{"id":"9f69d057-787c-4709-93e0-a9184ffa407d","arxiv_id":"2501.16320","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper computes the integer-coefficient Chow rings of the moduli stacks of hyperelliptic Prym pairs, for curves with one marked pair of Weierstrass points (any genus) and for odd genus with up to (g-1)/2 pairs. These rings record intersection-theoretic information about the geometry of these moduli spaces, and the computation is new beyond genus 2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.2 asserts a Chow-envelope property for the maps M without proof or reference; the discriminant ideal in Theorems 1.17–1.18 depends on it.","rationale":"The reader's weakest-assumption analysis identifies exactly the same point: Lemma 5.2 is the computational foundation for the ideal I2, and its proof is delegated. I agree that this is the most load-bearing unproven step. The concern is concrete and localized: the statement covers three families of maps, but the provided references and proof sketch address only F and G. The M family, which controls the common-factor component ∆1,2, is left without support. Since the final presentations in Theorems 1.17 and 1.18 are obtained by dividing by I2, any missing discriminant relation would directly invalidate the central claim. At the same time, the property is checkable and almost certainly true, because M_1 alone should be birational onto the irreducible hypersurface ∆1,2. Thus the appropriate disposition is conditional acceptance: the paper's argument is coherent and the gap is fillable, but the missing verification for M should be supplied before the computation is regarded as complete. I do not see a deeper structural error; the characteristic assumption, the root-gerbe formula, and the n=1 computations are consistent with the cited literature and with internal cross-checks.","tokens_in":42305,"tokens_out":34851,"duration_ms":294220,"concrete_test":"Give a direct proof that M_1: P^1 × P^{2a-1} × P^{2b-1} → P^{2a} × P^{2b}, (h,f,g) ↦ (hf,hg), is birational onto its image ∆1,2: over the open subset of pairs (F,G) with a single simple common root, the linear factor h is uniquely determined up to scalar, so the map is one-to-one. Since ∆1,2 is irreducible and M_1 is dominant, this yields a Chow envelope for r=1; the maps M_r for r>1 are then unnecessary for surjectivity. Checking this (or finding a counterexample) settles whether the missing step in Lemma 5.2 lands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 5.2 asserts that the disjoint unions F, G, M of the maps F_r, G_r, M_r form PGL2-equivariant Chow envelopes of the discriminant components ∆1, ∆2, ∆1,2, with surjective pushforwards. The proof is dismissed as 'standard', with references [Vis98], [EF09], [DL18] given only for F and G. No argument or reference is supplied for M, the envelope of the common-factor component ∆1,2. This is load-bearing: the subsequent computation of Im(i1,2∗) in §5.2 (Lemmas 5.4–5.6, Corollary 5.9, Lemma 5.19) assumes that the pushforwards M_r∗ generate the Chow groups of ∆1,2, and the ideal I2 in Theorem 1.17—and hence Theorem 1.18—is exactly the discriminant contribution. If the Chow-envelope property for M failed, the displayed presentations would have extra relations, and Theorems 1.17 and 1.18 would be incomplete. The property is very likely true (M_1 should be birational onto the irreducible hypersurface ∆1,2 over the locus of a single simple common root), but it is not proved or cited, and it is not covered by the references given.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the integral Chow rings of the moduli stacks RH^1_g of hyperelliptic Prym pairs with a single pair of Weierstrass points (all g), and of RH^n_g for odd g and 1<n<(g+1)/2. The main results are Theorem 1.10 (even g, n=1), Theorem 1.14 (odd g, n=1), and Theorems 1.17–1.18 (odd g, 1<n<(g+1)/2). The strategy is to give explicit presentations as quotient stacks, compute the relevant equivariant Chow rings, and then pass to µ2-root gerbes. The genus-2 specialization is checked against [CIL24], and the characteristic assumption is sharpened from char(k)>2g to char(k)>2g+2, with Example 1.1 motivating the refinement.","tokens_in":42563,"tokens_out":5905,"duration_ms":53014,"significance":"If the results are correct, the paper provides the first complete integral Chow ring presentations for these Prym moduli stacks, substantially extending the genus-2 work of [CIL24] and the prior hyperelliptic computations of [EF09, FV11, DL18]. The methods are well suited to the problem: explicit quotient presentations, torus-equivariant reduction via Lemma 3.3, GL3-counterparts following [DL18], and the root-gerbe formula of Proposition 3.5. The paper also recovers known results in the boundary case g=2 and improves the characteristic hypothesis. The main limitations are completeness of proof rather than apparent error: several load-bearing checks are delegated to the reader, and the Chow-envelope statement for the common-factor discriminant component is asserted without proof. These gaps are local but essential, and they affect the derivation of the discriminant ideal in Theorems 1.17–1.18.","major_comments":[{"comment":"The assertion that the disjoint unions F, G, M form PGL2-equivariant Chow envelopes of the discriminant components ∆1, ∆2, ∆1,2 with surjective pushforwards is load-bearing: §5.2 uses this property to define the ideal I2 through the images of M1∗ and M2∗ (Lemmas 5.4–5.6 and Corollary 5.9), and I2 is exactly the discriminant contribution in Theorem 1.17, hence in Theorem 1.18. The proof is dismissed as 'standard', and the cited references [Vis98], [EF09], [DL18] cover F and G but do not cover M. A proof or a precise reference for the envelope property of the maps M_r must be supplied. The property is very plausible—M_1 should be birational onto the generic locus of the irreducible hypersurface ∆1,2—but it is not proved, and the displayed presentations would be incomplete if it failed.","section":"§5.1, Lemma 5.2"},{"comment":"The proof of Theorem 1.14 is only a sketch: the statement that the relations coming from ∆1, ∆2 generate the ideal (31), and the statement that the projective-bundle relation p(ξ2g) vanishes modulo (31), are both left to the reader. Since Theorem 1.14 is one of the two main n=1 results and there is no external genus-2 check for the odd-parity case, these verifications should be written out, or at least reduced explicitly to the even-g computation with the parity substitutions made.","section":"§4, Proof of Theorem 1.14"},{"comment":"The final step of the induction asserts that the class α of equation (41) is non-zero in CH∗_{G3_m}((P(V_a) ×_S P(V_b)) \\ ∆′) for any non-zero polynomial p with coefficients 0 and 1. The argument restricts to a residual gerbe BG′ and then claims that p(c2,c3)c3 is non-zero there, but the injectivity of the pullback ι∗ used in this conclusion is not established. This step is load-bearing for the completion of the induction that all pushforwards M_{r∗} land in I2; it needs a fuller justification.","section":"§5.2, Lemma 5.19 and proof of Theorem 1.17"}],"minor_comments":[{"comment":"The word 'isomophism' should be 'isomorphism'.","section":"Definition 1.2"},{"comment":"The heading contains the typo 'equivarinat'; it should be 'equivariant'.","section":"§3.3"},{"comment":"The formula M′_{2r} = (ψ′_{r,r−a} × ψ′_{r,r−b}) ∘ Δ uses negative subscripts when a,b > r; it should presumably be ψ′_{r,a−r} and ψ′_{r,b−r}, matching the usage later in the section.","section":"§5.2, definition of M′_{2r}"},{"comment":"The classes s_r^j are used without definition; a sentence recalling their definition from [Lar21] would improve readability.","section":"Lemma 4.9"},{"comment":"The phrase 'On the other end' should be 'On the other hand'.","section":"Remark 5.12"},{"comment":"The symbol ∆ is used both for the discriminant locus in the quotient presentations of §1.2 and for the union of discriminant components in §5; the two uses are mathematically close but should be distinguished for clarity.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is well within the scope of a serious algebraic geometry journal, and I see no evidence of circularity or missing attribution: the input rings come from prior independent work, and the genus-2 specialization matches [CIL24]. The reason for major_revision is not a suspected error but the incompleteness of the proof of Lemma 5.2 and of several delegated computational checks. Once those are supplied, I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious computation paper, and the main results are new: for all g it computes CH*(RH^1_g), and for odd g and 1<n<(g+1)/2 it computes CH*(RH^n_g), plus the quotient presentations and the Chow ring of D_{2a,2b}. The genus-2 case recovers [CIL24], which gives an external check, and the paper is honest that the case n=(g+1)/2 is deferred. The characteristic assumption is sharpened from 2g to 2g+2, with a concrete counterexample for char 5 in Example 1.1; that is careful, real work.\n\nThe argument is built on the established equivariant Chow toolbox (Edidin-Graham, Vistoli, Di Lorenzo, Edidin-Hu). I checked the overall architecture: presentations as quotient stacks, root-stack formula, then excision into discriminant components. The algebra is heavy but mostly coherent. Theorems 1.10 and 1.17/1.18 have enough detail in the pushforward computations that a referee can verify them, though not painlessly.\n\nWhere I would push back: Lemma 5.2 is the real soft spot. It asserts that the maps F, G, and M form PGL2-equivariant Chow envelopes of Delta_1, Delta_2, and Delta_{1,2}, with surjective pushforward. The proof says 'standard' and points to [Vis98], [EF09], [DL18] only for F and G. M, the one controlling the common-factor component Delta_{1,2}, gets no argument or reference. Everything after it, including M2* and the discriminant ideal I2 in Theorem 1.17, assumes the envelope statement. I think it is true (M1 is birational onto the generic locus of Delta_{1,2}), so this looks fixable, but it is not a trivial remark: the characteristic hypothesis is exactly there to make such envelope arguments work. A referee should demand a proof of Lemma 5.2 or a precise citation.\n\nTwo smaller things. Theorem 1.14 is only sketched, with a 'we leave to the reader' check that the first relations generate the stated ideal; that check is routine but not instant. There are also several 'simple calculation' steps in Section 4. They are probably fine, but they slow verification.\n\nThe citation pattern looks healthy: prior integral Chow rings of hyperelliptic stacks are the scaffolding, and [CIL24] is used as a benchmark rather than as an answer. I did not find circularity.\n\nWho is this for? Specialists in moduli of curves, equivariant intersection theory, and enumerative geometry. It deserves a serious referee: the stakes are genuine, the architecture is sound, and the main gap is identifiable and likely repairable. I would send it to review, with instructions to focus the referee on Lemma 5.2 and on the verification of the delegated algebraic checks.","headline":"A genuinely new integral Chow ring computation for an infinite family of Prym moduli stacks, with one under-documented envelope lemma in the load-bearing part.","tokens_in":43125,"tokens_out":2615,"would_cite":true,"duration_ms":26060,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C15","14H10","14H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives explicit generator-and-relation presentations, with integer coefficients, of the Chow rings of the moduli stacks of hyperelliptic Prym pairs, covering a single Weierstrass pair in every genus and, when the genus is odd, up…","keywords":["integral Chow ring","moduli stack","Prym pair","hyperelliptic curve","equivariant intersection theory","quotient stack","root gerbe","Weierstrass points"],"falsifier":"Compute the degree-two Chow group of the quotient stack in Theorem 1.8 directly from the localization sequence for $\\Delta=\\Delta_1\\cup\\Delta_2$, using the explicit pushforwards in Lemmas 4.8, 4.9, and 4.12; the announced ring forces the class $\\beta_2$ to have order exactly $4g$, and any different torsion order would refute Theorem 1.10.","tokens_in":42076,"feed_emoji":"🧮","tokens_out":10762,"duration_ms":96018,"temperature":0.7,"pith_summary":"The paper aims to determine, completely and with integer coefficients, the Chow rings of the moduli stacks of hyperelliptic Prym pairs: smooth hyperelliptic curves equipped with a nontrivial square root of the trivial line bundle. It proves that the locus where the square root comes from one pair of Weierstrass points is a quotient stack by an explicit two-dimensional group, and from that presentation it derives the full integral Chow ring for every genus. For odd genus it also computes the rings when the square root comes from up to $(g-1)/2$ pairs of Weierstrass points, by viewing the stack as a $\\mu_2$-root gerbe over a stack of two disjoint even-degree divisors and combining equivariant Chow envelopes with a root-gerbe formula. These formulas record all additive torsion, not only rational information, and they extend the previously known genus-$2$ computation to all genera. If correct, they give the first complete integral Chow rings for these moduli stacks outside the genus-$2$ case.","feed_headline":"Exact Chow rings for hyperelliptic Prym moduli stacks","feed_subtitle":"Explicit generators and relations, with all 2-torsion, for n=1 in every genus and for odd genus up to (g−1)/2 pairs.","key_machinery":"The carrying mechanism is the presentation of $\\mathcal{RH}^n_g$ as a quotient stack together with the root-gerbe description. The key structural input is a lemma of Verra, quoted as Lemma 1.6, which decomposes the stack of hyperelliptic Prym pairs into loci $\\mathcal{RH}^n_g$ indexed by $n$: every nontrivial square root of the trivial bundle is exactly $H^{\\otimes n}(-e)$ for a reduced effective divisor $e$ supported on $2n$ Weierstrass points. The paper then computes the Chow rings of the relevant quotient stacks using equivariant intersection theory: the projective bundle formula, the excision sequence for the discriminant $\\Delta$, and three explicit $\\mathrm{PGL}_2$-equivariant Chow envelopes $F$, $G$, and $M$ for the three components of $\\Delta$; a Chow envelope is a collection of maps from simpler spaces whose pushforwards generate the Chow group of the closed locus. Finally, the root-gerbe result of Proposition 3.5 converts the $\\mu_2$-gerbe structure of $\\mathcal{RH}^n_g$ over $\\mathcal{D}_{2n,2g+2-2n}$ into the single additional relation $\\xi_{2n}+\\xi_{2g+2-2n}+2t=0$.","core_discovery":"The central discovery is that the integral Chow ring of these stacks has an explicit presentation by generators and relations. For $n=1$, the paper proves $\\mathrm{CH}^*(\\mathcal{RH}^1_g) = \\mathbb{Z}[\\beta_1,\\beta_2,\\gamma]/(2\\beta_1,2\\gamma,4g\\beta_2,\\gamma(\\gamma+\\beta_1),\\beta_1(\\beta_1+\\gamma))$ when $g$ is even, and $\\mathrm{CH}^*(\\mathcal{RH}^1_g) = \\mathbb{Z}[c_2,t,\\gamma]/(2\\gamma,4t,\\gamma^2+gc_2)$ when $g$ is odd. For odd $g$ and $1<n<(g+1)/2$, it proves $\\mathrm{CH}^*(\\mathcal{RH}^n_g) = \\mathbb{Z}[c_1,c_2,c_3,t,\\xi_{2n},\\xi_{2g+2-2n}]/(I+(\\xi_{2n}+\\xi_{2g+2-2n}+2t))$, where $I$ is the ideal of relations of the auxiliary stack $\\mathcal{D}_{2n,2g+2-2n}$ listed in Theorem 1.17. The formulas are obtained from quotient-stack presentations with group $(\\mathbb{G}_m\\times\\mathbb{G}_m)\\rtimes\\mu_2$ in the even case and $\\mathbb{G}_m\\times(\\mathbb{G}_m\\rtimes\\mu_2)$ in the odd case, from the identification of $\\mathcal{RH}^n_g$ as a $\\mu_2$-root gerbe over $\\mathcal{D}_{2n,2g+2-2n}$, and from exhaustive computation of all discriminant contributions through equivariant Chow envelopes.","pith_inferences":["A natural extension is the complementary case $n=(g+1)/2$ for odd $g$, where Verra's lemma says the map $\\beta_n$ is 2-to-1; one can test whether that 2-to-1 behavior introduces an extra $\\mathbb{Z}/2$ quotient or a new root-gerbe relation in the Chow ring.","Theorem 1.17 already computes $\\mathrm{CH}^*(\\mathcal{D}_{2a,2b})$ for all $a,b>1$, so the remaining boundary cases $a=1$ or $b=1$, treated ad hoc for $n=1$, could plausibly be unified into one formula that completes the odd-genus story.","The paper's stronger characteristic assumption, $\\mathrm{char}(k)=0$ or $>2g+2$, is used only in the proof of the Chow-envelope lemma; if that lemma can be reproved in lower characteristic, the same presentations and Chow-ring formulas should remain valid in positive characteristic.","The explicit order $4g$ forced on $\\beta_2$ in the even-genus ring suggests a concrete numerical check: evaluating the top Chern class on the quotient presentation of Theorem 1.8 for a small even genus should reproduce the known degree of the moduli stack, which would independently confirm the coefficient $4g$ in the relation."],"forward_implications":["For even $g$, the computed ring makes the torsion explicit: $\\beta_1$ and $\\gamma$ are 2-torsion classes and $\\beta_2$ satisfies $4g\\beta_2=0$, so the integral Picard group and the higher torsion are completely pinned down by the presentation.","For odd $g$ and $n=1$, the relation $\\gamma^2+gc_2=0$ ties the Weierstrass-pair class $\\gamma$ to the hyperelliptic class $c_2$, and $4t=0$ records a 4-torsion line-bundle class coming from the quotient presentation.","For odd $g$ and $1<n<(g+1)/2$, the two divisor classes $\\xi_{2n}$ and $\\xi_{2g+2-2n}$ are not independent on $\\mathcal{RH}^n_g$: their sum is forced to equal $-2t$, so the Chow ring is obtained from that of $\\mathcal{D}_{2n,2g+2-2n}$ by one clean relation.","The same geometric input yields a quotient-stack presentation of $\\mathcal{RH}^n_g$ for all odd $g$ and $1\\le n<(g+1)/2$, giving a uniform description of the whole family beyond the single-pair case.","Because the computations are integral, the resulting rings carry all additive torsion and are directly usable for enumerative intersection-theoretic questions on these moduli stacks, not merely for rational characteristic-class computations."],"supporting_citations":[{"why":"Gives the bijection between nontrivial square roots of the trivial bundle on a hyperelliptic curve and reduced effective divisors on Weierstrass points, which yields the decomposition of the Prym-pair stack into the loci $\\mathcal{RH}^n_g$.","marker":"[Ver13, Lemma 4.3]"},{"why":"Supplies the genus-2 quotient-stack presentation and several pushforward and class computations that the present proof adapts to all genera.","marker":"[CIL24]"},{"why":"Provides the Chow-envelope argument for the discriminant component of polynomials with a double root and the integral Chow ring of $\\mathcal{M}_2$ that this paper extends.","marker":"[Vis98]"},{"why":"Computes $\\mathrm{GL}_2$-equivariant pushforwards along multiplication maps and the integral Chow ring of the even-genus hyperelliptic stack, giving the image of the double-root locus for even $g$.","marker":"[EF09]"},{"why":"Introduces $\\mathrm{GL}_3$-counterparts for $\\mathrm{PGL}_2$-spaces, computes the Chow ring of the odd-genus hyperelliptic stack, and supplies the pushforwards $\\pi_{2*}(\\xi_2^2)$ and the classes $[W_{m;1,0}]$, $[W_{m;2,0}]$ used in Section 5.","marker":"[DL18]"},{"why":"Computes the Chow ring of $B((\\mathbb{G}_m\\times\\mathbb{G}_m)\\rtimes\\mu_2)$ and the relevant multiplication-map pushforwards, fixing the notation $\\beta_1,\\beta_2$.","marker":"[Lar21]"},{"why":"Provides pushforwards along multiplication maps on products of projective lines and the integral Chow ring of cyclic covers, used for $\\pi_{1*}(1)$, $\\pi_{1*}(\\tau)$ and the $\\mathbb{Z}_{(2)}$-local computations.","marker":"[FV11]"},{"why":"Gives the presentation of the hyperelliptic stack and of cyclic-cover stacks as quotients by $\\mathbb{G}_m\\times\\mathrm{PGL}_2$ or $\\mathrm{GL}_2$, including the parity-dependent isomorphism used in Lemma 2.1.","marker":"[AV04]"},{"why":"Establishes equivariant intersection theory with integer coefficients and the notion of Chow envelope used throughout the computation.","marker":"[EG98]"}],"fun_headline_variants":["Explicit Chow rings for hyperelliptic Prym stacks","Prym moduli Chow rings: all 2-torsion explicitly","Chow ring generators and relations for Prym stacks","Integral Chow rings of Prym stacks with torsion","Hyperelliptic Prym Chow rings: new explicit formulas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the claim, cited to earlier work rather than proved here, that the three explicit maps $F$, $G$, and $M$ form surjective Chow envelopes for the three discriminant components; if that claim failed, some discriminant relations could be missing from the computed ideals.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Chow rings for hyperelliptic Prym stacks","Prym moduli Chow rings: all 2-torsion explicitly","Chow ring generators and relations for Prym stacks","Integral Chow rings of Prym stacks with torsion","Hyperelliptic Prym Chow rings: new explicit formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000401,"raw_usage":{"total_tokens":2113,"prompt_tokens":985,"completion_tokens":1128,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":1045}},"tokens_in":601,"tokens_out":1128,"duration_ms":11015,"temperature":1.0,"reasoning_tokens":1045,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T13:31:58.704862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the degree-two Chow group of the quotient stack in Theorem 1.8 directly from the localization sequence for $\\Delta=\\Delta_1\\cup\\Delta_2$, using the explicit pushforwards in Lemmas 4.8, 4.9, and 4.12; the announced ring forces the class $\\beta_2$ to have order exactly $4g$, and any different torsion order would refute Theorem 1.10.","supporting_citations":[],"review_version":1}