Counting curves in a linear system with upto eight singular points

Ritwik Mukherjee; Somnath Basu

REVIEW 4 major objections 3 minor 53 references

Counting curves in a linear system with upto eight singular points

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that every curve count with δ nodes and one further singularity of total codimension at most eight is an explicit universal polynomial in Chern classes.

desk verdict Useful, likely-correct enumeration with new codim-8 numbers, but the load-bearing boundary multiplicity proofs are deferred to an unpublished companion. read the letter →

arxiv 1909.00772 v1 pith:KW63NDKK submitted 2019-09-02 math.AG

Somnath Basu , Ritwik Mukherjee This is my paper · ORCID

classification math.AG MSC 14N1014C1714H2014N35

keywords enumerativegeometrysingularplanecurvesEulerclasslocalintersectiontheoryrecursiveformulasChernclasseslinearsystemscodimensioneight

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 asks how many curves in a sufficiently positive linear system on a compact complex surface have a prescribed collection of singularities: δ ordinary nodes and one further singularity of a fixed type. The authors establish that for every singularity of codimension k, the number N(A1^δ X) is explicitly computable whenever δ+k≤8, and the answer is a universal polynomial in the Chern classes of the line bundle and the surface. This matters because enumerative counts of this kind were previously known only in fragments—up to six nodes, all codimension-seven cases, and a few codimension-eight cases—and the paper fills the remaining codimension-eight table with new numbers. The method is a systematic recursion: each count is expressed through the Euler class of a derivative bundle, and boundary contributions are interpreted as counts of curves with more degenerate singularities.

What carries the argument

The machinery is the Euler class of derivative bundles over spaces of marked curves. A singularity condition such as 'the curve has a cusp at q' is encoded by choosing coordinates and requiring certain directional derivatives—the quantities A_i^f and D_i^f defined in the paper—to vanish; these conditions define sections of line bundles over the space of curves with a marked point and, for degenerate singularities, a marked tangent direction. The number N(A1^δ X) is then the Euler class of the relevant bundle evaluated on the fundamental class of the open stratum A1^δ ∘ X. The paper's main technical work is to compute the boundary contribution to this Euler class: when marked points collide, the section vanishes on strata such as $A1^{{δ-2}}$ ∘ A3 (two nodes becoming a tacnode) or $A1^{{δ-3}}$ ∘ D4 (three nodes becoming a triple point), and the contribution from each such stratum is a fixed integer—4, 18, and so on—obtained from local intersection theory. Recursive formulas chain these boundary computations together to reach the base case.

What would settle it

Take the boundary stratum where two marked nodes of a plane quartic collide to form a tacnode, compute the local Euler-class contribution of the section directly on a one-parameter family, and check that it equals 4; if the direct calculation yields any other integer, every recursive formula built on that multiplicity, and hence all the final enumerative numbers, would have to change.

Watch

Extended reading notes

Core claim

The central claim is that the enumerative numbers N(A1^δ X)—the number of curves through the expected number of generic points with δ ordered nodes and one singularity of type X—are determined, for δ+k≤8, by an explicit recursion whose base case is the count of one nodal curve. The recursion writes N(A1^δ X) as a product of the base curve count with an insertion, minus boundary contributions coming from marked points colliding. Each boundary contribution is a fixed integer multiple of a count of curves with fewer nodes and a more degenerate singularity: for instance, two colliding nodes contribute 4 times the tacnode count, three colliding nodes contribute 18 times the ordinary triple-point count, and the stratum where more than three nodes collide contributes nothing once the generic point conditions are imposed. Iterating the recursion down to the base case yields an explicit universal polynomial in c1(L), c1(T^*X), and c2(T^*X). The same procedure reproduces every previously known formula in this range and produces the previously unknown codimension-eight numbers.

Load-bearing premise

The load-bearing premise is that the deferred proofs in the companion paper supply the stated closures and multiplicities for colliding singular points—two nodes contributing exactly 4 when they form a tacnode, three nodes exactly 18 when they form a triple point, and more-than-three-node collisions contributing nothing under the generic point constraints—because any one of these numerical claims being wrong would change the final formulas.

Editorial extensions

If this is right

For every compact complex surface and every line bundle that is sufficiently $(2\delta+C_X)$-ample, the number $N(A_1^\delta X)$ with $\delta+k\le 8$ is an explicit universal polynomial in $c_1(L)$, $c_1(T^*X)$, and $c_2(T^*X)$.
The recursion reproduces all previously known formulas in this range—up to six nodes, all codimension-seven counts, eight nodes, and one codimension-eight singular point—and supplies the remaining codimension-eight numbers, which are new.
The direct low-degree checks in $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$ confirm the formulas; for example, the count of quintics through 12 points with six nodes and one cusp matches a purely combinatorial count of reducible configurations.
When more than three marked points attempt to collide, the generic point condition cuts the boundary stratum away, so the recursion only needs boundary strata with at most three colliding points.

Reading between the lines

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

An inevitable next step, not taken in the paper, is to push the same recursion to $\delta+k=9$; the structure suggests that only finitely many new boundary multiplicities would need to be computed.
Because the formulas are universal polynomials, they can be specialized mechanically to any compact complex surface with computable Chern classes, so regenerating tables for other surfaces is a direct computational extension.
The paper's own remark that its ampleness bound is not optimal suggests that rerunning the transversality argument with the marked point allowed to move as well as the curve would yield a sharper bound on when the formulas are genuinely enumerative.
The corrected $\mathbb{P}^1\times\mathbb{P}^1$ checks reveal how easily reducible configurations can be double-counted in such tables; auditing other published low-degree enumerative numbers with the same subtraction logic is a cheap way to find similar oversights.

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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

Here is my take on Basu–Mukherjee, arXiv:1909.00772. The paper delivers something real: explicit formulas for N(A1^δ X), the number of curves in a sufficiently ample linear system with δ nodes and one singularity of type X, for all δ+k ≤ 8. The codimension eight numbers are mostly new, and the recursive framework subsumes prior results by Vainsencher, Kazarian, Kleiman–Piene, and Weber–Mikosz–Pragacz. The low degree checks in Sections 8 and 9 are genuine evidence—they match known counts from Kontsevich–Manin and Kleiman–Piene, and the authors correct a real oversight in Vainsencher's P1×P1 computations.

The soft spot is exactly where the stress-test note lands. The recursion depends on boundary multiplicity and closure constants—4 for the tacnode stratum, 18 for the D4 collision, 24 for the X9 stratum, and more—and nearly all of them are 'shown in [1]', where [1] is an unpublished companion paper. Remark 1.7 says this openly. This does not sink the paper, because the low degree checks exercise parts of the recursion independently, and the base case comes from their earlier published work. But the main result is not verifiable from the manuscript alone, and a referee cannot certify it without seeing [1] or at least the key lemmas.

There are also a few printed typos. Theorem 6.10 has 12·(δ choose 1)·N(A1^{δ−2}X9), which is undefined at δ=1; it should presumably be (δ choose 2). Theorem 6.16 appears to have a missing plus sign between the first two terms. These are minor but make one wary about the transcription from the Mathematica output.

All that said, the paper is clearly the work of people who know this area. The recursion is coherent, the citations to related work are in place, and the new numbers are exactly the kind of explicit data this field uses. I would not desk-reject it.

Who is the audience? Enumerative geometers working on singular curves on surfaces, and anyone needing explicit universal polynomials for low-codimension singularity counts. If the companion paper appears with the promised proofs, this becomes a solid reference. Without it, it is an extended announcement.

Recommendation: send it to peer review, but the referee should be instructed that the decisive multiplicity and closure claims live in the companion, and acceptance should be conditional on those proofs being supplied or published alongside. I would cite it once that happens.

Referee Report

4 major / 3 minor

Summary. The paper develops a systematic, Euler-class-based recursion to count curves in a sufficiently ample linear system on a compact complex surface that have δ nodes and one further singularity of type X, for all δ+k ≤ 8. The main result, Main Result 1.5, claims explicit universal formulas for N(A_1^δ X) in terms of the Chern classes of L and T^*X. The recursion is built stratum by stratum: Sections 6 and 7 reduce each count to boundary contributions coming from collisions of nodes, and the paper states the needed closure, multiplicity, and transversality results, deferring their proofs to a companion paper [1]. The paper then recovers, in Sections 8 and 9, several known numbers (Kleiman–Piene's 8-node quintic count, Kontsevich–Manin rational curve numbers, and selected codimension-seven and codimension-eight results) and reports new codimension-eight numbers.

Significance. If the asserted recursion and boundary contributions are correct, this is a substantial contribution: it gives a uniform treatment of singular curve counts through codimension eight, recovers and partly re-proves results of Vainsencher, Kazarian, Kleiman–Piene, and Weber–Mikosz–Pragacz, and produces new universal polynomials. The manuscript has genuine strengths: the recursive structure is explicit, the base case N(A_1) is taken from the published paper [4], and the low-degree checks in P^2 and P^1×P^1 are concrete and nontrivial. The paper also provides a Mathematica implementation, which helps reproducibility. However, the central technical content — the closure, multiplicity, and transversality assertions for the Euler class boundary contributions — is not proved here but is deferred to an unpublished companion paper [1]. Until those proofs appear, the main result cannot be fully verified from the manuscript alone.

major comments (4)

[Section 7, especially §§7.2, 7.4, 7.7, 7.10, 7.15–7.17] The load-bearing step of the paper is the computation of the Euler class boundary contributions, and nearly all of that computation is deferred to the companion paper [1], listed as “in preparation”. For example, §7.2 asserts that the tacnode stratum contributes 4 and the D4-collision stratum contributes 18; §7.4 asserts contributions 2 and 3; §7.7 asserts contribution 24 from X9; §7.10 asserts contributions 12 and 6; and §§7.15–7.17 assert contributions 3, 16, and 6. Each of these constants, together with the closure identifications such as B(q1,q2,qδ+1)≈A^{δ−2}_1∘A3 and B(q1,q2,q3,l_{qδ+1})≈A^{δ−2}_1∘D4, is announced as a result proved in [1]. Remark 1.7 explicitly acknowledges this split. These assertions are not cosmetic; if any one of them is incorrect, the recursive formulas and the final enumerative numbers change. The low-degree checks in Sections 8 and 9 sample only a few of the many asserted formulas and cannot substitute for the missing proofs.
[Section 6, before Theorem 6.1; end of Section 6] The paper states that “we have generated an output of all the numbers N(A_1^δ X) and have appended it at the end of this paper (after the Bibliography),” but the submitted manuscript contains no such appendix. Thus the promised explicit formulas are not actually present in the text; the only access to the final polynomials is a Mathematica program on the second author's homepage. This is a problem for the central claim “we obtain an explicit formula”: a reader of the paper cannot verify or use the claimed universal polynomials without consulting an external, version-uncontrolled file. The low-degree checks verify only a handful of cases, not the full set of codimension-eight formulas.
[Theorem 6.10 and §7.8] Theorem 6.10 contains the term 12*(δ choose 1)*N(A^{δ−2}_1 X9,n1,m1,m2,θ), which is undefined for δ=1 because N(A^{−1}_1 X9) is not defined. The surrounding text in §7.8 says “two points come together,” so the intended binomial is almost certainly (δ choose 2), which would make the term vanish for δ=1 as it should. This is not merely a notational slip in a peripheral display: it occurs in a theorem that is part of the recursive machinery, and as printed the recursion is not literally self-consistent.
[Theorem 6.16] Theorem 6.16 is printed with a missing operator: the right-hand side reads “N(A^δ_1PD6,...) − N(A^δ_1PD6,...,θ+1) 2N(A^δ_1PD6,...,θ+1)+...”, which is not a valid formula as written. Comparison with the derivation in §7.12 and with the companion formulas strongly suggests a missing “+” sign between the second and third terms. Since this theorem is used in the recursion, the printed statement should be corrected.

minor comments (3)

[Theorem 6.16 and Main Result 1.5] The ampleness hypothesis in Theorem 6.16 is stated as “sufficiently (2δ+3)-ample”, which appears inconsistent with Main Result 1.5 (where C_{E7}=4, giving 2δ+4) and with the proof in §7.12 which invokes (2δ+4)-ample or stronger. The authors should clarify which bound is intended.
[Throughout] There are numerous typographical errors, e.g., “signularity” in Section 5, “desried number” in §8.4, “Kontsevich-Manin s’s formula” in §9, and “upto” in the title and abstract. These do not affect the mathematics but should be corrected.
[Section 7.4] In the proof of Theorem 6.5, the sentence after equation (15) says “Hence the remaining stratum of B does not contribute to the Euler class, giving us Theorem 6.5,” but the displayed equation (15) concerns r≥3, whereas the boundary component with two nodes colliding has already been handled; the logical flow is clear but would benefit from a more explicit statement that the omitted strata are precisely those with r≥3.

Circularity Check

2 steps flagged · score 7.0 of 10

Main Result 1.5 is not derived in this paper: the recursive formulas in Theorems 6.3–6.22 are assembled from boundary multiplicity and closure constants whose proofs are deferred to the same authors' unpublished companion paper [1].

self citation load bearing [Section 7, opening paragraph; see also Remark 1.7]
"The relevant claims on closure, multiplicity and transversality are proved in our earlier papers [6], [5], [27] and [4], when δ = 0 or 1 and δ + k ≤ 7. For the remaining cases (namely δ > 1 or δ + k > 7), we will prove the relevant closure, multiplicity and transversality claims in [1]."

The proof of the central theorem for the main range of the paper is not contained in the manuscript. The key closure, multiplicity and transversality inputs for δ > 1 or δ + k > 7 are announced as results of [1], an in-preparation paper by the same two authors. The recursive formulas in Theorems 6.3–6.22 have no independent derivation in the present text because the constants and identifications that enter the Euler class computation are exactly the statements relegated to [1]. A wrong multiplicity in [1] would change the final enumerative numbers, so the claim 'we obtain an explicit formula' is, at present, supported only by an unverified self-citation.
self citation load bearing [Subsection 7.2, proof of Theorem 6.3]
"We will show in [1] that the contribution from each of the tacnodal points, namely B(q1,q2,qδ+1) ∩ μ is 4. Hence the total contribution from all the components of type B(qi1,qi2,qδ+1) is 4 (δ2) N(Aδ1A3,n1,m1,m2)."

This is a concrete instance of the self-citation chain: the coefficient 4 in Theorem 6.3, which is load-bearing for the recursion for N(Aδ1 A1), is not computed in this paper but asserted to be proved in [1]. The same pattern repeats throughout Section 7 with the multiplicities 18, 2, 3, 24, 12, 6, 16, and 6. The formula for the enumerative number is therefore literally constructed from the same authors' forthcoming claims rather than from a proof contained in the paper.

full rationale

This is not a clean no-circularity case. The paper has substantial independent ingredients: Theorem 6.1, the base case N(A1), is cited to the authors' available paper [4], which is not in preparation; the low-degree checks in Sections 8 and 9 compare several final values with externally known enumerative numbers (Kleiman–Piene, Kontsevich–Manin, Vainsencher) and these checks are genuine evidence. However, Main Result 1.5 is not self-contained. Its recursive proof for essentially the whole range δ > 1 or δ + k > 7 delegates the decisive closure, transversality and multiplicity statements to companion paper [1], which is listed as in preparation and is by the same authors. The text repeatedly says 'We will show in [1]' for the exact constants used in the formulas; no proof or external verification is supplied. Under the rules, an in-preparation self-citation is not independent support, so this is load-bearing self-citation rather than a mere correctness concern. The low-degree checks cover only a handful of cases (e.g., N(A1^8), N(A1^6 A2), N(A1^4 D4) in P2 and two P1 × P1 checks) and cannot validate all numbers through codimension eight. Score 7 rather than 8 because the announced boundary claims are geometric statements with plausible content rather than a definitional identification, and because some external benchmarks are met; score 2 would be appropriate only if the companion paper [1] were supplied or if the boundary contributions were proved here.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters; the formulas are universal polynomials in Chern classes. The main assumptions are the deferred technical claims in the companion paper [1] and the correctness of the previously published base case.

assumptions (2)

ad hoc to paper All multiplicity, closure and transversality statements for the Euler class boundary contributions hold as stated (deferred to companion paper [1]).
Section 7 repeatedly invokes 'we will show in [1]' for critical contributions, e.g., the factor 4 in Theorem 6.3 and the vanishing statements in equations (12), (15), (22).
standard math The base case N(A1,n1,m1,m2) from Theorem 6.1 is correct, as proved in [4].
The recursion rests on this base case; it is cited from the authors' earlier paper rather than re-proved here.

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Pith. "Pith review of Counting curves in a linear system with upto eight singular points." pith.science (2026). https://pith.science/paper/KW63NDKK

@misc{pith2026190900772,
  author       = {Pith},
  title        = {Pith review of: Counting curves in a linear system with upto eight singular points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KW63NDKK}},
  note         = {Machine review of arXiv:1909.00772}
}

abstract

In this paper, we develop a systematic approach to enumerate curves with a certain number of nodes and one further singularity which maybe more degenerate. As a result, we obtain an explicit formula for the number of curves in a sufficiently ample linear system, passing through the right number of generic points, that have $\delta$ nodes and one singularity of codimension $k$, for all $\delta+k \leq 8$. In particular, we recover the formulas for curves with upto six nodal points obtained by Vainsencher. Moreover, all the codimension seven numbers we have obtained agree with the formulas obtained by Kazarian. Finally, in codimension eight, we recover the formula of A.Weber, M.Mikosz and P.Pragacz for curves with one singular point and we also recover the formula of Kleiman and Piene for eight nodal curves. All the other codimension eight numbers we have obtained are new.

Figures

Figures reproduced from arXiv: 1909.00772 by the authors.

**Figure 1.** Two nodes colliding into a tacnode That is indeed the case. We give a rigorous proof of (10) in [5, Lemma 6.3(2)]. We will show in [1] that the contribution from each of the tacnodal points, namely B(q1, q2, qδ+1) ∩ µ is 4. Hence the total contribution from all the components of type B(qi1 , qi2 , qδ+1) is 4 δ 2 N(A δ 1A3, n1, m1, m2). Next, let us focus on B(q1, q2, q3, qδ+1). We claim that B(q1, q2, q3, qδ+1) … view at source ↗

**Figure 2.** Three nodes colliding into a triple point [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Three nodes colliding into an A5-singularity 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Four nodes collapsing to a D6-singularity [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 5.** Figure 5: Six nodes collapsing to an ordinary quadruple poin [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: Two nodes and one cusp collapsing to a D5-singularity We also show that the contribution from each of the points of A δ−2 1 ◦ PD5 ∩ µ is 2. Hence the total contribution from all the components of type B(qi1 , qi2 , lqδ+1) equals 2 δ 2 N(A δ−2 1 PD5, n1, m1, m2, θ).…

**Figure 7.** Figure 7: Two nodes and one tacnode collapsing to a [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

**Figure 8.** Figure 8: Three nodes and one D4-singularity collapsing to a quadruple point We also show in [1] that the contribution of the section from each of the points of Xˆ 9 ∩µ is 24. That gives us Theorem 6.8 for θ = 1. 7.8 Proof of Theorem 6.10: computing N(Aδ 1PD5, n1, m1, m2, θ) Rec…

**Figure 9.** Figure 9: Three nodes and one triple point collapsing to a qua [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: Two nodes and nodes and one A4-singularity collapsing to a D7-singularity We also show that the contribution from each of the points of A δ−2 1 ◦ PD7 ∩ µ is 6. Hence the total contribution from all the components of type B(qi1 , qi2 , lqδ+1) equals 6 δ 2 N(A δ−2 1…

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