{"id":"44210e2c-bf54-4e41-8b06-2e0fbaa992b8","arxiv_id":"2506.09598","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Herzog classes of codimension-3 licci ideals are parametrized by pairs of partitions via a graph of direct links, with applications to Tor algebra structures.","lead":"This paper classifies the deformation families, called Herzog classes, of codimension-3 licci ideals by purely combinatorial pairs of partitions, and describes the graph of direct links between them. It also uses this classification to settle a conjecture about the multiplication structure in Tor algebras of such ideals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The graph theorem depends on Theorem 2.1's generic-specialization framework, imported from a preprint and not proved here; the only-if direction of Theorem 3.2 further assumes the single-weight support of w^(1)⊗k survives GL(F_1)-perturbation with the predicted Weyl-group image, an assertion not…","rationale":"The paper's central claim is that the graph of Herzog classes of codimension-3 licci ideals is exactly the combinatorial graph Licci_3 described by pairs of partitions and the partition link formula. The authors provide substantial independent evidence: explicit tables, an implemented algorithm, infinite families, and a squares formula. I checked several entries of the tables against Theorem 3.1 and found them consistent. The combinatorial half, Theorem 3.1, appears self-contained and internally coherent. The algebraic bridge, however, is Theorem 3.2, whose proof is a sketch that imports the existence and uniqueness of a single-weight specialization from [15] via Theorem 2.1. The reader's weakest assumption identifies this same dependency; my stress-test sharpens it to the unproved claim that the single-weight support transforms rigidly under the GL(F_1)-perturbation used in the only-if direction. Section 8 is explicitly conjectural and does not affect the codimension-3 claim. I do not see an internal contradiction in the codimension-3 arguments, so rejection is not warranted; but the imported framework and the perturbation step need a full proof or independent verification before the central claim is unconditionally accepted. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":45930,"tokens_out":21071,"duration_ms":249728,"concrete_test":"Verify Theorem 2.1 and the linkage transformation in the exact form used: for the non-Dynkin format (1,6,8,3), instantiate the explicit ideals I_2 and J_2 from Section 4.3 using the Macaulay2 package LicciExamples, realize the smallest minimal link from I_2 to J_2 by a regular sequence chosen according to Theorem 3.1, and compute the linked ideal's decoration with the package. Then repeat for the generic link with λ'=(0,0,0). A mismatch in either case would falsify Theorem 3.2. Independently, enumerate the Weyl-group double-coset graph for small (d,t), e.g. d=t=2, and compare its edges to the partitions produced by Theorem 3.1, to rule out a purely combinatorial error in the formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Theorem 3.2, which converts the combinatorial graph into a statement about actual direct links. Its only-if proof fixes a good specialization w for I, i.e. one with w^(1)⊗k nonzero only on the lowest weight space of S_λF_1⊗S_μF_3^*, and then replaces w^(1) by (g+ε)·w^(1), where g is a permutation matrix for a y-arm Weyl element τ and ε is a general matrix with entries in m. The proof asserts that the reduced map is then supported only on the weight −τσω_{x_1}, and that the y_1-restricted specialization gives an ideal whose Herzog class is the prescribed σ'. The first half is plausible since ε vanishes modulo m, but the second half is exactly the content of Theorem 2.1, which is only paraphrased from [15] and not proved here. In particular, nothing in this paper shows that the lowest nonzero component in Bruhat order of the perturbed map is unique and transforms by the same τ for non-minimal links (λ'_i=0). If the generic-ring/higher-structure-map framework of [15] fails for some non-Dynkin format, or if the perturbation produces another extremal weight modulo m, the constructed linked ideal need not lie in the asserted Herzog class. Since the remaining description of Licci_3 inherits this step, this is the most load-bearing point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies licci ideals of codimension 3 (and, conjecturally, arbitrary codimension) through a representation-theoretic parametrization of Herzog classes by pairs of partitions associated with Schur functors. The central object is the infinite graph Licci_3 whose vertices are decorations (λ,µ) and whose edges are determined by a partition-theoretic link formula (Theorem 3.1); Theorem 3.2 asserts that this combinatorial adjacency is equivalent to the existence of two directly linked licci ideals in the corresponding Herzog classes. The paper also provides tables of decorations for classical families (complete intersections, Gorenstein ideals, almost complete intersections, hyperplane sections, Dynkin formats), an infinite family for the smallest non-Dynkin format, results on Tor algebra multiplication, explicit free resolutions for the family S_{I_k}, and a conjectural extension to arbitrary codimension. A substantial part of the theory is imported from the authors' preprint [15], especially Theorem 2.1 on the classification of Herzog classes through higher structure maps.","tokens_in":46231,"tokens_out":4411,"duration_ms":54073,"significance":"If the framework of [15] is accepted, the paper offers a striking and useful description: the infinite graph of Herzog classes of codimension-3 licci ideals becomes a purely combinatorial object governed by Weyl-group data. The paper has several genuine strengths: the proof of Theorem 3.1 is self-contained and the resulting partition formula is checked against many independent families; the tables in Section 4 unify known results (Watanabe, Brown, Kustin-Miller, E6/E7/E8 formats); the squares formula in Theorem 8.7 is a clean invariant; and the explicit free resolutions in Section 7 are concrete and reproducible. The algorithmic construction in Section 3.4 and the implemented Macaulay2 code are additional assets. However, the identification of the combinatorial graph with the graph of Herzog classes under direct linkage rests on Theorem 2.1 and on a perturbation argument in the proof of Theorem 3.2 that is not fully proved; this is the main correctness risk.","major_comments":[{"comment":"The 'only if' direction of Theorem 3.2 is the load-bearing bridge from the combinatorial graph Licci_3 to actual direct links, but its proof relies on two unproved assertions. First, it uses Theorem 2.1, which is only paraphrased from the preprint [15] and not proved here; in particular, the claim that every σ in z1W(d,t)x1 is realized by a codimension-3 licci ideal is an imported input. Second, after replacing w^(1) by (g+ε)·w^(1), the proof asserts that the reduced map is nonzero only on the weight −τσω_{x1} and that the y1-restricted specialization gives an ideal whose Herzog class is the prescribed σ′. No argument is given that the lowest nonzero Bruhat component remains unique and transforms by the same τ, especially when the link is non-minimal (some λ′_i = 0). Since the rest of the paper inherits this identification of Licci_3 with Herzog classes, this gap should be closed by a full proof or, failing that, the statement should be made explicitly conditional on the framework of [15] rather than presented as a theorem established here.","section":"Section 3.2, Theorem 3.2"},{"comment":"The proof of Theorem 3.12 says 'We omit the details' and refers to [15, §3], and Lemma 3.13 then uses the same generic example to identify S_I from the graded Betti numbers. These statements are used in Algorithm 3.16 and in Theorem 6.1, so they are not merely cosmetic. The manuscript should either include a complete proof of Theorem 3.12 or state precisely, in the form of a citable theorem, the result from [15] that supplies the graded free resolution (3.3). As written, a central part of the constructive side of the paper is delegated to an unpublished preprint of the same authors.","section":"Section 3.4, Theorem 3.12 and Lemma 3.13"}],"minor_comments":[{"comment":"In the displayed formula for S_J in Definition 8.3, the second Schur functor is written as S_{λ′′}G∗_3; for arbitrary codimension c this should be G∗_c.","section":"Definition 8.3"},{"comment":"In the displayed matrix for d_2 in the even case, the entry 'x 12k' appears to be a typo for 'x_{1k}' or similar; please correct the typesetting.","section":"Section 7.1"},{"comment":"The phrase 'by Theorem, 8.7' contains a stray comma and should read 'by Theorem 8.7'.","section":"Proof of Proposition 5.1(3)"},{"comment":"The notation (1^{2k+1}) for a partition consisting of repeated 1s is used without an explicit explanation; a short sentence defining exponential notation for partitions would improve readability.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem depends essentially on the authors' own preprint [15], which is paraphrased (Theorem 2.1) and used as the source of the generic examples (Theorem 3.12). The editor may wish to verify that [15] is available in a form suitable for citation and that the statements used here are indeed proved there. If the journal expects a paper to be self-contained for its central claims, the dependency on Theorem 2.1 and the perturbation step in Theorem 3.2 should be made explicit or proved in full. These are fixable within the scope of a revision, but they are not purely presentational."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the right paper to referee if you work in liaison or free resolutions. It converts the Herzog-class classification for codim-3 licci ideals into explicit partition combinatorics, gives a link formula, a graph description, and a constructive algorithm, and then uses all that to settle a Tor-algebra conjecture. The main results are genuinely new relative to the cited literature, and the examples check out against the known cases (CI, Gorenstein, ACI, hyperplane sections, Dynkin formats). I believe the substance is real, not a repackaging.\n\nThe centerpiece is Theorem 3.1's partition-link formula and Theorem 3.2's statement that the combinatorial graph edges exactly match direct links. The proof of 3.1 is self-contained Weyl-group combinatorics and looks correct. The only-if direction of 3.2 rests on the generic-ring/specialization framework imported from the authors' preprint [15]: you fix a good specialization, perturb by g+ε, and rely on Theorem 2.1 to know the lowest nonzero component transforms the way you want. That is a genuine dependency, not a hole in this paper; Theorem 2.1 is quoted rather than proved here. Anyone building on this paper should be aware that if the [15] framework fails for some non-Dynkin format, the parametrization of Herzog classes by partitions fails with it. I don't think that is likely, but it should be stated as a hypothesis or proved here.\n\nSmaller soft spots: Theorem 3.12's proof is deferred to [15, §3] and to a Macaulay2 implementation; the output is standard-looking but the reader has to trust the Schubert-cell grading. The higher-codimension section is honestly labeled conjectural, so I don't hold that against it. The only real quality issue is that the paper could do more to isolate which results are fully proved here and which inherit from [15].\n\nBottom line: send it to a serious referee. The combinatorics and the applications are substantial; the imported framework is the right thing to scrutinize, and a referee should ask for the dependency to be made explicit and for a proof or machine-checked verification of the Betti-number construction. I'd cite it.","headline":"A substantial, mostly convincing combinatorial classification of codim-3 Herzog classes; the refereeing should focus on the imported generic-ring framework and the omitted Betti-number proof.","tokens_in":46815,"tokens_out":2086,"would_cite":true,"duration_ms":22951,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","13C05","13C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every Herzog class of codimension-3 licci ideals is indexed by a pair of partitions, and that direct linkage between classes is computed by a simple reordering formula.","keywords":["licci ideals","Herzog classes","linkage","Schur functors","free resolutions","Kac-Moody Lie algebras","Tor algebra","codimension three"],"falsifier":"Compute the decoration of a codimension-3 licci ideal from its minimal free resolution, link it by a chosen regular sequence, compute the decoration of the linked ideal, and compare with Theorem 3.1's formula; a mismatch for any single ideal would disprove the claim.","tokens_in":1842,"feed_emoji":"🔗","tokens_out":5281,"duration_ms":124605,"temperature":0.7,"pith_summary":"The paper establishes a one-to-one correspondence between Herzog classes of codimension-3 licci ideals and pairs of partitions $(\\lambda,\\mu)$ with $\\sum \\lambda_i = 2\\sum \\mu_j + 1$, and it proves that direct linkage between classes is captured by a simple reordering of the partition parts. Concretely, Theorem 3.1 gives the formula for the decoration of a directly linked ideal: remove three parts of $\\lambda$, add a fixed integer $p$ to them, and re-sort them together with the parts of $\\mu$. Theorem 3.2 shows that two decorations are adjacent in the graph $\\mathrm{Licci}_3$ exactly when the corresponding Herzog classes contain directly linked ideals. If these results are correct, the classification of licci ideals in codimension 3 reduces to finite combinatorial checks on partitions.","feed_headline":"Two partitions classify every codim-3 licci class","feed_subtitle":"Direct links between Herzog classes become a reordering rule, making the infinite licci graph purely combinatorial.","key_machinery":"The key object is the decoration $S_{\\lambda}F_1\\otimes S_{\\mu}F_3^*$, a pair of Schur functors attached to the lowest nonzero component of the specialized higher structure map of an ideal. The linkage formula of Theorem 3.1—choose three parts of $\\lambda$, remove them, shift by a constant $p$, and re-sort with the parts of $\\mu$—carries the entire argument, converting Weyl-group double-coset overlaps into explicit partition operations. The paper also uses minimal links, tight double links, and special generating systems (Definition 3.3) to give an algorithm (Algorithm 3.16) for constructing representatives of every Herzog class from a complete intersection, and for establishing that every pair of partitions satisfying the conditions is realized.","core_discovery":"The paper's central claim is that the graph $\\mathrm{Licci}_3$, whose vertices are Herzog classes and whose edges represent direct links, is exactly the graph on the set $^{z_1}W^{x_1}$ with edges defined by the partition formula of Theorem 3.1. In particular, two decorations $S_{\\lambda}F_1\\otimes S_{\\mu}F_3^*$ and $S_{\\lambda_{\\mathrm{link}}}F_1\\otimes S_{\\mu_{\\mathrm{link}}}F_3^*$ are adjacent if and only if there are directly linked ideals in the two classes. The proof translates the Weyl-group double-coset condition into the partition operation, and the converse uses a special generating system to make a three-generated regular sequence produce the desired linked decoration. Along the way, the paper shows that the graded Betti numbers of a representative ideal are read off from the partitions, and that the Tor-algebra multiplication is described by the same data: $e_i e_j$ is nonzero modulo the maximal ideal exactly when $\\lambda_i+\\lambda_j=k+1$, and $e_i f = g_j$ exactly when $\\lambda_i+\\mu_j=k+1$.","pith_inferences":["If the combinatorial description is correct, a perfect ideal of codimension 3 that is not licci should have no decoration linking back to a complete intersection; this could give a finite certificate of non-licci-ness.","The quadratic identity $\\sum \\lambda_i^2 + \\sum \\mu_j^2 = (k+1)^2$ could serve as a fast necessary condition in computer searches for valid decorations.","If the higher-codimension conjectures hold, the doubling construction of Section 8 predicts infinite families of Herzog classes for Gorenstein ideals of codimension 4, which could be checked by explicit free resolutions."],"forward_implications":["Every link between Herzog classes in codimension 3 can be written down as a partition reordering, so the infinite graph $\\mathrm{Licci}_3$ is fully described in finite terms.","For each decoration, Theorem 3.12 gives an explicit graded free resolution with shifts determined by the partitions, so the graded Betti numbers of a licci ideal are an invariant of its Herzog class and can be read off directly.","Theorem 6.1 characterizes the Tor-algebra class of a licci ideal from its decoration, giving a simple criterion for nonzero multiplications and for classes such as $G(r)$.","The bounds in Section 6.2 settle a conjecture: a non-Gorenstein perfect ideal of codimension 3 generated by $b$ elements cannot be of Tor-algebra class $G(r)$ with $r\\ge b-2$.","In the smallest non-Dynkin format $(1,6,8,3)$ the paper lists infinite families of distinct decorations, so there are infinitely many Herzog classes in that format."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the result that a codimension-3 ideal is licci exactly when its specialized higher structure map is nonzero modulo the maximal ideal, with the Herzog class read off from the lowest extremal Schur-functor component.","marker":"[15]"},{"why":"Establishes the generic rings and Kac-Moody representation framework in which the higher structure maps are defined.","marker":"[44]"},{"why":"Provides the Dynkin-type structure theorems for licci ideals that the present combinatorial description extends to all Herzog classes.","marker":"[13]"},{"why":"Gives the existence and uniqueness of rigid deformations that underlie the definition of Herzog classes.","marker":"[17]"},{"why":"Shows licci ideals are strongly unobstructed, so every licci ideal admits a rigid deformation.","marker":"[5]"},{"why":"Develops the representation-theoretic treatment of free resolutions and linkage that the paper uses for the decoration point of view.","marker":"[33]"}],"fun_headline_variants":["Two partitions draw the entire licci graph","Partitions turn licci links into combinatorial moves","A pair of partitions classifies all codim-3 Herzog classes","Licci graph: purely partition-based","Schur functors pin down the licci link structure"],"cache_read_input_tokens":48768,"weakest_assumption_plain":"The classification relies on the existence and good behavior of generic rings and higher structure maps: every codimension-3 licci ideal must be detected by the lowest nonzero component of $w^{(1)}$ modulo the maximal ideal, and every pair of partitions must be realized by some licci ideal; if this framework fails for a given format, the partition description fails there.","fun_headline_variants_meta":{"raw":{"variants":["Two partitions draw the entire licci graph","Partitions turn licci links into combinatorial moves","A pair of partitions classifies all codim-3 Herzog classes","Licci graph: purely partition-based","Schur functors pin down the licci link structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001297,"raw_usage":{"total_tokens":5292,"prompt_tokens":944,"completion_tokens":4348,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":4273}},"tokens_in":560,"tokens_out":4348,"duration_ms":28399,"temperature":1.0,"reasoning_tokens":4273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:45:03.992806+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the decoration of a codimension-3 licci ideal from its minimal free resolution, link it by a chosen regular sequence, compute the decoration of the linked ideal, and compare with Theorem 3.1's formula; a mismatch for any single ideal would disprove the claim.","supporting_citations":[{"cited_title":"The linkage class of a grade three complete intersection","cited_arxiv_id":"2412.00399","evidence_quote":"Supplies Theorem 2.1, the result that a codimension-3 ideal is licci exactly when its specialized higher structure map is nonzero modulo the maximal ideal, with the Herzog class read off from the lowest extremal Schur-functor component."},{"cited_title":"Weyman,Free resolutions and root systems.Annales de l’Institute Fourier 68 (3) (2018) 1241-1296","cited_arxiv_id":null,"evidence_quote":"Establishes the generic rings and Kac-Moody representation framework in which the higher structure maps are defined."},{"cited_title":"Herzog,Deformationen von Cohen-Macaulay algebren,J","cited_arxiv_id":null,"evidence_quote":"Gives the existence and uniqueness of rigid deformations that underlie the definition of Herzog classes."},{"cited_title":"Buchweitz,Contributions á la thèorie des singularités,Thesis, University of Paris, 1981","cited_arxiv_id":null,"evidence_quote":"Shows licci ideals are strongly unobstructed, so every licci ideal admits a rigid deformation."},{"cited_title":"Ni,Free resolutions, linkage, and representation theory","cited_arxiv_id":null,"evidence_quote":"Develops the representation-theoretic treatment of free resolutions and linkage that the paper uses for the decoration point of view."}],"review_version":1}