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BPS polynomials and Welschinger invariants

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Pith's one-line read The paper proves that for blow-ups of $\mathbb{P}^2$ at up to six general points, the BPS polynomial at $q=-1$ equals the Welschinger invariant, tying real curve counts to Gromov–Witten theory.

desk verdict Valuable BPS/Welschinger bridge undercut by a load-bearing theorem used outside its stated hypothesis — the n=6 proof has a real gap. read the letter →

arxiv 2506.02770 v1 pith:XSXJGS23 submitted 2025-06-03 math.AG hep-thmath.SG

Hülya Argüz , Pierrick Bousseau This is my paper
classification math.AGhep-thmath.SG MSC 14N3514N1014P0514J26
keywords BPSinvariantsWelschingerBlock–GöttschepolynomialsGromov–WittentheoryfloordiagramsrealalgebraicgeometrydelPezzosurfacesrefinedcurvecounting
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

The paper introduces BPS polynomials for any smooth projective surface $S$, organizing the BPS invariants of the threefold $S \times \mathbb{P}^1$ into a Laurent polynomial in $q$. These polynomials refine counts of complex rational curves: at $q=1$ they give the genus-zero Gromov–Witten invariant, and for toric del Pezzo surfaces they coincide with the tropical Block–Göttsche polynomials. The paper's central conjecture is that for the surfaces $S_n$ obtained by blowing up $\mathbb{P}^2$ at $n$ general real points, evaluating the BPS polynomial at $q=-1$ gives the Welschinger invariant, a signed count of real rational curves. The paper proves this for every $n \leq 6$ and every curve class with $m_\beta \geq 0$, establishing an interpolation between complex and real enumerative geometry via higher-genus Gromov–Witten theory. This matters because Welschinger invariants are subtle real-geometric quantities that now appear as a specialization of complex curve-counting data.

What carries the argument

The key object is the BPS polynomial of a surface $S$, defined as the BPS polynomial of the threefold $S \times \mathbb{P}^1$ in class $(\beta, 0)$ with $m_\beta$ point insertions lifted from $S$ and one point in $\mathbb{P}^1$. Lemma 3.3 converts it into a generating series of higher-genus Gromov–Witten invariants of $S$ with insertion of $(-1)^g \lambda_g$, which is what makes the polynomial a refinement of the genus-zero count. The proof machinery then brings in relative BPS polynomials of the pair $(\widetilde{S}_n, \widetilde{C})$: Theorem 4.17 and Corollary 4.18 compute these from refined counts of marked floor diagrams, using $q$-integer multiplicities $[w]_q^2$, and Theorem 4.20 shows that at $q=-1$ these refined counts become Welschinger signs. Finally, the refined Abramovich–Bertram–Vakil formula (Theorem 5.1) writes the BPS polynomial of the cubic surface $S_6$ as a weighted sum of relative BPS polynomials, and its real counterpart matches the analogous decomposition of Welschinger invariants.

What would settle it

A decisive check would be to compute the BPS polynomial of the cubic surface in the class $\beta = 2c_1(S_6)$ from the defining higher-genus Gromov–Witten invariants using a method that does not invoke the refined Abramovich–Bertram–Vakil formula; if its value at $q=-1$ is not 1000, the main theorem is false.

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Extended reading notes

Core claim

The central discovery is Theorem 5.2: for every $n \leq 6$ and every class $\beta \in H_2(S_n, \mathbb{Z})$ with $m_\beta = -1 + c_1(S_n)\cdot \beta \geq 0$, the specialization at $q=-1$ of the BPS polynomial equals the Welschinger invariant, $\mathrm{BPS}_{S_n,\beta}(-1) = W_{S_n,\beta}$. The argument first proves a relative version for the pair $(\widetilde{S}_n, \widetilde{C})$, where $\widetilde{S}_n$ is the blow-up of $\mathbb{P}^2$ at $n$ points on a conic and $\widetilde{C}$ is the strict transform of the conic: the relative BPS polynomial at $q=-1$ equals, up to an explicit factor, a relative Welschinger count. The $n \leq 6$ statement then follows by combining this relative statement with a refined Abramovich–Bertram–Vakil formula expressing BPS polynomials of $S_6$ as weighted sums of relative BPS polynomials, and the analogous real formula for Welschinger invariants. The paper also shows that BPS polynomials agree with Block–Göttsche polynomials on toric del Pezzo surfaces, so the $q=-1$ phenomenon genuinely extends the earlier toric interpolation.

Load-bearing premise

The $n=6$ proof relies on a degeneration formula that is stated under a transversality condition (the curve class must meet the chosen conic at least once), but it is applied to the class $\beta = 2c_1(S_6)$, which does not meet the conic; if the formula does not extend to such classes, the $n=6$ case is not established as written.

Editorial extensions

If this is right

  • For every $n \leq 6$ and every class $\beta$ with $m_\beta \geq 0$, the Welschinger invariant of $S_n$ is a specialization of the BPS polynomial of $S_n \times \mathbb{P}^1$; the signed real count is therefore fixed by complex Gromov–Witten data.
  • BPS polynomials provide a single Laurent polynomial interpolating between Gromov–Witten counts at $q=1$ and Welschinger counts at $q=-1$; the paper computes the explicit example $\mathrm{BPS}^{S_6}_{2c_1(S_6)}(q) = q^{-4} + 13q^{-3} + 100q^{-2} + 547q^{-1} + 1918 + 547q + 100q^2 + 13q^3 + q^4$.
  • For toric del Pezzo surfaces, BPS polynomials coincide with Block–Göttsche polynomials, so the new invariants extend the tropical refined counts to arbitrary surfaces.
  • The relative version holds for all $n$: relative BPS polynomials of the pair $(\widetilde{S}_n, \widetilde{C})$ at $q=-1$ equal relative Welschinger counts up to an explicit factor depending only on the contact orders $\nu$.
  • The proof yields an effective algorithm for computing BPS polynomials of $S_n$ for $n \leq 6$, since each class is handled by finite sums of refined floor-diagram counts.

Reading between the lines

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

  • One could test the conjecture at $n=7$ and $n=8$, where the surfaces are still del Pezzo and Welschinger invariants are defined, by running the floor-diagram algorithm the paper describes; agreement at $q=-1$ would support the conjecture beyond the proved range.
  • If Conjecture E (BPS polynomial equals a K-theoretic refined BPS invariant of the canonical bundle) is combined with the $q=-1$ result, it would predict that a purely sheaf-theoretic count also specializes to Welschinger invariants.
  • The paper's K3 discussion suggests a broader principle: for real surfaces whose real locus has Euler characteristic equal to the signature, the $q=-1$ specialization of the BPS polynomial may always admit a real-curve interpretation; testing this on real K3 surfaces with $e_R = -16$ is already possible from the formulas in the paper.
  • A direct check of the $n=6$ proof's edge case would be to verify Theorem 5.1 for $\beta = 2c_1(S_6)$ by an independent computation of both sides; Example 5.3 is precisely the case that exercises the formula beyond its stated hypothesis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper defines BPS polynomials for any smooth projective surface S from the Gromov-Witten theory of S × P^1, with BPS_{S,β}(1) equal to the genus-zero Gromov-Witten count of rational curves. It proves that for toric del Pezzo surfaces these polynomials agree with the Block-Göttsche polynomials, and it establishes a relative version stating that the q = -1 specialization of relative BPS polynomials for (eS_n, eC) equals, up to explicit factors, relative Welschinger counts. The central conjecture, that BPS_{S_n,β}(-1) equals the Welschinger invariant W_{S_n,β} for blow-ups of P^2 at n general real points, is claimed for all n ≤ 6 via a refined Abramovich-Bertram-Vakil formula, the real ABV formula, and the relative floor diagram computations.

Significance. If correct, the main result gives a striking bridge between real enumerative geometry and higher-genus Gromov-Witten theory, and provides an effective algorithm for computing Welschinger invariants of del Pezzo surfaces of degree at least 3. The toric comparison and the relative floor-diagram theorem are proved in detail, and the explicit Example 5.3 gives a concrete interpolation between complex and real counts. No free parameters are fitted, and the overall architecture is coherent. However, the n = 6 case of the main theorem currently rests on an unproved extension of the refined ABV formula, so the central claim is not fully established as written.

major comments (2)
  1. [§5.1–5.2, Theorem 5.1 and Eq. (5.1)] Theorem 5.1 is stated for every β with β · eC ≥ 1, but the proof of Theorem 5.2 applies Eq. (5.1) to every β with m_β ≥ 0, and Example 5.3 explicitly uses it for β = 2c_1(S_6), for which β · eC = 0. The proof of Theorem 5.1 is only a one-sentence reference to an 'analogous degeneration argument' citing [13, Theorem 8.3]; no argument is given that the refined ABV formula extends to classes with zero intersection with the conic. Since the cases n ≤ 5 are reduced to n = 6, this is a load-bearing gap. Please either prove the extension to β · eC = 0, state and prove a separate boundary-case formula, or restructure the argument so that this range is not needed.
  2. [§5.2, proof of Theorem 5.2] After proving the n = 6 case, the text concludes the result for all n ≤ 6 'by Lemma 3.8.' Lemma 3.8 concerns equality of Gromov-Witten invariants and BPS polynomials under blow-up at a point; it does not address the Welschinger side. The needed equality W_{S_6, π^*β} = W_{S_n, β} under blowing down exceptional curves is standard, but it must be stated and justified or given a precise reference, because the theorem compares BPS polynomials with Welschinger invariants.
minor comments (6)
  1. [§4.3.1, Lemma 4.12] The proof refers to 'Theorem 3.4,' but no Theorem 3.4 exists in the manuscript; the intended reference is likely Theorem 3.9 or a result from [12].
  2. [§4.4.1, Definition 4.19] The tuple x^{eC} is written with entries indexed up to ℓ(ν), but the surrounding text says it is a set of ℓ(µ) points; the indexing should be ℓ(µ).
  3. [§4.2.2, Remark 4.6] Remark 4.6 says vertices in V_2(Γ) are drawn as white disks; the second occurrence should refer to V_4(Γ).
  4. [§4.5, Example 4.22] The text says the floor diagrams are represented in 'Figures 4.8,' but the relevant figure is Figure 4.9.
  5. [§2.3] The q = -1 specialization of the Block-Göttsche polynomial is described as a 'number' of real genus zero stable maps; since it equals the signed Welschinger count, the wording should say 'signed count.'
  6. [§4.2.3 and §4.5] The manuscript contains duplicated passages and repeated figure captions in these subsections; the editorial cleanup should remove the duplicates before publication.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantive circularity: the BPS(-1)=Welschinger result is a chain of independently published theorems, with one scope gap in Theorem 5.1 that is a correctness risk rather than a circular reduction.

full rationale

The paper's central comparison BP S_{S_n,beta}(-1) = W_{S_n,beta} is not obtained by fitting, renaming, or defining the output into the input. BPS polynomials are defined from Gromov-Witten invariants of S x P^1 (Definition 3.2 and Lemma 3.3); for toric del Pezzo surfaces they are identified with Block-Gottsche polynomials (Theorem 3.11) using the independently published result [12] together with a degeneration proof (Theorem 3.9) contained in this paper. For the relative setting, Theorem 4.17 and Corollary 4.18 compute relative BPS polynomials as refined counts of marked floor diagrams, and Theorem 4.20 converts the q = -1 specialization into relative Welschinger counts via Brugalle's independently proved [18, Theorem 3.12]. The final step, Theorem 5.2, combines this with the real Abramovich-Bertram-Vakil formula from [22, 23, 50] and the refined Abramovich-Bertram-Vakil formula stated as Theorem 5.1. No free parameter is fitted to the Welschinger side, and none of the equalities is definitionally identical to the target statement. The one caveat worth flagging is a scope gap: Theorem 5.1 is stated under the hypothesis beta*eC >= 1, and its proof is only described as 'an analogous degeneration argument' citing [13, Theorem 8.3], yet equation (5.1) is applied in the proof of Theorem 5.2 and in Example 5.3 to beta = 2c_1(S_6), for which beta*eC = 0. This is a missing justification or correctness risk, not a circular reduction, so it does not by itself raise the circularity score.

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

The central claim has no free parameters and no invented entities. It rests on a chain of published theorems, including GV integrality, the GW/pairs correspondence, [12], [18], the real ABV formula, and the paper's Theorem 5.1. Theorem 5.1 is the least externally anchored item because it is stated with a proof sketch and used outside its stated range.

assumptions (6)
  • domain assumption Gopakumar-Vafa BPS invariants are integers and vanish for large genus for fixed class and insertions.
    Invoked in Section 3.1.1 to define BPS polynomials as Laurent polynomials; proved in [45,92] and [31,32].
  • domain assumption The Gromov-Witten/pairs correspondence holds for Sn × P1.
    Used in Lemma 3.6; the paper cites [65] because Sn × P1 is deformation equivalent to a toric 3-fold.
  • domain assumption Block-Göttsche polynomials equal higher-genus log Gromov-Witten invariants with insertion of (-1)^g λ_g.
    From [12, Theorem 1]; used in proofs of Theorem 3.11 and Lemma 4.12.
  • domain assumption Refined counts of marked floor diagrams at q=-1 equal relative Welschinger counts.
    From [18, Theorem 3.12]; used in Theorem 4.20.
  • domain assumption Real Abramovich-Bertram-Vakil formula.
    Used in Theorem 5.2, quoted from [22, Theorem 2.2], [23, Theorem 7], and [50, Section 4].
  • domain assumption Refined Abramovich-Bertram-Vakil formula, Theorem 5.1.
    Stated as a theorem with a one-sentence proof by analogy to [13, Theorem 8.3]; load-bearing for n=6 and used for β·eC = 0 outside its stated hypothesis.

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Pith. "Pith review of BPS polynomials and Welschinger invariants." pith.science (2026). https://pith.science/paper/XSXJGS23

@misc{pith2026250602770,
  author       = {Pith},
  title        = {Pith review of: BPS polynomials and Welschinger invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSXJGS23}},
  note         = {Machine review of arXiv:2506.02770}
}
abstract

We generalize Block-G\"ottsche polynomials, originally defined for toric del Pezzo surfaces, to arbitrary surfaces. To do this, we show that these polynomials arise as special cases of BPS polynomials, defined for any surface $S$ as Laurent polynomials in a formal variable $q$ encoding the BPS invariants of the $3$-fold $S \times \mathbb{P}^1$. We conjecture that for surfaces $S_n$ obtained by blowing up $\mathbb{P}^2$ at $n$ general points, the evaluation of BPS polynomials at $q=-1$ yields Welschinger invariants, given by signed counts of real rational curves. We prove this conjecture for all surfaces $S_n$ with $n \leq 6$.

Figures

Figures reproduced from arXiv: 2506.02770 by the authors.

Figure 3.1
Figure 3.1. The polyhedral decomposition PS. We equip the total space S of the degeneration with the divisorial log structure defined by the central fiber S0, and C with the divisorial log structure defined by {0} ⊂ C, so that the morphism π : S → C naturally lifts to a log smooth morphism. The tropicalization of S is the cone over the compact polygon P ⊂ R 2 with vertices the points uρ. Finally, we specialize the mβ point cons… view at source ↗
Figure 4.1
Figure 4.1. Two floor diagrams of degree 2 We will later make use of the following elementary result on floor diagrams in the proof of Theorem 4.17: Lemma 4.7. A floor diagram Γ of degree d satisfies |V2(Γ)| + 2|V4(Γ)| = d. Proof. We evaluate P v∈V (Γ) div(v) in two different ways. Since V (Γ) = V2(Γ) ⨿ V4(Γ), we obtain P v∈V (Γ) div(v) = 2|V2(Γ)| + 4|V4(Γ)|. On the other hand, expressing div(v) as a difference of weights, all … view at source ↗
Figure 4.2
Figure 4.2. Two floor diagrams of degree 4 of class β = 4H − P6 i=1 Ei ∈ H2(Se6,Z) and type (∅,(1, 1)). 4.2.3. Curves in the degeneration of Sen and floor diagrams. In this section we provide exam￾ples of marked floor diagrams and briefly explain how they encode topological information about curves in the central fiber eϵ −1 (0) = P 2 ∪ F (mβ,(µ,ν) ) 4 ∪ . . . F (1) 4 ∪ Bln(F4) of the degeneration eϵ of Sen described in §4.2.1.… view at source ↗
Figures from the paper (7 more)
Figure 4.3
Figure 4.3. Figure 4.3: Different markings on a floor diagram There exists a natural correspondence between the topology of irreducible components of curves in the central fiber, and vertices, edges, and legs of floor diagrams. White vertices marked by i, for 1 ≤ i ≤ mβ,(µ,ν) , correspond t…
Figure 4.3
Figure 4.3. Figure 4.3: Different markings on a floor diagram There exists a natural correspondence between the topology of irreducible components of curves in the central fiber, and vertices, edges, and legs of floor diagrams. White vertices marked by i, for 1 ≤ i ≤ mβ,(µ,ν) , correspond t…
Figure 4.5
Figure 4.5. Figure 4.5: Degeneration used in the proof of Lemma 4.12. 4.3.2. Lines tangent to the conic. In this section, we compute another particular case of the relative Gromov–Witten invariants GWSen/Ce g,β,(µ,ν) defined in (4.3). We assume that n = 0, that is Sen = P 2 and Ce = C. More…
Figure 4.6
Figure 4.6. Figure 4.6: Degeneration used in the proof of Lemma 4.13. 4.3.3. Relative BPS polynomials from refined counts of floor diagrams. In this section, we prove that the relative Gromov–Witten invariants GWSen/Ce g,β,(µ,ν) and the corresponding BPS [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 4.7
Figure 4.7. Figure 4.7: Curve contributing to GWF4/C−4∪C4 g,C4+|mv|F,(mv,nv) Corollary 4.18. Let β ∈ H2(Sen, Z) such that H · β ≥ 1, and µ, ν as in (4.1). Then, the relative BPS polynomial BP SSen/C β,(µ,ν) (q) is equal to the refined count with multiplicity Nfloor β,(µ,ν) (q) of marked flo…
Figure 4.8
Figure 4.8. Figure 4.8: All floor diagrams of degree 4 of class β = 4H − P6 i=1 Ei ∈ H2(Se6, Z) and type (∅,(1, 1)). Diagram C-count R-count Refined count 1) 16 0 q −3 + 2q −2 + 3q −1 + 4 + 3q + 2q 2 + q 3 2) 54 6 6q −2 + 12q −1 + 18 + 12q + 6q 2 3) 60 0 15q −1 + 30 + 15q 4) 20 20 20 5) 36 …
Figure 4.9
Figure 4.9. Figure 4.9: All floor diagrams of degree 6 of class β = 6H − 2 P6 i=1 Ei ∈ H2(Se6, Z) and type (∅, ∅). 5. BPS polynomials and Welschinger invariants of del Pezzo surfaces In §5.1, we prove a refined version of the Abramovich–Bertram–Vakil formula which relates the BPS polynomial…

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