on a self-citation chain, and no ansatz is smuggled via prior work by the same author. The computation is self-contained against external geometric benchmarks and does not rely on any of the enumerated circular patterns.
Assumptions & free parameters
assumptions (1)
- standard math Standard properties of integral Chow rings for smooth stacks and the universal curve over the hyperelliptic stack hold without extra torsion.
Cite this review
Pith. "Pith review of The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves." pith.science (2026). https://pith.science/paper/2404.15873
@misc{pith2026240415873,
author = {Pith},
title = {Pith review of: The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves},
year = {2026},
howpublished = {\url{https://pith.science/paper/2404.15873}},
note = {Machine review of arXiv:2404.15873}
}
abstract
We study the integral Chow ring of the stack $\mathcal{H}_{g,n}$ parametrizing $n$-pointed smooth hyperelliptic curves of genus $g$. We compute the integral Chow ring of $\mathcal{H}_{g,n}$ for $n=1,2$ completely, while for $3\leq n\leq2g+2$ we compute it up to the additive order of a single class in degree 2. We obtain partial results also for $n=2g+3$. In particular, taking $g=2$ and recalling that $\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, our results hold for $\mathrm{CH}^*(\mathcal{M}_{2,n})$ for $1\leq n\leq7$.
Forward citations
Cited by 2 Pith papers
-
The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs I
The integral Chow rings of the moduli stacks RH^1_g and RH^n_g (odd g, 1<n<(g+1)/2) are explicitly computed as quotients of polynomial rings.
-
The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs II
For odd genus g, the integral Chow rings of the balanced hyperelliptic Prym component and of the unordered divisor stack are explicitly computed as quotient rings.
Reviewed May 24, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.