On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$

Sayanta Mandal

REVIEW 2 major objections 4 minor 25 references

On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For sheaves on the projective plane with coprime rank and first Chern class, the paper proves that the 2N-th Betti number of the moduli space becomes independent of the second Chern class once $c_2 \geq N + \left\lfloor \frac{r-1}{2r}a^2…

desk verdict Explicit stabilization threshold is new and plausible, but the load-bearing S1 bound in Lemma 18 is asserted rather than proved; should go to referees with a requirement to fix it. read the letter →

arxiv 1908.09977 v2 pith:LSMCEKBT submitted 2019-08-27 math.AG

Sayanta Mandal This is my paper · ORCID

classification math.AG MSC 14D2014J6014J2614F45

keywords modulispacesofsheavesBettinumbersstabilizationprojectiveplanemotivicgeneratingfunctionswall-crossingHilbertschemepointsslopestability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes an explicit numerical threshold beyond which the even Betti numbers of the moduli space of slope-semistable sheaves on the projective plane stop changing. For rank $r \geq 2$ and first Chern class $aH$ with $\gcd(r,a)=1$, the theorem says that the $2N$-th Betti number depends only on $N$ once $c_2 \geq N + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2+1) \right\rfloor$. Because the stable values are already known through a universal generating function, the result turns infinitely many previously unknown Betti numbers into explicit numbers. The argument works by showing that all terms of a motivic generating function that could contribute to variation vanish above the threshold.

What carries the argument

The engine is the motivic generating function $G_{r,c}(q)=\sum_{\Delta \geq 0}[\mathcal{M}_{\mathbb{P}^2,H}(r,c,\Delta)]L^{r^2(1-2\Delta)}q^{r\Delta}$, together with its blow-up counterpart on the one-point blow-up of $\mathbb{P}^2$. A criterion from earlier work identifies stabilization of Betti numbers with the vanishing of the coefficient of $L^{-N}q^{\Delta}$ in $(1-q)G_{r,c}(q)$ for $\Delta$ large relative to $N$. The paper obtains such a vanishing bound by combining a formula for stack classes on ruled surfaces, a wall-crossing identity that relates two polarizations on the blow-up, and a blow-up formula connecting $\mathbb{P}^2$ to its blow-up; the final constant $C_0=\frac{1}{2}(r^2+1)$ comes from bounding a quadratic expression in the first Chern classes of the direct-summand characters.

What would settle it

A finite computer search over integer tuples $(l, r_i, a_i)$ satisfying $\sum r_i = r$, $\sum r_i a_i = a$, and the slope inequalities of the wall-crossing sign rule, evaluating the expression $S_1$ in equation (25), would settle the bound. If any tuple gives $S_1 < -r + 3 - \frac{4}{r}$, the claimed $\kappa \geq -(r-1)$ and hence $C_0=\frac{1}{2}(r^2+1)$ would fail; if no such tuple exists for all $r$ and $a$, the bound is confirmed.

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Extended reading notes

Core claim

The paper's central claim is Theorem 26: for $r \geq 2$ and $a$ coprime to $r$, the $2N$-th Betti number of $M_{\mathbb{P}^2,H}(r,aH,c_2)$, the moduli space of slope-$H$-semistable torsion-free sheaves with those Chern classes, stabilizes in $c_2$ once $c_2 \geq N + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2+1) \right\rfloor$. Stabilization means the value of $b_{2N}$ is independent of $c_2$ above the threshold, and it coincides with the coefficient predicted by the stable generating function for sheaves on $\mathbb{P}^2$. The paper proves this by reducing the question to a vanishing statement for coefficients of $L^{-N} q^{\Delta}$ in $(1-q)G_{r,aH}(q)$, a motivic generating function built from stack classes.

Load-bearing premise

The whole explicit bound rests on an unproved combinatorial assertion inside Lemma 18: that a certain collection of non-negative summands is jointly bounded below by $2r-4$ because the relevant inequalities cannot be satisfied simultaneously; if this fails, the stated constant $\frac{1}{2}(r^2+1)$ is not justified, though a weaker threshold could still hold.

Editorial extensions

If this is right

For every pair $(r,a)$ with $\gcd(r,a)=1$, all Betti numbers $b_{2N}$ of the corresponding moduli spaces become explicitly computable once $c_2$ crosses the stated line, since the stable generating function is known.
The rank-one case is recovered as a special case, with stabilization for $c_2 \geq 2N$.
For the known rank-two example $r=2$, $a=-1$, the method yields the improved threshold $c_2 \geq N+1$, matching the previously tabulated Betti numbers.
For $r=4$, $a=1$, the bound is improved to $c_2 \geq N+5$, showing the general threshold is not sharp in every example.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The general threshold is almost certainly not optimal: the worked examples show case-by-case improvements, so a sharper uniform constant could exist, possibly closer to $\frac{r-1}{2r}a^2$ plus a smaller rank-dependent term.
The same coefficient-vanishing strategy should extend to other rational surfaces or to stabilization statements for higher-degree Betti numbers; the only surface-specific inputs are the blow-up formula and the wall-crossing data.
The unproved combinatorial assertion in Lemma 18 could be tested by a finite minimization over integer tuples; if it fails for some $r$, a weaker explicit threshold would still follow from the surrounding method.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper gives the first explicit stabilization threshold for the Betti numbers of MP2,H(r,aH,c2) for all ranks r≥2 with gcd(r,a)=1: if c2 ≥ N + floor((r−1)/2r a^2 + (r^2+1)/2), then b_{2N} stabilizes, with the stable value determined by Coskun–Woolf. That is a real new result; prior work had qualitative stabilization and low-rank examples. The strategy is coherent: motivic generating functions, Mozgovoy's stack formula, Joyce's wall-crossing, and the blow-up formula are assembled in the expected way, and the rank-1 section cleanly recovers the known Hilbert-scheme stabilization. The worked examples are also honest: Proposition 27 reduces the rank-2 bound to the known c2 ≥ N+1, and Proposition 28 gets a better bound for rank 4. These checks give me confidence that the machinery is fundamentally right.

The problem is real, though. Lemma 18 is load-bearing for the constant C0=(r^2+1)/2, and its proof of the lower bound for S1 is a single sentence: 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r−4) because all the inequalities in the summations cannot be simultaneously compatible.' That is not a proof. The expression in (27) has position-dependent coefficients and the compatibility claim depends on the order of the partition and the sizes of the r_i; the reader and the stress-test note are right to flag it. If that bound fails, κ≥−(r−1) fails and the stated threshold is unjustified, though a weaker one might still hold. Proposition 28 inherits the same gap through its S1+S2 case analysis, and its claims like 'we see that S1≥−3/4' are asserted without the supporting computation.

I don't think this is fatal: the architecture is sound, the bound matches all known data, and the missing piece is a combinatorial estimate that may be fixable with a real argument. But it is exactly the kind of assertion that a referee should not let through as is. The reliance on the unpublished Coskun–Woolf preprint is secondary but a stable reference or an appendix stating the needed results would help.

Who is this for? People working on moduli spaces of sheaves on surfaces, stable cohomology, and Donaldson-type invariants. A serious referee will get value from engaging, and after a revision that proves Lemma 18 and clarifies the dependence on [CW], this would be a solid paper.

My recommendation: send it to peer review rather than desk-reject; the referee should require a real proof of the S1 bound and a stable reference for the input results. With those fixed, I'd be happy to cite it.

Referee Report

2 major / 4 minor

Summary. The paper studies stabilization of the Betti numbers of the moduli space M_{P^2,H}(r,aH,c_2). Its main result, Theorem 26, claims that for r >= 2 and gcd(r,a)=1, the 2N-th Betti number stabilizes once c_2 >= N + floor( (r-1)/(2r) a^2 + (r^2+1)/2 ). The method follows the Coskun--Woolf framework: it encodes the moduli spaces in the ring A^- through generating functions, uses Mozgovoy's formula for ruled surfaces and Joyce's wall-crossing formula on F_1, proves a coefficient-vanishing estimate for (1-q)G_{r,c}(q), and returns to P^2 via the blow-up formula. The key new ingredient is a combinatorial estimate, Lemma 18, that bounds an expression S_1+S_2 and yields the constant C_0=(r^2+1)/2 in Proposition 17. The paper also gives improved bounds for ranks 2 and 4 in Propositions 27 and 28.

Significance. If the main theorem is established, it is a genuinely useful result: it gives an explicit linear-in-N stabilization threshold for all ranks and first Chern classes, rather than only the qualitative statement that stabilization occurs. The approach is systematic, the architecture of the proof is coherent, and the final bound is not fitted but derived from a chain of known structural results. The paper also demonstrates the method by recovering the known rank-2 threshold after a separate calculation and by improving the general bound in a rank-4 example. However, the correctness of the explicit threshold currently rests on an unproved combinatorial assertion in Lemma 18, so the central claim is not yet fully justified. The heavy reliance on the unpublished Coskun--Woolf preprint is a secondary obstacle to independent verification.

major comments (2)

[Lemma 18, Eq. (24)-(27)] The lower bound for S_1 is not established. After deriving Eq. (27), the proof states: 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r-4) because all the inequalities in the summations cannot be simultaneously compatible.' No argument is supplied for this compatibility claim, and it is not apparent from the displayed sums: whether the positive terms are forced to contribute depends on the order and sizes of the r_i and on the a_i values. Moreover, even accepting the separate bounds on the positive summand (2r-4) and on the negative summand (-(r^2-r)) as stated, the conclusion would only give S_1 >= (2r-4) - (r^2-r), which is not the claimed bound -r+3-4/r; an additional interaction between the positive and negative parts is needed. Because this estimate feeds directly into kappa in Eq. (32), then into C_0 in Proposition 17, and finally into the explicit threshold in Theorems 25 and 26, the main quantitative claim is not yet proven.
[Proposition 28] The improved rank-4 bound inherits the same gap. The proof asserts a sequence of statements of the form 'we see that S_1 >= ...' and 'S_1+S_2 may attain the least possible value ...' for the partitions (r_1,...,r_l) = (3,1), (1,3), (2,1,1), (1,2,1), (1,1,2), and (1,1,1,1), without showing the computations that rule out all remaining integer vectors satisfying the linear constraints. This is a finite check that could be made explicit, but as written it is not verifiable and it depends on the same unproved S_1 estimate from Lemma 18. The text also contains a typo: the second displayed formula for 'S_1' in this proof is in fact the definition of S_2, which makes the case analysis harder to follow.

minor comments (4)

[Lemma 18, partition argument] The reduction to a_i in {-1,0,1,2} should be justified. The Lagrange multiplier calculation gives a/r - 1/2 <= a_i <= a/r + 1/2, and with 0 <= a <= r-1 these intervals are contained in (-1/2, 3/2); the text should explain why checking { -1,0,1,2 } suffices.
[Section 2 and Proposition 10] The same symbol M is used for both the moduli stack and the coarse moduli space; Proposition 10 relates them by a factor (L-1), so the two objects must be typographically distinguished throughout.
[References] Several foundational statements are quoted from the unpublished preprint [CW] (Propositions 2 and 10, Theorem 9, Remark 16, and the sign conditions used in Lemma 18). The author should state the status of this preprint and, if possible, include the necessary statements or proofs so that the main argument can be independently checked.
[Theorem 13 reference] There is a typographical error in the citation: '[Mo][Theorem 1.1)' should be '[Mo][Theorem 1.1]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stabilization threshold is derived from external theorems and independent combinatorial estimates, not from its own conclusion.

full rationale

The paper's derivation chain for Theorem 26 is: use Mozgovoy's Theorem 13 for the motivic generating function of the F-framed moduli stack; use Joyce's Theorem 15 and Coskun-Woolf's extension to nef polarizations to express the E+F-framed stack in terms of F-framed stacks; use Coskun-Woolf's Theorem 9 for the stable generating function; use Mozgovoy's blow-up formula to pass from P^2 to F_1; and then prove coefficient-vanishing estimates for (1-q)G_{r,c}(q). The novel explicit threshold c2 >= N + floor((r-1)/(2r) a^2 + (r^2+1)/2) is not assumed anywhere in the input. No fitted parameter is renamed as a prediction, and no equation defining the conclusion is used as the premise. The cited Coskun-Woolf result supplies the qualitative stable generating function and the stack/space comparison, but its assumptions do not include the bound being proved, so it is an external input rather than a circular restatement of the paper's target. The main weakness is a correctness gap, not circularity: Lemma 18's lower bound for S1 is asserted with the sentence 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r-4) because all the inequalities in the summations cannot be simultaneously compatible,' with no detailed proof; if that bound fails, the stated constant C0 is not established. Similarly, the heavy reliance on the unpublished Coskun-Woolf preprint is a verifiability concern. These issues affect the soundness of the explicit constant but do not make the derivation equivalent to its inputs. The rank-one section explicitly re-derives Ellingsrud-Stromme's known threshold using Göttsche's generating function, but presents it as a re-derivation, not as an independent prediction, so it is not circular. Overall, no load-bearing circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the constant C0 is derived via upper bounds and improved case-by-case, never fitted to the Betti number tables. The Betti tables of Yoshioka and Manschot are used only as checks after the fact, not as inputs. The axioms are the external theorems the derivation imports: Mozgovoy's formula for H_{r,c}, Joyce's wall-crossing with a nef extension, Mozgovoy's blow-up formula, Göttsche's Hilbert scheme generating functions, Coskun-Woolf's stack-to-space comparison, and Yoshioka's discriminant identity. No new entities are postulated.

assumptions (6)

domain assumption Mozgovoy's motivic formula for the stack of F-semistable sheaves on F1 (Theorem 13): H_{r,c}(q) = 1/(L-1) times products of Z_{P1}(L^i) and Z_{P1}(L^{rk+i} q^k).
Imported in Section 4 to expand the generating function; Proposition 14 and the estimate for tilde G depend on it.
domain assumption Joyce's motivic wall-crossing formula (Theorem 15) applied with H1=F, H2=E+F, including the nef extension for non-ample H1 asserted in Remark 16 via Coskun-Woolf Corollary 5.3.
Converts classes for polarization F into those for E+F (equation 11); convergence of the sum is assumed from [CW].
domain assumption Mozgovoy's blow-up formula (equation 33, Proposition 7.3) relating P2 and F1 stack generating functions.
Basis of equation (36) connecting G_{r,c} with tilde G_{r,c-mE}; the P2 bound flows from this relation.
domain assumption Göttsche's generating functions for Hilbert schemes of points on P2 and F1 (Go01, Examples 4.9.1 and 4.9.3).
Used in Section 3 for the rank one case and in the stable limit product.
domain assumption Coskun-Woolf Proposition 10: the class of the moduli stack equals (L-1) times the class of the moduli space in A.
Used to pass from stack generating functions to moduli space Betti numbers in Sections 3 and 6; relies on gcd(r, c.H)=1 so all semistable sheaves are stable.
standard math Yoshioka's discriminant identity (equation 14) expressing r*Delta - sum r_i*Delta_i in terms of first Chern classes.
Used in the Joyce expansion (equations 14-16) to eliminate discriminants.

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Cite this review

Pith. "Pith review of On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$." pith.science (2026). https://pith.science/paper/LSMCEKBT

@misc{pith2026190809977,
  author       = {Pith},
  title        = {Pith review of: On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbbP^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSMCEKBT}},
  note         = {Machine review of arXiv:1908.09977}
}

abstract

Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$, then the $2n$th Betti number of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Coskun, I., Woolf, M., The stable cohomology of moduli spaces of sheaves on surfaces, preprint

work page

[2] [2]

Arild, On the homology of the Hilbert scheme of points in the plane, Invent

Ellingsrud, G., Str mme, S. Arild, On the homology of the Hilbert scheme of points in the plane, Invent. math. 87 (1987), pp. 343-352

work page 1987

[3] [3]

Math., 106 (1977), pp

Gieseker, D., On the moduli space of vector bundles on an algebraic surface, Ann. Math., 106 (1977), pp. 45-60

work page 1977

[4] [4]

G \" o ttsche, L., The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math Ann. 286, pp. 193 - 207

work page

[5] [5]

Z., 223 no

G \" o ttsche, L., Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces, Math. Z., 223 no. 2 (1996), pp. 247-260

work page 1996

[6] [6]

G \" o ttsche, L., Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces, Comm. Math. Phys., 206 no. 1 (1999), pp. 105-136

work page 1999

[7] [7]

G \" o ttsche, L., On the motive of the Hilbert scheme of points on a surface, Math. Res. Lett., 8 no. 5 (2001), pp. 613-627

work page 2001

[8] [8]

a hler cone of a compact hyperk \

Huybrechts, D., The K \" a hler cone of a compact hyperk \" a hler manifold , Math. Ann., 326 no. 3 (2003), pp. 499-513

work page 2003