Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

Dusa McDuff; Kyler Siegel

REVIEW 2 major objections 4 minor 11 references

Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Exact formula for stabilized ellipsoid embeddings into the round ball: Fibonacci staircase below τ⁴, then the rational fold 3a/(a+1).

desk verdict A major result: the stabilized ellipsoid embedding function for the ball is now known exactly, via a new scattering-diagram bridge to singular curves, but one internal lemma is asserted without proof and needs a referee's attention. read the letter →

arxiv 2412.00561 v2 pith:M33HXAW2 submitted 2024-11-30 math.AG math.SG

Dusa McDuff , Kyler Siegel This is my paper · ORCID

classification math.AGmath.SG MSC 53D3514H5014J2614N35

keywords symplecticnonsqueezingstabilizedellipsoidembeddingsembeddingcapacityFibonaccistaircasesesquicuspidalcurvesscatteringdiagramsminimaldegreeplanedelPezzosurfaces

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 claims to settle the stabilized symplectic ellipsoid embedding problem for the round four-ball: after increasing the ambient dimension by an arbitrary amount, the sharp embedding capacity is an explicit function that repeats the known four-dimensional Fibonacci staircase up to the accumulation point $\tau^4=(7+3\sqrt{5})/2$ and then becomes the rational fold $3a/(a+1)$. The route is indirect: stabilized ellipsoid embeddings are obstructed by rational algebraic curves in the complex projective plane that carry a prescribed $(p,q)$ cusp and only node-like auxiliary singularities, so the embedding result is deduced from a purely algebro-geometric existence theorem for such curves. The existence theorem is proved via scattering diagrams, converting each candidate curve into a term in a completed diagram, with the needed terms shown to be nonzero using positivity results for basic scattering diagrams. If correct, this gives the first complete solution of Problem 1.0.2 for the round ball and explains the phase transition between the staircase and the simple folding tail as a transition from unicuspidal to sesquicuspidal obstructions.

What carries the argument

The load-bearing bridge is a bijection between $(p,q)$-well-placed rational curves in a uninodal Looijenga pair and curves in a toric model that meet one distinguished toric divisor with contact order one, together with the theorem that existence of such curves is detected by nonvanishing of a coefficient in the minimal scattering diagram $S(D_T)_{\min}$. A scattering diagram is a collection of rays in the plane labeled by power series; the completion algorithm adds outgoing rays until the monodromy around every loop is trivial. For rigid del Pezzo surfaces the paper exhibits toric models whose diagrams are basic scattering diagrams with two or three initial rays, and a change-of-lattice reduction turns the relevant two-ray cases into the standard diagrams $D^{\ell_1,\ell_2}_{e_1,e_2}$. Known positivity results for these standard diagrams then supply the required nonzero coefficients, yielding the curve existence theorems and hence the embedding obstructions.

What would settle it

Evaluate the stabilized capacity at any $a>\tau^4$ by an independent method, for example by computing the relevant higher symplectic capacities at $a=7$ or $a=8$, and compare with $3a/(a+1)$; any value below $3a/(a+1)$ would refute Theorem A. On the algebraic side, run the completion of the scattering diagram $D^{3,3}_{e_1,e_2}$ to sufficiently high order in $t$ and check that the coefficient at the relevant lattice point is nonzero for every coprime $p,q$ with $p+q$ divisible by $3$ and $p/q>\tau^4$; a single zero coefficient, or a failure of the predicted Fibonacci condition below $\tau^4$, would refute Theorem B and with it the curve construction behind the embedding theorem.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem A: for every $N\ge 1$, the stabilized ellipsoid embedding function of the round ball is $c_{B^4\times\mathbb{R}^{2N}}(a) = \frac{1}{\sqrt{\alpha_k}}\,a$ on each interval $[\alpha_k,\beta_k]$, is $\sqrt{\alpha_{k+1}}$ on each interval $[\beta_k,\alpha_{k+1}]$, and is $3a/(a+1)$ for all $a\ge\tau^4$, where the numbers $\alpha_k,\beta_k$ are built from Fibonacci numbers and accumulate at $\tau^4$. The paper derives this from Theorem B, a complete answer to the minimal-degree problem for rational plane curves with a $(p,q)$ cusp when $p+q$ is divisible by $3$: such a curve of degree $(p+q)/3$ exists exactly for the Fibonacci pairs $(\mathrm{Fib}_{k+4},\mathrm{Fib}_k)$ with $k$ odd, or when $p/q>\tau^4$. Because the symplectic obstruction mechanism turns these curves into lower bounds and known folding constructions give matching upper bounds, the two theorems together determine the capacity. The same framework is extended to del Pezzo surfaces, yielding a complete description of their stabilized embedding functions in the rigid cases and a universal rational lower bound in the non-rigid cases.

Load-bearing premise

The paper assumes as a black box that a singular rational sphere in a symplectic four-manifold, with one cusp whose parameters satisfy the rigidity condition $p+q=c_1([C])$ and only node-like other singularities, always obstructs stabilized ellipsoid embeddings in the stated way; the present paper does not reprove the compactness and transversality behind that implication.

Editorial extensions

If this is right

The stabilized embedding capacity of the round four-ball is now known for every $a\ge 1$ and every $N\ge 1$, replacing the previous piecemeal lower bounds with one explicit formula.
Hind's folding embeddings are sharp in the whole tail region $a\ge\tau^4$, so no stronger stabilized obstruction can exist beyond the accumulation point.
For plane curves with a prescribed $(p,q)$ cusp and $p+q$ divisible by $3$, the minimal degree is $(p+q)/3$ except in finitely many cases of low singularity excess, and the curves realizing the minimum are rational and can be chosen well-placed with respect to any nodal cubic.
For rigid del Pezzo surfaces, the same framework computes the stabilized capacity as the unstabilized staircase up to the accumulation point and $a/(a+1)$ beyond; for non-rigid del Pezzo surfaces it proves the lower bound $a/(a+1)$ for all $a$.
The ratio set $S_X$ of cusp types realized by rational curves is dense beyond the accumulation point for rigid del Pezzo surfaces and dense in $[1,\infty)$ for non-rigid ones, giving an abundance of algebraic obstructions.

Reading between the lines

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

Editorial inference: if the scattering-diagram bridge is as robust as the paper suggests, the same coefficient-nonvanishing criterion should compute stabilized capacities for other monotone targets beyond del Pezzo surfaces, where the rational tail $a/(a+1)$ would be governed by the same folding construction.
Editorial inference: the byproduct equality $c_{B^4\times\mathbb{R}^{2N}} = c_{\mathbb{CP}^2\times\mathbb{R}^{2N}}$ suggests that in the stabilized regime the capacity depends mainly on the symplectic area class and the minimal cusp ratios available, not on finer features of the four-dimensional target.
Editorial inference: the conjectural exact count of well-placed curves carrying the tail obstructions, if proved, would give a quantitative refinement of Theorem A and a sharp check on the scattering coefficients at $t=1$.
Editorial inference: the refined multi-variable scattering coefficients described in the paper should detect the auxiliary singularity counts and homology classes of well-placed curves, potentially linking the stabilized embedding problem to the subtle combinatorial phenomena seen in the classification of unicuspidal curves.

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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

Bottom line: this paper solves a central open problem in quantitative symplectic geometry — the stabilized ellipsoid embedding function for B^4 is now known for all a, and the answer is as clean as conjectured: Fibonacci staircase up to tau^4, then 3a/(a+1). The novelty and significance are real, and the scattering-diagram bridge to singular algebraic curves is clearly the right framework.

What is new: Theorem A's lower bound 3a/(a+1) on the whole tail (previously only isolated values), the iff characterization of minimal-degree (p,q) cuspidal curves in Theorem B, the general correspondence Theorem G (well-placed curves iff nonzero scattering coefficient), and the density results in Theorem F. There are no free parameters; the staircase constants come from quiver moduli and prior honest results.

Soft spots: two. First, the paper's internal bridge Lemma 6.3.2 (discrete rays vs outer corners, dense region vs (a_acc,∞)) is stated and used as the linchpin for Theorems B and F, but I could not find a proof anywhere in §6.3; it is not deferred, it is simply missing. This is the load-bearing step that the stress-test identifies, and the concern is valid. The statement is plausible and probably follows from GP10's classification plus the known staircase numerology, but as written it is an assertion, not a proved lemma. The second soft spot is the black-box obstruction theorem 2.0.1 imported from [McS23]; if that theorem has hidden assumptions, the symplectic consequences don't go through. The authors acknowledge the dependency and sketch the argument, so this is a normal (if heavy) external dependence rather than an internal error. Both soft spots are fixable in principle; the first definitely needs a proof in any final version.

Citation pattern: heavy self-citation, but to papers that actually prove the needed tools (e.g. staircase stability). Not a flaw by itself.

Audience: symplectic topologists, algebraic geometers interested in cuspidal curves and scattering diagrams. Deserves a serious referee. Recommendation: send to peer review; the referee should treat Lemma 6.3.2 as the main thing to verify.

Referee Report

2 major / 4 minor

Summary. The paper computes the stabilized symplectic embedding function c_{B^4 \times \mathbb{R}^{2N}}(a) for ellipsoids into the round ball, proving an explicit piecewise formula: an infinite Fibonacci staircase for a below the accumulation point \tau^4, followed by the rational tail 3a/(a+1) for a \ge \tau^4. The proof proceeds by converting symplectic embedding obstructions into statements about existence of (p,q)-well-placed rational algebraic curves, then using a correspondence with scattering diagrams, the change-of-lattice trick, and positivity results for basic scattering diagrams from Gross--Pandharipande and Gr\"afnitz--Luo. The paper also proves analogous results for del Pezzo surfaces and gives new families of sesquicuspidal plane curves, including a resolution of the minimal-degree problem in many cases.

Significance. If the proof is completed, the paper fully solves the stabilized ellipsoid embedding problem for the round ball, a central open problem in quantitative symplectic geometry after McDuff--Schlenk. The explicit formula is parameter-free and exhibits a sharp phase transition from the unstabilized staircase to a rational folding tail. Conceptually, the paper builds a bridge from singular algebraic curve theory and scattering diagrams to symplectic nonsqueezing, and it gives a large new supply of low-degree cuspidal rational plane curves. The exposition is largely careful and the main reductions are clearly laid out; the paper is explicit that it avoids unproved symplectic field theory virtual techniques. The main reservations are the unproved internal Lemma 6.3.2 and the heavy reliance on the imported obstruction theorem from [McS23].

major comments (2)

[Section 6.3, Lemma 6.3.2] Lemma 6.3.2 is stated without proof, and it is the hinge for Theorem B and for the J=2 cases of Theorem F(a), and hence, through Section 3 and Section 2, for Theorem A. The lemma identifies, under the bijection W_X, the discrete rays of S(D^{\ell_1,\ell_2}_{m_1,m_2})_{min} with outer corners of the infinite staircase c_X|_{[1,a_X^{acc}]} and the dense region with (a_X^{acc},\infty). After the statement the text moves directly to Corollary 6.3.3 and the proofs of Theorem B and Theorem F, with no further argument. If this identification is incorrect at any boundary value, or if a non-outer fraction maps to a discrete ray, the dense lower bounds c_{X\times\mathbb{R}^{2N}}(a)\ge a/(a+1) would not follow, and Theorems A and E would fail. Please supply a complete proof of both bullets, or reduce them explicitly to published results such as [GP10, Thm. 5] together with the staircase classification in [MS24]/[Cri+25], including the congruence conditions stated in Corollary 6.3.3.
[Section 2, Theorem 2.0.1] All symplectic embedding obstructions in the paper pass through Theorem 2.0.1, imported from [McS23] (Cor. 2.7.2, Cor. 2.3.8, Thm. D, Thm. E), which is described only by a short sketch involving moduli of J-holomorphic curves. This is the unique bridge from existence of algebraic curves to the lower bounds c_{X\times\mathbb{R}^{2N}}\ge a/(a+1). The manuscript should state the publication status of [McS23] and either give a complete proof of Theorem 2.0.1 or a precise reference to a published version with all hypotheses verified. In particular, the hypotheses of Corollary 2.0.2 (monotonicity of X or N\le 1, semipositivity, index-zero condition) should be checked explicitly for the unimonotone del Pezzo surfaces used in Theorem E(b), and the perturbation step from algebraic sesquicuspidal curves to symplectic sesquicuspidal curves in the proof of Theorem E should be justified in the presence of auxiliary singularities.

minor comments (4)

[Corollary D] The monotonicity inequalities in the proof of Corollary D appear reversed. Since U\subset B^4(3), one has c_{U\times\mathbb{R}^{2N}}\ge c_{B^4(3)\times\mathbb{R}^{2N}}, and since X\subset U, one has c_{U\times\mathbb{R}^{2N}}\le c_{X\times\mathbb{R}^{2N}}. The displayed chain of inequalities should be corrected so that the two directions give the claimed equality.
[Corollary 6.3.3] The sentence 'Inspecting Table 4.2.1' refers to the wrong table; the relevant data appears in Table 4.3.1.
[Various] There are several typographical errors: 'especicially' in Remark 1.0.9, 'geoemtry' and 'resuls' in Remark 1.0.10, 'Cantour' in Remark 5.2.10, 'necesarrily' in Remark 3.0.6, and 'if and only of one if' in Corollary 6.3.1.
[Section 6.2, Theorem 6.2.1] The sentence 'an inspection of their argument shows that we can take \kappa=1' would benefit from a precise pointer to the relevant part of [GP10], since the sharp 'if and only if' in Theorem B depends on nonvanishing for the primitive ray, not merely for some positive multiple of it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation detected; the staircase formula is not hardwired into the inputs, though the proof leans on prior same-author theorems and an unproved bridge lemma.

full rationale

The claimed derivation is not circular by construction. Theorem A's upper bound on [τ^4, ∞) comes from explicit folding embeddings [Hin15; CHS22], while the lower bound is obtained by combining the general obstruction theorem 2.0.1 (imported from [McS23]) with existence of (p,q)-well-placed curves certified by scattering-coefficient nonvanishing theorems [GP10, Thm 6.2.1; Gro+18, Prop C.13; GL23, Thm 1]. The staircase constants α_k, β_k, τ^4 are not fitted to c; they are generated by the Fibonacci recursion and by the roots ξ of R(t)=t^2/ℓ2 - t + 1/ℓ1, matched to the staircase via the explicit bijection W_X in Table 4.3.1 and Proposition 4.2.1. The same-author citations [McS23, MS24] are load-bearing, but they are used as general theorems whose stated assumptions (semipositivity, index-zero sesquicuspidal curves, unimonotone del Pezzo surfaces) do not include Theorem A; hence they are not 'uniqueness imported from authors' in the forbidden sense. Caveat, not circularity: Lemma 6.3.2, which identifies discrete scattering rays with outer staircase corners and the dense region with (a_acc,∞), is stated without proof and is essential for Theorems B and F; if that identification failed the density argument would collapse. This is an omitted proof/correctness risk rather than a reduction of the conclusion to its inputs; no equation in the paper defines the staircase corners as the scattering rays.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or invented entities; the paper introduces no new constants, forces, or dimensions. It relies on a network of external theorems, some by the same authors, which are cited transparently.

assumptions (8)

standard math Kontsevich-Soibelman: every scattering diagram has a unique minimal consistent completion
Used throughout Section 5.1; cited [KS06].
domain assumption GPS10 Thm 5.4: scattering coefficients of basic diagrams equal relative Gromov-Witten invariants
Bridges scattering diagrams to curve counts; cited [GPS10].
domain assumption GP10 and Reineke quiver nonvanishing for equal weights
Proves dense region nonvanishing for S(D^{ell,ell}_{e1,e2}); cited [GP10, Rei10].
domain assumption GL23 Thm 1: dense region positivity for unequal weights
Handles CP^1 x CP^1 case; cited [GL23].
domain assumption McS23 obstruction theorem: index-zero sesquicuspidal curves yield embedding obstructions (Thm 2.0.1 black box)
The bridge from curves to symplectic nonsqueezing; imported without proof in Section 2.
standard math GHK15 Prop 1.3: existence of toric models for Looijenga pairs
Used in Section 4.1 to set up the fundamental bijection.
domain assumption Classification of rational unicuspidal curves for Fibonacci pairs (Orevkov, Kashiwara, BLMN)
Provides the curves in Theorem B case (a).
standard math K3 surfaces are not uniruled (used in Lemma 5.2.8)
Rules out positive-dimensional families of well-placed curves.

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Pith. "Pith review of Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing." pith.science (2026). https://pith.science/paper/M33HXAW2

@misc{pith2026241200561,
  author       = {Pith},
  title        = {Pith review of: Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M33HXAW2}},
  note         = {Machine review of arXiv:2412.00561}
}

read the original abstract

We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces.

Figures

Figures reproduced from arXiv: 2412.00561 by the authors.

**Figure 1.0.** 1: The stabilized ellipsoid embedding function [PITH_FULL_IMAGE:figures/full_fig_p003_1_0.png] view at source ↗

**Figure 4.3.** 1: A toric model for ℂℙ2 , with its (essentially unique) uninodal anticanonical divisor Nℂℙ2 . (up to biholomorphism). In the following, given any uninodal anticanonical divisor N ⊂ X, we will give a prefered toric model TX for the Looijenga pair (X, N ) and explicitly describe the corresponding function WX : ℤ2 ≥1 → ℤ2 appearing in Proposition 4.2.1. The toric model TX turns out to be essentially independent of the… view at source ↗

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Works this paper leans on

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Reviewed August 12, 2026 · model on record in the stance chip above.

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[EG22] John B Etnyre and Marco Golla. “Symplectic hats”.Journal of topology15.4 (2022), pp. 2216–2269. [Eva23] Jonny Evans.Lectures on Lagrangian torus fibrations. Vol

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[Sym] Margaret Symington. “Four dimensions from two in symplectic topology. Topology and geometry of manifolds (Athens, GA, 2001), 153–208”.Proc. Sympos. Pure Math. Vol

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