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This paper proves that intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces are governed, in a range of degrees, by the Betti numbers of Hilbert schemes of points, and that on Enriq

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2026-08-03 20:45 UTC pith:JQEVR5QP

load-bearing objection Genuinely new stabilization results for Enriques and bielliptic surfaces, but the proof leans on black-box citations that a referee must verify. the 3 major comments →

arxiv 2511.18426 v3 pith:JQEVR5QP submitted 2025-11-23 math.AG

Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces

Fei Si , Feinuo Zhang This is my paper
classification math.AG MSC 14D2014C0514F4514J28
keywords intersection cohomologymoduli spaces of sheavesone-dimensional sheavesHilbert scheme of pointsBetti number stabilizationEnriques surfacesbielliptic surfacesperverse filtration
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a bridge between two a priori unrelated families of moduli spaces: the moduli space M_{β,χ} of one-dimensional semistable sheaves on a smooth projective surface, and the Hilbert scheme of points on the same surface. It proves that, assuming M_{β,χ} is irreducible and its most singular part is of large codimension, the intersection Betti numbers of M_{β,χ} in low degrees match the Hilbert-scheme Betti numbers given by the standard product formula. The key geometric input is a new smoothness result: for sufficiently positive curve classes, a large open subset of the relative compactified Jacobian over integral curves is smooth. On Enriques and bielliptic surfaces, this leads to a stabilization theorem: for any fixed degree k, the k-th intersection Betti number of M_{dβ0} is eventually constant, equal to the Hilbert-scheme number. A reader should care because it gives the first stabilization result for these moduli spaces in the Enriques and bielliptic cases, and it isolates the two conditions—irreducibility and codimension of the singular locus—that control the phenomenon.

Core claim

The central claim is Theorem 1.1: if M_{β,χ} is irreducible, β is sufficiently very ample, and the locus over non-integral curves has codimension c with k≤c−1 and k≤(2/3)dim|β|, then dim_Q IH^k(M_{β,χ})=b∞_k, where b∞_k is the Hilbert-scheme Betti number given by the product formula. The proof compares perverse filtrations on a smooth open part h^{-1}(U) with Betti numbers of relative Hilbert schemes of points; a new smoothness theorem shows such an open part exists with complement of codimension at least k+1 when β is (2k−2)-very ample. In the Enriques and bielliptic cases this yields stabilization for all sufficiently positive multiples of any ample class, and in the generic Enriques case

What carries the argument

The relative Hilbert scheme C^{[ℓ]}_U of ℓ points on the universal family of integral curves, together with the perverse filtration induced by the Hilbert–Chow morphism h:M_{β,χ}→|β|. A support-theoretic identity expresses Betti numbers of C^{[ℓ]}_U as sums of graded pieces of that perverse filtration; the k-very ampleness of β makes C^{[ℓ]} a projective bundle over the Hilbert scheme S^{[ℓ]}, so its Betti numbers are known. The new smoothness result is obtained by showing the universal family of curves is locally versal on a large open set, using the fact that (2ℓ−2)-very ampleness forces the relevant restriction maps, controlled by Tjurina numbers of singularities, to be surjective.

Load-bearing premise

The main theorem is conditional on the moduli space M_{β,χ} being irreducible; for Enriques and bielliptic surfaces the paper proves this only under gcd(β·H,χ)=1, and without that coprime condition the equality of codimensions in (5.19) can fail, so the stabilization proof would break.

What would settle it

Take a bielliptic or Enriques surface S, an ample class β0, and an integer χ such that dβ0·H and χ are not coprime for all d. If M_{dβ0,χ} admits strictly semistable sheaves and its intersection Betti number IH^1(M_{dβ0,χ}) differs from b∞_1—or if M_{dβ0,χ} has an irreducible component supported entirely over non-integral curves—then the stabilization claim as stated fails, because the proof's equality (5.19) would be false in that case.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • For any smooth projective surface satisfying the two hypotheses, intersection Betti numbers of M_{β,χ} in degrees up to roughly (2/3)dim|β| are independent of χ and of the fine structure of β; they equal the Hilbert-scheme numbers b∞_k.
  • On Enriques and bielliptic surfaces, each fixed-degree intersection Betti number IH^k(M_{dβ0}) is eventually constant as d→∞, equal to b∞_k.
  • When M_{dβ0} is smooth, as in the generic Enriques case with β0 not divisible by 2, the same stabilization applies to ordinary Betti numbers, so all odd Betti numbers of fixed degree vanish for sufficiently large odd multiples.
  • In the generic Enriques case, the refined perverse Hodge numbers n^{i,j}_{dβ0} stabilize to the coefficients n^{i,j}_∞ of the product formula (1.2), matching the conjectural refined Gromov–Witten/Pandharipande–Thomas invariants of the local surface.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the equality c = codim(|β| \ |β|^int, |β|) could be proven without the coprimality assumption, the same stabilization would likely hold for all Euler characteristics χ on Enriques and bielliptic surfaces; the paper's Remark 5.4 identifies this as the only missing piece.
  • The explicit threshold d(β0,i,j) in Remark 5.9 scales roughly like a constant times sqrt(i+j)/β0^2, suggesting stabilization may begin quite early; computing an explicit low-degree example on a bielliptic surface could test how sharp the bound is.
  • Because the argument uses only the asymptotic growth of N(dβ) and k-very ampleness, it may extend to other surfaces with numerically trivial canonical class, such as abelian or K3 surfaces, provided the two hypotheses—irreducibility and constant fiber dimension—are verified.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies intersection Betti numbers of the coarse moduli space M_{β,χ} of H-semistable one-dimensional sheaves with fixed determinant O_S(β) and Euler characteristic χ on a smooth projective surface. The main theorem (Thm. 1.1) states that, assuming M_{β,χ} is irreducible and β is sufficiently positive, the k-th intersection Betti number equals the stable Hilbert-scheme Betti number b∞_k for k ≤ min(c−1, 2/3 dim |β|). The proof introduces a big open locus U_ℓ ⊂ |β|^int over which the relative compactified Jacobian is smooth (Thm. 1.2), obtained from local versality of the universal family of curves, and then uses the perverse filtration comparison of Maulik–Yun and Migliorini–Shende together with Göttsche's formula. For Enriques and bielliptic surfaces the authors verify irreducibility (Prop. 5.1) and unboundedness of N(dβ0) (Prop. 5.2), yielding stabilization IH^k(M_{dβ0}) = b∞_k for fixed k and large d (Thm. 5.3). For a generic Enriques surface with 2∤β0, a refined stabilization of perverse Hodge numbers n^{i,j}_{dβ0} is also proved (Thm. 5.7).

Significance. The result, if correct, provides a unified explanation of the phenomenon observed for P^2 and del Pezzo surfaces and confirms the stabilization conjecture of [33] in two new surface classes. The proof has no free parameters: the stable value is fixed by Göttsche's product formula and the argument is a direct comparison through relative Hilbert schemes and perverse filtrations. The main strengths are the clean local-versality argument for smoothness in high codimension and the induction using relative Hilbert scheme Betti numbers. The principal risks are the paper's reliance on unstated black-box inputs — [34, Prop. 2.2] for irreducibility over the integral locus and [36, Cor. 1.3] for constant fiber dimension of the Hilbert–Chow morphism — and the terse handling of preimage-codimension assertions. These points are load-bearing for the stabilization theorem, so the paper needs a revised version that states and verifies the hypotheses of these cited results.

major comments (3)
  1. [§5, Prop. 5.1 and Thm. 5.3, esp. (5.17), (5.19)] The constant-fiber-dimension assertion that 'the fibers of h have the same dimension p_a(β)' is imported from [36, Cor. 1.3] and is used to bound dim Y' in (5.17) and to obtain the first equality in (5.19). The precise statement of [36, Cor. 1.3] is not given, and its hypotheses are not checked for all dβ0 on Enriques/bielliptic surfaces with χ=1. Since the stabilization conclusion collapses if this corollary has an extra condition (e.g., a primitivity or reduced-support restriction), please quote the corollary and verify the hypotheses in this setting. Also justify the codimension equality in (5.19) explicitly: equal fiber dimensions imply the equality only via the dimension formula for proper equidimensional morphisms (or flatness), and the manuscript should spell this out.
  2. [§5, Prop. 5.1] The irreducibility of h^{-1}(|β|^int), including the existence of the smooth open h^{-1}(|β|^sm), is taken from [34, Prop. 2.2]. This is the only input that makes M_{β,χ} irreducible in the applications. Please state [34, Prop. 2.2] and verify that its hypotheses hold for the base-point-free ample divisors β satisfying gcd(β·H,χ)=1 on Enriques and bielliptic surfaces. If [34, Prop. 2.2] requires additional conditions (e.g., β not divisible by 2 in Num(S), or a different stability convention), those conditions must be stated and included in Theorem 5.3.
  3. [§4, proof of Thm. 1.1 and Prop. 4.4] The proof uses preimage-codimension equalities such as codim(C^[ℓ]\C^[ℓ]_U, C^[ℓ]) = codim(|β|\U, |β|) (Prop. 4.4) and codim(h^{-1}(|β|^int)\h^{-1}(U_k), h^{-1}(|β|^int)) = codim(|β|^int\U_k, |β|^int) (proof of Thm. 1.1). These are true for proper morphisms with equidimensional fibers, but the manuscript does not justify them. Since these equalities control the degree range in the main theorem, please add a lemma or reference making the equidimensionality/flatness hypothesis explicit.
minor comments (4)
  1. [§3.2, Prop. 3.6] The definition of 'ℓ-nodal' is not fully formal ('no singularities other than ℓ nodes'). Please clarify whether it means exactly ℓ nodes or at most ℓ nodes, since the subsequent genus bound and Tjurina-number bound depend on this.
  2. [§5.1.2, inequalities (5.20)–(5.24)] Some estimates in the proof of Proposition 5.2 (especially Case 1.3) are terse. A few words explaining the inequalities such as C_1·C_2 ≥ d√(2β^2−2) and s ≤ d^2β^2/2 would improve readability.
  3. [§5.1.3, Lemma 5.6] The vanishing H^k(ι_*ι^!Q_M)=0 for k≤N(β)+1 is asserted in one sentence. Since this is a technically important point, a brief explanation (e.g., via the dimension bound on W) would be helpful.
  4. [§1, Theorem 1.3] The introduction labels the stabilization result for Enriques and bielliptic surfaces as Theorem 1.3, while Section 5 presents it as Theorem 5.3. This is harmless but should be cross-referenced consistently in the final version.

Circularity Check

0 steps flagged

No demonstrated circularity; auxiliary self-citations are not shown to encode the target stabilization result.

full rationale

The paper's central target b_infty^k is fixed externally by Goettsche's formula for Hilbert-scheme Betti numbers, and the main proof derives the intersection Betti numbers through perverse filtrations, relative Hilbert schemes, and the support theorem, rather than by fitting or renaming the target quantity. The cited results [29, Lemma 2.4] and [34, Prop 2.2] are used as auxiliary black boxes: the former concerns Betti numbers of large open subsets of projective bundles, and the latter supplies irreducibility of h^{-1}(|beta|_int). The paper does not exhibit any equation in which the claimed stabilization is equivalent by construction to an input, nor does it call a fitted parameter a prediction. The applications rely on [36, Cor 1.3] for constant fiber dimension, which is an external input and is flagged in Remark 5.4 with its hypothesis. There are self-citations, but nothing in the provided text shows that these prior results contain the stabilization statement being proved, so the step does not meet the standard required to flag circularity. Overall, the derivation appears self-contained once the stated irreducibility and constant-fiber-dimension hypotheses are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central theorems rely on standard tools (Göttsche, perverse filtrations, local versality) and on several results from the authors' prior work ([29], [34]) and from [36]. No new entities are introduced.

axioms (7)
  • standard math Göttsche's formula (G(z,w) in [8]) determines Betti numbers of S^[n]
    Used throughout (Theorem 1.1, 4.1, Remark 4.2) as the benchmark b_∞^k; it is a published theorem.
  • standard math Maulik–Yun / Migliorini–Shende support theorem (Theorem 2.4, IS 2.5)
    Gives the isomorphism between cohomology of relative Hilbert schemes and perverse filtration on h^{-1}(U); used in Theorem 4.1.
  • standard math Kool–Shende–Thomas [16, Prop 2.1] controls non-ℓ-nodal curves
    Used in Proposition 3.6 to construct V_ℓ with codim of bad locus; this is a cited result.
  • domain assumption [36, Cor 1.3] gives constant fiber dimension p_a(β) for h over all of |β|
    Load-bearing for Proposition 5.1 and (5.19); only guaranteed under gcd(β·H,χ)=1 (Remark 5.4).
  • domain assumption [34, Prop 2.2] gives irreducibility of h^{-1}(|β|^int)
    Used in the proof of irreducibility of M_{β,χ} (Proposition 5.1). Self-cited.
  • domain assumption [29, Lemma 2.4] gives Betti comparison for open subsets
    Used in Proposition 4.4 to compare b_m(C^[ℓ]_U) with b_m(C^[ℓ]). Self-cited.
  • domain assumption Saccà [30]: generic Enriques with 2∤β gives smooth M_β
    Used in Theorem 1.4 for the refined perverse Hodge numbers.

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Cite this review

Pith. "Pith review of Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces." pith.science (2026). https://pith.science/paper/JQEVR5QP

@misc{pith2026251118426,
  author       = {Pith},
  title        = {Pith review of: Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQEVR5QP}},
  note         = {Machine review of arXiv:2511.18426}
}
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read the original abstract

In this paper, we develop a unified approach to study the intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces. Assuming the irreducibility of such moduli spaces, we prove that their intersection Betti numbers in a certain range of degrees coincide with the stable Betti numbers of Hilbert schemes of points. As an application, for surfaces with nef anticanonical divisor, we show that these intersection Betti numbers stabilize in each fixed degree, which fits into the broader context of stable cohomology for moduli spaces of sheaves; if in addition the moduli spaces are smooth, we also prove a refined stabilization result on perverse Hodge numbers.

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Reference graph

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