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REVIEW 3 major objections 6 minor 25 references

A new addition to the zoo of isolated symplectic singularities

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Pith's one-line read An affine chart in a blow-up of $\mathbb{C}^4/G_5$ is a new isolated symplectic singularity, distinguished by its non-reduced projective tangent cone.

desk verdict A credible new isolated symplectic singularity with a genuinely new invariant, but the load-bearing isolatedness proof rests on an undocumented computer check. read the letter →

arxiv 2505.24524 v1 pith:PDXMPMHD submitted 2025-05-30 math.AG math.RT

Callum Berry This is my paper
classification math.AGmath.RT MSC 14E1514B0513A3053D20
keywords isolatedsymplecticsingularityprojectivetangentconenon-reducedcrepantpartialresolutionQ-factorialterminalisationcomplexreflectiongroupG5affinechartblow-uphyperplanearrangement
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 constructs a new 4-dimensional isolated symplectic singularity, denoted $\ddot{Y}$, as an affine chart in the blow-up of the quotient space $\mathbb{C}^4/G_5$ along its reduced singular locus. The singularity is normal, Cohen–Macaulay, and locally simply connected at its unique singular point, so it cannot be a finite quotient of another symplectic singularity. Its distinguishing feature is that the projective tangent cone at the singularity is non-reduced: the associated graded ring of the local ring contains a nonzero element whose square is zero. The paper proves that every previously known locally simply-connected isolated symplectic singularity has a reduced projective tangent cone, so $\ddot{Y}$ is a new example. It then computes all 12 distinct $\mathbb{Q}$-factorial terminalisations of $\ddot{Y}$ and uses them to obtain 24 of the 92 $\mathbb{Q}$-factorial terminalisations of $\mathbb{C}^4/G_5$.

What carries the argument

The load-bearing object is the affine chart $X_{4,4}=\ddot{Y}$ of the blow-up $S_I\to\mathbb{C}^4/G_5$ at the reduced singular locus $I$; its coordinate algebra is generated by $h, c_1,c_0,c_{-1},d_1,d_0,d_{-1},z_2,z_1,z_0,z_{-1},z_{-2}$ subject to 35 explicit relations. The mechanism that proves isolatedness is a computer-verified radical-membership statement: every generator of $\mathbb{C}[\mathbb{C}^4/G_5]$ lies in the radical of the ideal generated by the 35 relations together with $k_{9,3}, k'_{9,3}, k_{3,9}, k'_{3,9}, k_{13,1}, k_{1,13}, k_{24,0}, k_{0,24}$, which says every point of $\ddot{Y}$ outside the origin lies in a chart that is smooth or has only an $a_2$ singularity. The invariant that carries the separation from known examples is the projective tangent cone $PT_0(\ddot{Y})=\operatorname{Proj}(\operatorname{gr}_{\mathfrak{m}}\mathbb{C}[\ddot{Y}])$; being an analytic invariant, its non-reducedness persists under analytic isomorphism. The terminalisation count is controlled by the hyperplane arrangement attached to $\mathbb{C}^4/G_5$: a codimension-two face meets 2, 3, 4, or 6 hyperplanes, giving at most 12 maximal crepant partial resolutions, and the paper exhibits 12 distinct ones.

What would settle it

Recompute, over $\mathbb{Q}$ and over $\mathbb{C}$, the radical of the ideal generated by the 35 relations of $\mathbb{C}[X_{4,4}]$ together with $k_{9,3}, k'_{9,3}, k_{3,9}, k'_{3,9}, k_{13,1}, k_{1,13}, k_{24,0}, k_{0,24}$; the assertion of Section 3.3 is that this radical contains every generator of $\mathbb{C}[\mathbb{C}^4/G_5]$, so any nonzero solution of those equations would be an uncovered singular point of $\ddot{Y}$ and would disprove isolatedness.

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Extended reading notes

Core claim

The central claim is that the affine chart $X_{4,4}$, written $\ddot{Y}$, is a new isolated symplectic singularity. The proof goes through four checkable statements: (1) $\ddot{Y}$ is normal and Cohen–Macaulay, with the explicit regular sequence $h, z_2, c_{-1}, c_1+z_{-2}$; (2) it is smooth away from the origin, by covering $\ddot{Y}\setminus\{0\}$ with smooth affine charts or charts with only an $a_2$ point; (3) the pullback of the symplectic form is non-degenerate because the exceptional fiber over the origin has dimension at most 2, making the blow-up semi-small; and (4) the local fundamental group at the singular point is trivial, proven via Seifert–van Kampen on the open cover $X_{9,3}\cup X_{13,1}$. The newness is established by the projective tangent cone $PT_0(\ddot{Y})=\operatorname{Proj}(\operatorname{gr}_{\mathfrak{m}}\mathbb{C}[\ddot{Y}])$: in the associated graded ring one has $z_1^2=0$ with $z_1\neq 0$, so the tangent cone is non-reduced, whereas the known locally simply-connected isolated symplectic singularities—the quasi-minimal ones, the dihedral family $Y(d)$, and the toric hyper-Kähler family—all have reduced tangent cones. The final section enumerates 12 distinct $\mathbb{Q}$-factorial terminalisations of $\ddot{Y}$ and, by combining them with the resolutions of the two $a_2$ points, produces 24 of the 92 $\mathbb{Q}$-factorial terminalisations of $\mathbb{C}^4/G_5$.

Load-bearing premise

The proof that $\ddot{Y}$ has no singular points outside the origin depends on the computer calculation in Section 3.3 that every generator of $\mathbb{C}[\mathbb{C}^4/G_5]$ lies in the radical of the ideal generated by the 35 relations together with the eight $k$-variables; if that calculation is wrong or cannot be reproduced, $\ddot{Y}$ might have additional singular points and fail to be an isolated symplectic singularity.

Editorial extensions

If this is right

  • If $\ddot{Y}$ is correct, the known catalogue of 4-dimensional locally simply-connected isolated symplectic singularities gains an entry whose tangent cone is non-reduced, so the catalogue is not exhausted by the quasi-minimal, dihedral, and toric hyper-Kähler families.
  • The non-reduced tangent cone immediately rules out analytic isomorphisms between $\ddot{Y}$ and any singularity whose tangent cone is reduced, including all previously known locally simply-connected examples.
  • The 12 $\mathbb{Q}$-factorial terminalisations of $\ddot{Y}$ are exhaustive and pairwise distinct, and each is reached by two further blow-ups; this is a richer partial-resolution graph than the two terminalisations known for other isolated examples.
  • The same blow-up construction, together with the two terminalisations of each $a_2$ chart, yields 24 distinct $\mathbb{Q}$-factorial terminalisations of $\mathbb{C}^4/G_5$, a concrete step toward the full list of 92.

Reading between the lines

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

  • One could test the same construction on the next exceptional reflection groups $G_6$ and $G_7$: if a non-reduced projective tangent cone appears there too, it would suggest that these blow-up charts form a systematic source of new locally simply-connected examples.
  • The paper's Hilbert-series computation and the 12 symmetries of $\ddot{Y}$ suggest that $\ddot{Y}$ may carry a dihedral-symmetric deformation family; checking whether the non-reduced tangent cone persists in nearby fibres would clarify how rigid the example is.
  • Because the Coulomb-branch and toric hyper-Kähler families all have reduced tangent cones, future searches for isolated symplectic singularities might profitably look at non-toric Coulomb branches, where no reducedness theorem of the kind proved in Section 5.2 is known.
  • The enumeration of terminalisations via face counts of the hyperplane arrangement could become a predictive rule: a partial resolution whose singular point lies on a codimension-two face meeting $r$ hyperplanes should have at most $2r$ $\mathbb{Q}$-factorial terminalisations, and $\ddot{Y}$ realizes the maximal case $r=6$.
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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

3 major / 6 minor

Summary. The paper studies the exceptional complex reflection group G5 acting on C4 and the blow-up of C4/G5 at its reduced singular locus. In the affine chart X4,4 the author writes an explicit 12-generator algebra with 35 relations, denotes the corresponding singularity by ÿ, and claims it is a 4-dimensional isolated locally simply-connected symplectic singularity. The main invariant used to distinguish ÿ from previously known examples is that its projective tangent cone at the singular point is non-reduced, whereas the known locally simply-connected isolated symplectic singularities have reduced projective tangent cones. The paper also computes the Hilbert series, describes the symmetries of ÿ, proves triviality of the local fundamental group, and lists 12 Q-factorial terminalisations of ÿ (and 24 for C4/G5), with exhaustiveness argued via the associated hyperplane arrangement.

Significance. If the computational assertions are correct, the paper provides a genuinely new example in a small and actively studied class. The non-reducedness of the projective tangent cone is an elegant and easy-to-state invariant, and the comparison with Beauville's theorem, the Bellamy--Bonnafé--Fu--Juteau--Levy--Sommers family Y(d), and Namikawa's toric hyper-Kähler varieties is the right kind of external benchmarking. The explicit coordinate algebra, the local-fundamental-group computation, and the 12 terminalisations are substantial contributions. The main weakness is that several load-bearing facts are delegated to unreproduced computer calculations; the paper would be considerably stronger if those assertions were accompanied by certificates, code, or fully detailed proofs.

major comments (3)
  1. [Section 3.3] The proof that ÿ is an isolated singularity reduces to the assertion that all generators of C[C4/G5] lie in the radical of the ideal generated by the 35 relations together with k9,3, k'9,3, k3,9, k'3,9, k13,1, k1,13, k24,0, and k0,24. The paper states only that this 'can be verified via computer calculation' and gives no code, no certificate, no algorithm, and no base field. This is not a cosmetic omission: if the radical-membership assertion is false, then X4,4 has points outside the smooth charts whose singularities are undetected, so the central claim that ÿ is isolated fails. Please provide a reproducible certificate, such as a script that computes a radical decomposition over Q together with explicit membership witnesses, or a hand proof.
  2. [Section 3.2 and Appendix B] The definition of C[ÿ] as the quotient of the free algebra on h,c1,...,z−2 by the 35 listed relations depends on the assertion that this ideal is prime of dimension 4. The only detailed argument offered is in Appendix B, but the appendix works out the ρ-lemma explicitly for C[C4/G5] and for C[ÿ] says only that the same justification applies because h is not contained in the ideal being blown up. Please state the hypotheses of Lemma B.0.1 for R = C[ÿ] explicitly, exhibiting the ideal I, the elements g_i and h_i, or supply the computational certificate. As written, the completeness of the relation list is asserted rather than established, and this completeness is needed for every later geometric claim.
  3. [Section 3.4] Normality is established by asserting that h, z2, c−1, c1 + z−2 is a regular sequence, with the statement 'We can verify using a computer.' This is another load-bearing computation: without normality, ÿ is not known to be a symplectic singularity. Please provide a proof or an explicit certificate (including the term order, the base field, and the Gröbner-basis computation used) so that the depth-4 claim can be checked independently.
minor comments (6)
  1. [Section 3.1.2] The list of affine charts given in the paragraph before Section 3.1.1 repeats X13,1 twice; the fifth chart should be X1,13.
  2. [Section 4.2] In the paragraph on U3, the map κ2 : U2 → C× × C× should be κ3 : U3 → C× × C×, and the displayed pushout arrows are garbled. Also, because f4,4 = g4,4 g13,1−1 in U3, the loop γ4 maps to −[γ2] in π1(U2), not +[γ2]; the sign does not affect the conclusion that the pushout is trivial.
  3. [Section 4.3] The Hilbert series is stated to be 'calculated (for example with Magma)' without specifying the method or providing code; a reader cannot reproduce the value from the text alone.
  4. [Section 5.2, Proposition 5.2.1] The proof of Proposition 5.2.1 is quite compressed, especially the reduction to an element of the form f ∈ r_λ · Sym(t*) and the lifting of freeness from gr_I(gr_m(R)) to gr_m(R). Since this proposition is used to exclude the Namikawa family, please expand the argument or give a reference where the full proof is written out.
  5. [Section 6.2] The Sage code fragment for the hyperplane-arrangement computation is printed inline and the referenced diagram of the 2-dimensional slice is missing from the text; please provide the code as supplementary material and include the diagram it refers to.
  6. [General] The paper would benefit from a data-availability statement listing the exact Magma/Sage scripts used for the claims in Sections 3.2, 3.3, 3.4, 4.3, and 6.2, together with version information and the relevant input files.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new singularity and its invariants are derived by explicit computation against independent external benchmarks, not by fitting or self-referential definition.

full rationale

The paper's central construction is an explicit blow-up computation: X4,4 is defined as an affine chart in the blow-up of C4/G5 at the reduced singular locus, and its coordinate algebra is presented with 35 explicit relations. The claim that ÿ is a new isolated symplectic singularity is not built from a fitted parameter renamed as a prediction, nor from a normalization chosen to force the outcome. The non-reduced projective tangent cone is derived from the displayed relation c1^2 c0 + d1^2 d0 + 2 z1^2 = 0, which becomes z1^2 = 0 in the associated graded ring; the assertion that z1 is nonzero follows from the absence of linear terms, so the invariant is computed rather than assumed. Distinguishing ÿ from known families is done against independent external results: Beauville's theorem for smooth tangent cones, the Bellamy–Bonnafé–Fu–Juteau–Levy–Sommers family Y(d) whose reducedness is proved in the paper by embedding the tangent cone into an integral domain, and Namikawa's toric hyper-Kähler singularities whose reduced tangent cones are proved via the Coulomb-branch presentation from Braverman–Finkelberg–Nakajima. None of these citations are self-citations of the present author in a load-bearing way; the only self-citation, [7], is used for peripheral context about G4 and G6 and does not support the main theorem. The weakest step is the radical-membership computation in Section 3.3, stated only as 'This can be verified via computer calculation' with no certificate or code; that is an under-specification and a correctness risk, not circularity, because the assertion is an external computational fact that could in principle be checked independently. Similarly, the exhaustive count of Q-factorial terminalisations depends on Thiel's hyperplane-arrangement data and a Sage computation, but that again is an independent computational input, not a definitional recycling of the conclusion. No equation in the paper is equivalent to its own input by construction, and no fitted quantity is relabeled as a prediction. Overall circularity score: 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on explicit computations and on the completeness of a short list of known families. No continuous parameters are fitted. The main external inputs are published theorems, a Magma-computed ideal, and Thiel's hyperplane arrangement data. The most fragile entries are the computational verifications that are not shipped as code or certificates.

assumptions (5)
  • domain assumption The singular locus of C4/G5 is the image of the fixed subspaces of symplectic reflections, with radical ideal I as computed in Magma.
    Used throughout Section 3 to define the blow-up center; the computation is stated without proof or code.
  • ad hoc to paper The 35 relations listed for C[ÿ] form a complete prime ideal defining a 4-dimensional variety.
    Foundation for all properties of ÿ; completeness is verified by computer and by the conceptual argument in Appendix B, but the computational certificate is not shipped.
  • standard math Beauville's classification of isolated symplectic singularities with smooth projective tangent cone holds.
    Used in Section 5 to dismiss closures of minimal nilpotent orbits; this is an external published theorem [2].
  • domain assumption The known locally simply-connected isolated symplectic singularities are exhausted by minimal nilpotent orbit closures, the Y(d) family, and Namikawa's toric hyper-Kähler varieties Y(A,0).
    Used for the 'all other known' distinctness claim in Section 5; no survey citation supporting this exhaustiveness is given.
  • domain assumption Thiel's hyperplane arrangement data for C4/G5, loaded from G5.sage, is correct.
    Used in Section 6.2 to bound the number of Q-factorial terminalisations of ÿ by 12; the data is available online but not bundled with the preprint.

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Cite this review

Pith. "Pith review of A new addition to the zoo of isolated symplectic singularities." pith.science (2026). https://pith.science/paper/PDXMPMHD

@misc{pith2026250524524,
  author       = {Pith},
  title        = {Pith review of: A new addition to the zoo of isolated symplectic singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDXMPMHD}},
  note         = {Machine review of arXiv:2505.24524}
}
abstract

We give details of a new isolated symplectic singularity found in an affine chart in a crepant partial resolution of $\mathbb{C}^4/G_5$, which is 4-dimensional, isolated, and locally simply-connected. We distinguish the new singularity among all known such by the fact that the projective tangent cone at the singularity is non-reduced. We also find all 12 of its $\mathbb{Q}$-factorial terminalisations, in the process finding 24 for the quotient singularity $\mathbb{C}^4/G_5$.

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Reference graph

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