{"id":"b4cf5f7e-0f0c-43a1-8184-15985a0285e5","arxiv_id":"2507.05796","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper computes jet closure and jet support closure invariants for monomial, homogeneous and simple curve singularities, and introduces a filtration and jet index measuring when jets recover the ideal.","lead":"Jet closures and jet support closures are ideals attached to the 'jet scheme' approximations of a singularity. This paper shows these closures give computable algebraic invariants, calculates them for monomial, homogeneous and weighted homogeneous examples, and defines a 'jet index' measuring how many jets recover the singularity.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem D(2)/5.4(2) is false as stated: it omits the reducedness hypothesis required in Proposition 5.3; for f=x^2, m=2 the stated formula gives (x^2,xy^2,y^3) while the actual jet support closure is (x^2,xy,y^3).","rationale":"The reader's weakest assumption correctly identifies the missing reducedness hypothesis in Theorem D(2)/5.4(2). This is the most load-bearing concern because it is not a mere omission of a detail but a false statement as written: we can exhibit a non-reduced weighted homogeneous polynomial where the theorem's formula gives the wrong ideal. The proof of Proposition 5.3 explicitly uses reducedness at the step where membership in sqrt(f(x_a,y_b)) is promoted to membership in (f), so the theorem cannot be salvaged merely by re-reading the proof. The central classification Theorem D(3) survives because it only applies to simple curve singularities, all of which are reduced; however, the overbroad statement of Theorem D(2) is a correctness risk for any application outside that class, and the introduction's claim should carry the same reducedness hypothesis as the proposition. I agree with the reader's conditional verdict: the paper's main classification may be correct, but the statements need correction before they can be used as written. The garbled table is a secondary reproducibility concern, but the reducedness gap is the sharper, more easily settled issue.","tokens_in":27808,"tokens_out":48639,"duration_ms":476255,"concrete_test":"Run the paper's own Singular routine jetsuppclosure on R=k[[x,y]], I=(x^2), m=2 and compare the output with (x^2, xy^2, y^3). If the computed ideal is (x^2, xy, y^3) rather than (x^2, xy^2, y^3), the stated formula fails. As a control, run the same code for the reduced weighted homogeneous polynomial f=x^2+y^2 with a=b=1 and m=2; the expected output is (f)+m^3 = (x^2+y^2, x^3, x^2y, xy^2, y^3), confirming the reduced case is not affected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.4(2) (restated as Theorem D(2)) claims a formula for (f)^{m-jsc} for all weighted homogeneous f in k[[x,y]] with gcd(a,b)=1, with no reducedness assumption. The only proof of the m≥d case is Proposition 5.3, which begins by assuming f is reduced and uses that assumption to conclude g(x_a,y_b) ∈ (f) from g ∈ sqrt(f(x_a,y_b)). The theorem as stated therefore covers non-reduced f where the proof is unavailable, and the statement is actually false there. Example: f=x^2, weights a=b=1, degree d=2, m=2. Direct computation gives I_2+mR_2 ≡ (x_1^2) modulo (x_0,y_0), so sqrt(I_2+mR_2) = (x_0,y_0,x_1); the resulting closure is (x^2, xy, y^3). The theorem's m≥d formula, interpreting the empty second-intersection condition as the whole ring, yields (f, x^i y^j : i+j≥3) = (x^2, xy^2, y^3), which contains xy^2 but not xy. Thus the two ideals differ. The classification theorem D(3) is not directly invalidated because all simple curve singularities are reduced, but Theorem D(2) as stated is incorrect and must either be restricted to reduced f or be proved without that hypothesis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28137,"tokens_out":21758,"duration_ms":239793,"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":[{"comment":"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.","section":"Theorem D(2) / Theorem 5.4(2) / Proposition 5.3"},{"comment":"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.","section":"Theorem 5.6 / proof of Theorem D(3)"}],"minor_comments":[{"comment":"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":"Throughout"},{"comment":"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.","section":"Section 2"},{"comment":"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.","section":"Theorem D(2) / Theorem 5.4(2)"},{"comment":"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.","section":"Theorem 3.9"},{"comment":"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.","section":"Proposition 6.2"}],"recommendation":"major_revision","confidential_remarks":"The missing reducedness hypothesis in Theorem D(2) is a genuine mathematical error, but it is localized and fixable: the classification theorem D(3) concerns simple curve singularities, which are reduced, so the main classification claim may survive the correction. I recommend major revision rather than rejection. The paper would also be strengthened by a more complete proof of Theorem 5.6 and by making the notation in Theorem 5.4(2) precise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nBottom 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.","headline":"Real new content in jet closures, but Theorem D(2) as stated is false without a reducedness assumption.","tokens_in":28669,"tokens_out":10718,"would_cite":true,"duration_ms":101276,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E18","14B05","13A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["jet closure","jet support closure","jet schemes","simple singularities","local isomorphism problem","monomial ideals","Milnor number","jet index"],"falsifier":"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.","tokens_in":1896,"feed_emoji":"🎯","tokens_out":2885,"duration_ms":104279,"temperature":0.7,"pith_summary":"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.","feed_headline":"Milnor number plus one jet orders identifies a simple singularity","feed_subtitle":"Jet support closure algebras through order M+1 are a complete isomorphism invariant for simple curve singularities.","key_machinery":"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\\}.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces jet closure and jet support closure and the local isomorphism problem; the paper's definitions and basic properties come from here.","marker":"[3]"},{"why":"Supplies the classification of simple singularities as the ADE list, which Theorem D(3) refines to a finite jet-order criterion.","marker":"[1]"},{"why":"Provides the theorem that every ideal in a Noetherian local k-algebra is arc closed, used to prove the existence of the jet index for Artinian quotients.","marker":"[5]"},{"why":"Computes the radical of the jet ideal of a monomial ideal, the key input for the monomial jet support closure criterion.","marker":"[4]"},{"why":"Gives the Hasse-Schmidt derivation description of jet schemes, used throughout the explicit computations.","marker":"[6]"},{"why":"Supplies the standard existence and functoriality properties of jet schemes and arc spaces.","marker":"[7]"},{"why":"Defines filtrations and the passage to an associated homogeneous filtration, which the paper applies to jet closures.","marker":"[12]"}],"fun_headline_variants":["Jet closures up to Milnor max plus one classify simple singularities","M+1 jet support closures are a complete isomorphism invariant","New jet index tells when jets recover the base scheme","For ADE singularities, M+1 jet orders settle isomorphism","Jet support closures: a pinpoint invariant for curve singularities"],"cache_read_input_tokens":30720,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Jet closures up to Milnor max plus one classify simple singularities","M+1 jet support closures are a complete isomorphism invariant","New jet index tells when jets recover the base scheme","For ADE singularities, M+1 jet orders settle isomorphism","Jet support closures: a pinpoint invariant for curve singularities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000529,"raw_usage":{"total_tokens":2513,"prompt_tokens":873,"completion_tokens":1640,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":1556}},"tokens_in":489,"tokens_out":1640,"duration_ms":12979,"temperature":1.0,"reasoning_tokens":1556,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:19:34.986934+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"de Fernex, L","cited_arxiv_id":null,"evidence_quote":"Introduces jet closure and jet support closure and the local isomorphism problem; the paper's definitions and basic properties come from here."},{"cited_title":"Arnold, A","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of simple singularities as the ADE list, which Theorem D(3) refines to a finite jet-order criterion."},{"cited_title":"Mallory, Triviality of arc closures and the local isomorphism problem, J","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that every ideal in a Noetherian local k-algebra is arc closed, used to prove the existence of the jet index for Artinian quotients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the radical of the jet ideal of a monomial ideal, the key input for the monomial jet support closure criterion."},{"cited_title":"Ishii, Jet schemes, arc spaces and the Nash problem","cited_arxiv_id":null,"evidence_quote":"Supplies the standard existence and functoriality properties of jet schemes and arc spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines filtrations and the passage to an associated homogeneous filtration, which the paper applies to jet closures."}],"review_version":1}