On jet closures of singularities

Huaiqing Zuo; Yifan Chen

REVIEW 2 major objections 5 minor 16 references

On jet closures of singularities

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that simple curve singularities are distinguished up to isomorphism by their jet support closure algebras of orders up to one plus the larger Milnor number.

desk verdict Real new content in jet closures, but Theorem D(2) as stated is false without a reducedness assumption. read the letter →

arxiv 2507.05796 v1 pith:SBB6W5OW submitted 2025-07-08 math.AC math.AG

Yifan Chen , Huaiqing Zuo This is my paper · ORCID

classification math.ACmath.AG MSC 14E1814B0513A15

keywords jetclosuresupportschemessimplesingularitieslocalisomorphismproblemmonomialidealsMilnornumberindex

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 argues that the jet closure and jet support closure operations, which record what information is lost when a singularity is replaced by its finite jet schemes, can be packaged into local algebras that are isomorphism invariants and are computable in important cases. The main structural result is that an ideal generated by homogeneous polynomials has m-th jet closure equal to I plus the (m+1)-st power of the maximal ideal; the same equality holds for jet support closures when f is a reduced homogeneous polynomial. For monomial ideals, membership in the m-th jet support closure has an explicit combinatorial criterion. In the case of simple curve singularities, the paper proves that two such singularities are isomorphic exactly when their jet support closure algebras agree for every order m up to M+1, where M is the larger of their two Milnor numbers. It also introduces a filtration and a jet index measuring the smallest jet order that recovers an Artinian quotient.

What carries the argument

The m-th jet closure $I^{{m-jc}}$ is the largest ideal J whose m-th jet fiber over the closed point coincides with that of I, and the m-th jet support closure $I^{{m-jsc}}$ is the analogous construction using reduced fibers. The paper studies the quotient algebras R/$I^{{m-jc}}$ and R/$I^{{m-jsc}}$ as invariants, computing them through Hasse-Schmidt derivations and explicit jet ideals. The key identities are $I^{{m-jc}}$=I+\mathfrak{m}^{m+1} for ideals extended from homogeneous ideals; the monomial membership criterion of Theorem C; and the two-variable weighted-homogeneous formula for $I^{{m-jsc}}$ with the auxiliary set A_m = \{(u,v) \in \mathbb{N}^2 : u i_l + v j_l \geq m+1 for all l\}.

What would settle it

Compute ($x^{2}$)^{m-jsc} in k[[x,y]] for m=2 and m=3 using the paper's own Hasse-Schmidt method. Since $x^{2}$ is weighted homogeneous but not reduced, Theorem D(2) applies to it as stated; if the resulting ideal is not the claimed intersection (f, $x^{{p_1}}$$y^{{q_1}}$) \cap ($x^{{p_2}}$$y^{{q_2}}$), then the theorem is false as stated, while a match would leave only a proof gap.

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Extended reading notes

Core claim

The central claim is Theorem D(3): if R/I_1 and R/I_2 are simple curve singularities and M is the maximum of their Milnor numbers, then R/I_1 is isomorphic to R/I_2 if and only if R/$I_1^{{m-jsc}}$ is isomorphic to R/$I_2^{{m-jsc}}$ for all m \leq M+1. The proof works by computing explicit generators and k-dimensions of these jet support closure algebras for the ADE curve singularities, using a two-variable weighted-homogeneous formula for $I^{{m-jsc}}$ together with a table of small-order closures for E6, E7, and E8. The paper also establishes Theorem A, that any ideal extended from a homogeneous ideal satisfies $I^{{m-jc}}$ = I + \mathfrak{m}^{m+1}, and Theorem C, a membership criterion for the jet support closure of a monomial ideal. A separate contribution is the jet index: for any ideal with Artinian quotient, there is a smallest s such that I is s-jet closed, and this index is computed for diagonal sums and for the ADE singularities.

Load-bearing premise

The paper states the weighted-homogeneous support closure formula for all weighted homogeneous f in two variables, but proves the m\geq d half only under the extra assumption that f is reduced; the theorem as stated therefore depends on non-reduced weighted homogeneous polynomials behaving like reduced ones.

Editorial extensions

If this is right

A simple curve singularity can be identified by finitely many jet support closure algebras, namely those of order up to M+1, replacing the infinite arc-space data used in earlier local isomorphism criteria.
For ideals extended from homogeneous ideals, all jet closures are equal to I+\mathfrak{m}^{m+1}, so the k-dimension of R/I^{m-jc} can be read off from a Hilbert series.
For monomial ideals, deciding membership in I^{m-jsc} reduces to checking finitely many tuples t_1,...,t_n against the monomial generators of I.
The jet index j(f) exists whenever R/J(f) is Artinian and gives the smallest jet order at which the ideal is recovered; for f=x_1^{a_1+1}+\cdots+x_n^{a_n+1}, it equals a_1+\cdots+a_n-n.
The jet closures form a filtration with f_I(xy) \geq f_I(x)+f_I(y), and for homogeneous or monomial ideals the associated homogeneous filtration is described by \sqrt{I}+\mathfrak{m}^m.

Reading between the lines

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

My inference: the finiteness pattern in Theorem D(3) suggests testing whether the Milnor number gives a general upper bound on the jet order needed to distinguish isolated curve singularities beyond the simple ones; the paper's bound M+1 is tailored to the ADE list, so non-simple modality classes are a natural next test.
My inference: because the m\geq d weighted-homogeneous formula is proved only for reduced f, the stated theorem has a proof gap for non-reduced weighted homogeneous polynomials; checking whether the formula fails for examples such as f=x^2 would determine whether Theorem D(2) needs a reducedness hypothesis or a new argument.
My inference: the jet index and the example j_\mu(f)>N(J(f))-1 indicate that jet closures carry information beyond the nilpotency index of the maximal ideal, so the conjecture j_\tau(f)+1=N(f,J(f)) could be probed computationally on a broader family of isolated singularities.
My inference: the monomial criterion of Theorem C, together with the supplied procedures, makes it feasible to survey how often 'good' equality I^{m-jsc}=I+\mathfrak{m}^{m+1} holds for arbitrary monomial ideals and to locate the precise failures.

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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

First thing you should know: this paper contains genuinely new results and a clean proof of one of them, but one of its headline theorems is stated too broadly. Theorem D(2) (also 5.4(2)) gives a formula for jet support closures of weighted homogeneous polynomials in two variables without assuming f is reduced, and Proposition 5.3, the only proof of the m>=d part, explicitly needs reducedness. The counterexample in the stress-test is correct: for f=x^2, m=2, the actual jet support closure is (x^2, xy, y^3), while the theorem's formula gives (x^2). So the statement is false as written. The fix is to add 'reduced'; the example is non-reduced, and the classification of simple curve singularities only uses reduced ones, so Theorem D(3) is likely safe.

What's actually new: Theorem A—for ideals extended from homogeneous polynomials, I^{m-jc}=I+m^{m+1}—is proved carefully with a neat injectivity argument. The monomial jet support closure criterion (Theorem C) is a useful computational tool and is new. The jet index in Section 6 is a sensible invariant, and the filtration properties (monotonicity, mI^{m-jc} subset I^{(m+1)-jc}) are correct and worth having.

The soft spots beyond the missing hypothesis: the classification theorem D(3) leans on a table for E6/E7/E8 that is garbled in the write-up—there are missing rows (e.g., m=8 for E6) and some entries look inconsistent. The appendix supplies Singular code, so the table is in principle reproducible, but as printed it can't be checked line-by-line. The An and Dn computations in Section 3 are detailed, though I didn't verify every coefficient in the case analysis. The conjecture in Section 6 (jet Tjurina index vs nilpotency) is speculative but clearly labeled.

Bottom line: a serious referee should be sent this. The main ideas are sound and the new invariants are worth publishing, but the weighted homogeneous formula must be corrected to include reducedness, and the computational table needs to be cleaned up or generated by a script that is included. This is a revise-and-resubmit kind of paper, not a reject.

Referee Report

2 major / 5 minor

Summary. The paper studies the jet closure and jet support closure of ideals in formal power series rings over an algebraically closed field of characteristic zero. It introduces the quotient algebras R/I^{m-jc} and R/I^{m-jsc}, proves that their isomorphism types are invariants of the singularity, and computes them in several families: homogeneous ideals (Theorem A), simple curve singularities for small m (Theorem B), monomial ideals (Theorem C), and weighted homogeneous polynomials in two variables (Theorem D(2)). The paper also introduces a filtration and a 'jet index' associated with jet closure, and proves some properties of this index. An appendix provides Singular code for computing the two closures.

Significance. If the main results were correct, the paper would add useful invariants to the study of jet schemes and singularity classification, in particular the statement that finitely many jet-support-closure quotients distinguish simple curve singularities (Theorem D(3)). The proof of Theorem A is clean and self-contained, and the monomial criterion of Theorem C is a concrete computational tool. The supplied Singular code is a useful resource. However, the weighted-homogeneous formula of Theorem D(2)/Theorem 5.4(2) is false as stated because a necessary reducedness hypothesis is omitted, so the paper cannot be accepted in its current form.

major comments (2)

[Theorem D(2) / Theorem 5.4(2) / Proposition 5.3] The statement of Theorem D(2) (and Theorem 5.4(2)) omits the reducedness hypothesis on f that Proposition 5.3 explicitly assumes. This is not a cosmetic omission: the theorem is false as stated. Take f = x^2 in R = k[[x,y]], with weights a = 2, b = 1 and m = 2. Then a direct computation gives I_2 + mR_2 = (x_0^2, 2x_0x_1, 2x_0x_2 + x_1^2, x_0, y_0), so sqrt(I_2 + mR_2) = (x_0, y_0, x_1). The monomial xy lies in (x^2)^{2-jsc}: its Hasse-Schmidt coefficients x_0y_1 + x_1y_0 and x_0y_2 + x_1y_1 + x_2y_0 are both contained in this radical. On the other hand, the formula in Theorem D(2) contains an intersection with a monomial ideal whose defining inequalities force every monomial x^p y^q in that second factor to satisfy p ≥ 2 (the condition from (u,v) = (2,0) in A_2), so xy is not a member of the ideal produced by the theorem. Thus the stated formula is not the jet support closure. The theorem must either be restricted to reduced f (with the a ≠ b condition of Proposition 5.2, or an explicit reduction of the case a = b to Theorem D(1)) or be supplied with a genuinely new proof that does not use reducedness.
[Theorem 5.6 / proof of Theorem D(3)] The proof of Theorem 5.6 relies on comparing the k-dimensions of the quotients R/I_i^{m-jsc} using Corollary 5.5 and a displayed table, but the table as printed is malformed and incomplete: the m = 8 row appears to run three columns together, and no entry is shown for E8 at m = 8, although m = M+1 can require m = 9 for E8. Moreover, the proof only states that the dimension data distinguish the singularities without giving the needed case analysis. As printed, the proof of the 'if' direction of Theorem D(3) cannot be checked from the text. Please replace the table with a complete and correctly formatted table, and include the comparison argument for all ADE types up to the stated bound.

minor comments (5)

[Throughout] There are numerous typos and inconsistent notations: for example, 'homogeeous' in the introduction, 'F act' for 'Fact', 'Noetherian' is misspelled, and the local ring is sometimes written C{x,y} and sometimes k[[x,y]] (compare Corollary 3.7 with Theorem 3.9). These should be cleaned up.
[Section 2] Several cross-references are wrong: Example 2.1 should refer to Example 2.7, and the references to 'Proposition 2.1' in the proof of Proposition 3.1 and to 'Fact 2.4' and 'Fact 2.6' in later sections do not match the numbering in Section 2. Please renumber the internal cross-references.
[Theorem D(2) / Theorem 5.4(2)] The notation '(f, x^{p1}y^{q1}) ∩ (x^{p2}y^{q2})' is ambiguous: it is not clear whether the intended ideals are generated by all monomials satisfying the stated inequalities or by particular minimal monomials. Corollary 5.5 uses a different convention, writing ideals such as '(f, x^i y^j)(ni + j ≥ m+1)'. Please state the convention explicitly and make the two presentations consistent.
[Theorem 3.9] The E6, E7 and E8 rows of Theorem 3.9 are justified by an unrecorded computer calculation ('Using the code we have given in the appendix...'). Since the code is included, this could be acceptable, but for verifiability the authors should provide the explicit outputs or the commands that produce the displayed closures for the stated ranges of m.
[Proposition 6.2] In the proof of Proposition 6.2, the sentence describing (I_{m+1}, m_e) should specify that the generators g_{i,j} range over j = 1, ..., m+1, and the argument that f_1, ..., f_m lie in (I_m, m_e) because they do not involve the variables x_i^{(m+1)} should be written out more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main claims are derived from definitions and external results, while the missing reducedness hypothesis in Theorem 5.4(2) is a correctness gap, not a circular step.

full rationale

The paper is self-contained: jet closure and jet support closure are defined from Hasse-Schmidt data (Definitions 2.8 and 2.24), Theorem 3.3 is proved by constructing an injective map rather than assuming its conclusion, monomial jet support closures are computed from external radical-jet descriptions (Lemmas 4.2 and 4.3, cited to Goward and Smith), and the weighted-homogeneous formulas in Propositions 5.2 and 5.3 are obtained by explicit induction on the intersection-of-ideals decomposition of sqrt(I_m + mR_m). Theorem 5.6 then compares the independently computed k-dimensions of the support-closure algebras, so the classification is not fitted to the desired isomorphism type; the jet index (Definition 6.5) is also an independent invariant computed from the preceding jet-closure formulas. Citations to de Fernex-Ein-Ishii, Mallory, Vojta, and Ishii are used for background properties and are not load-bearing self-citations. One flagged issue is non-circular: Proposition 5.3 begins 'assume that f is reduced' and uses that assumption to conclude g in (f), but Theorem 5.4(2) and Theorem D(2) state the weighted-homogeneous formula without that hypothesis, so the theorem overreaches (e.g., f = x^2, where direct computation gives (x^2, xy, y^3), not the stated ideal); this is a correctness gap in the statement, not a reduction of the result to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The paper relies on standard background theorems on jet schemes, radical jet ideals of monomial ideals, arc closures, and the ADE classification. The jet index is a definition, not a postulated entity.

assumptions (4)

standard math Arc closure is trivial for Noetherian local k-algebras with separable residue field ([5, Theorem 5.1]).
Invoked in Proposition 6.4 to guarantee I = I^{∞-jc}.
standard math Radical of the m-th jet ideal of a monomial ideal has the generator descriptions in [4, Theorems 2.1 and 3.1].
Basis for Propositions 4.5, 4.7 and Theorem C.
standard math Arnold's ADE classification lists all simple curve singularities in characteristic zero ([1]).
Used to frame and organize Theorems B, D and Corollary 5.5.
domain assumption k is algebraically closed of characteristic 0 and R = k[[x1,...,xn]] is a formal power series ring with closed point.
This is the ambient setting throughout Sections 3-6.

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Pith. "Pith review of On jet closures of singularities." pith.science (2026). https://pith.science/paper/SBB6W5OW

@misc{pith2026250705796,
  author       = {Pith},
  title        = {Pith review of: On jet closures of singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SBB6W5OW}},
  note         = {Machine review of arXiv:2507.05796}
}

read the original abstract

The jet closure and jet support closure were first introduced by de Fernex, Ein and Ishii to solve the local isomorphism problem. In this paper, we introduce two local algebras associated to jet closure and jet support closure respectively. We show that these two algebras are invariants of the singularities. We compute and investigate these invariants for some interesting cases, such as the cases of monomial ideals and homogeneous ideals. We also introduce a new filtration and jet index to jet closures. The jet index describes which jet scheme recover the information of base scheme. Moreover, we obtain some properties of the jet index. Keywords:jet closure, jet support closure and filtration.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Arnold, A

V. Arnold, A. Varchenko, and S. Gusein-Zade, Singularities of differentiable mappings, I , 2nd ed. MCNMO, Moskva, 2004

work page 2004

[2] [2]

Durfee, Fifteen characterizations of rational double points and simple critical points , Enseign

A. Durfee, Fifteen characterizations of rational double points and simple critical points , Enseign. Math., II. Sr. 25 (1979), 132-163

work page 1979

[3] [3]

de Fernex, L

T. de Fernex, L. Ein and S. Ishii, Jet closures and the local isomorphism problem, J. Algebra 501 (2018), 166-181

work page 2018

[4] [4]

R. A. Goward and K. E. Smith, The jet scheme of a monomial scheme, Commun. Algebra 34 (2006), 1591- 1598

work page 2006

[5] [5]

Mallory, Triviality of arc closures and the local isomorphism problem, J

D. Mallory, Triviality of arc closures and the local isomorphism problem, J. Algebra 544 (2020), 47-61

work page 2020

[6] [6]

Vojta, Jets via Hasse-Schmidt derivations, January 2013

P. Vojta, Jets via Hasse-Schmidt derivations, January 2013. preprint, http://arxiv.org/abs/math/0407113

arXiv 2013

[7] [7]

Ishii, Jet schemes, arc spaces and the Nash problem

S. Ishii, Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc. R. Can. 29 (2007), no. 1, 1-21

work page 2007

[8] [8]

Ishii, Mather discrepancy and the arc spaces

S. Ishii, Mather discrepancy and the arc spaces. Ann. Inst. Fourier (Grenoble) 63 (2013), no.1, 89-111

work page 2013