{"id":"2371966d-08ca-45b0-a62e-24a1d6e1c85f","arxiv_id":"1908.03939","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For hyperplane arrangements satisfying a mild condition, the top-dimensional part and radical of the Jacobian ideal are arithmetically Cohen-Macaulay, and liaison constructions produce arrangements whose singular curves have prescribed Hartshorne-Rao modules.","lead":"This paper studies the singular locus of hyperplane arrangements and shows that several natural algebraic ideals attached to an arrangement are often Cohen-Macaulay, a strong regularity property. It introduces a geometric gluing technique called liaison addition to arrangement theory, which also yields new curves with prescribed algebraic failure behavior.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.2's induction depends on an unverified computer computation of the building block's Hartshorne-Rao module; the manuscript provides no code or certificate.","rationale":"The reader's weakest-assumption analysis identifies exactly the same point I would stress: the second main theorem depends on a computer check that is reported but not made reproducible. The concern is load-bearing in the sense that if the computation were wrong, Theorem 6.2 and Corollary 6.5 would fail. However, the computation is a finite, explicit algebraic verification, the Betti diagram is displayed, and the rest of the proof is a coherent application of standard liaison-addition techniques. There is no evidence of an actual error, and computational checks of this kind are common in the field. Therefore the correct response is not to reject or to mark the paper unverified, but to regard the claim as accepted subject to routine independent verification. The reader's ACCEPT verdict with moderate confidence already reflects this balance, so I do not change the verdict.","tokens_in":17718,"tokens_out":38576,"duration_ms":424737,"concrete_test":"Run a fresh Macaulay2 (or CoCoA) computation over Q for F = x*y*z*w*(x+y)*(y+z)*(z+w)*(w+x)*(w+x+y+z). Compute the minimal free resolution of R/Jbar and the Hartshorne-Rao module M(C) = direct sum over t of H^1(P3, I_C(t)), for example via prune HH^1(SheafOfModules I_C). Check that the Betti table is exactly 1,4,4,1 with nonzero rows 0, 7, 9, and that M(C) is a one-dimensional k-vector space concentrated in degree 8, with the four linear forms in its minimal presentation generating the maximal ideal. If this computation reproduces the reported table and module structure, the base of the induction in Theorem 6.2 is sound; if it does not, the inductive construction in Section 6 lacks its starting block.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction behind the second main theorem (Theorem 6.2 and Corollary 6.5) rests on a single computer-algebra verification that is not independently reproducible from the manuscript. The building block is the 9-plane arrangement F = xyzw(x+y)(y+z)(z+w)(w+x)(w+x+y+z). The proof displays the Betti diagram of R/Jbar (total 1,4,4,1, with nonzero rows 0, 7, 9) and then uses Rao's theorem to conclude that the Hartshorne-Rao module M(C) is a one-dimensional k-vector space supported in one degree. That one-dimensional module is the engine of the inductive liaison-addition construction producing, for every r >= 1, an arrangement whose top-dimensional part (and analogously whose radical) fails to be Cohen-Macaulay in exactly one degree with failure dimension r. If the displayed Betti table or the inferred module structure were wrong, the induction would have no starting block and Theorem 6.2 would collapse. The manuscript cites CoCoA elsewhere but provides no script, output file, or certificate for this specific computation. Moreover, the Rao-theorem step from the Betti table to 'M(C) is k' requires that the four linear forms in the minimal presentation generate the maximal ideal; this is not checked in the text. This is a reproducibility gap rather than a detected mathematical error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces liaison-theoretic tools (liaison addition and basic double linkage) to the study of hyperplane arrangements. For an arrangement defined by a product of linear forms F, the authors consider three unmixed ideals associated to the Jacobian ideal J: the top-dimensional part Jbar, the radical sqrt(J), and an intersection of powers of the associated primes. The first main result (Theorem 3.2 and Corollaries 3.5, 3.6, 3.7) states that under the hypothesis that no hyperplane of the arrangement contains the supports of two distinct non-reduced components of the Jacobian scheme, all of these ideals define arithmetically Cohen-Macaulay schemes, in P^3 and then in P^n by hyperplane sections. The second main result (Theorem 6.2 and Corollary 6.5) shows that for every r >= 1 there are arrangements in P^3 whose top-dimensional part (respectively radical) has Hartshorne-Rao module of dimension r supported in exactly one degree, and that arbitrary shifts of the supporting degree can be achieved while staying in the same even liaison class. The proofs rely on a building-block example whose Hartshorne-Rao module is claimed to be a one-dimensional vector space, verified by a displayed Betti diagram computed with CoCoA.","tokens_in":18009,"tokens_out":16417,"duration_ms":176718,"significance":"If the results hold, the paper makes a substantial contribution by importing liaison addition into arrangement theory, giving a broad sufficient condition for Cohen-Macaulayness of the top-dimensional part and the radical of the Jacobian ideal, independent of freeness. The second theorem is striking: it shows that the failure of ACM-ness can be concentrated in one degree with arbitrarily large multiplicity, and it connects these curves to known irreducible even liaison classes. The paper is clearly written, and the main non-computational proof of Theorem 3.2 is convincing and well structured. The novel use of liaison addition and basic double linkage for arrangements is a genuine strength. The principal weakness is the reproducibility gap for the computer-algebra verification that underpins Theorem 6.2 and Corollary 6.5, together with a compressed inference from a Betti diagram to the structure of the Hartshorne-Rao module.","major_comments":[{"comment":"The proof of the building-block claim rests entirely on a computer calculation that is not reproducible from the manuscript. For the 9-plane arrangement F = xyzw(x+y)(y+z)(z+w)(w+x)(w+x+y+z), the displayed Betti diagram is stated without providing the CoCoA or Macaulay2 code, the input file, or a verification certificate. If this computation is incorrect, the inductive construction has no starting block and Theorem 6.2 collapses. The same issue affects part (iii) of Theorem 6.2 (\"one can check on the computer\") and the building block of Corollary 6.5 (\"One can check\"). Please provide machine-readable code or an independent certificate for each of these computations, or replace the computational verification with a human-checkable derivation.","section":"Section 6, Theorem 6.2"},{"comment":"The step from the displayed Betti diagram to the conclusion M(C) is isomorphic to k is too compressed. The text asserts that Rao's theorem gives a minimal presentation R(-c-1)^4 -> R(-c) -> M(C)^vee -> 0 and hence M(C)^vee is isomorphic to k; however, the displayed Betti numbers alone do not determine the module structure. One needs to know that the presentation matrix is the 1 x 4 matrix of the four variables generating the maximal ideal, not some other linear forms. Please write out the full minimal free resolution of R/I_C with all degrees, state the precise form of Rao's theorem being used, and show explicitly how the Betti table forces the claimed Hartshorne-Rao module. As written, this load-bearing step is not checkable.","section":"Section 6, Theorem 6.2 proof"},{"comment":"The proof does not explicitly treat the case where every hyperplane of the arrangement is a factor of one of the F_i, so that no linear form L exists outside all F_i. In that case the final basic-double-linkage step is vacuous, and the proof should say that all components have already been accounted for. This is a minor gap in an otherwise sound argument, but it should be addressed in the revision.","section":"Theorem 3.2"}],"minor_comments":[{"comment":"The overline on Jbar is frequently lost in the typeset text, so that Jbar and J are indistinguishable in several displayed formulas. Please ensure consistent notation throughout.","section":"Notation and typography"},{"comment":"The term \"dual arrangement\" is used without definition. Please define the dual construction for a set of points Z in P^n.","section":"Corollary 3.9"},{"comment":"The phrase \"n-tuple\" in the discussion before Theorem 6.2 conflicts with the use of n for the dimension of projective space. Rename the tuple length to avoid confusion.","section":"Theorem 6.2 proof"},{"comment":"The Betti diagrams reported in Examples 4.1, 4.3, 4.5, 4.6, 5.4, and 6.6 are stated as computer checks. While individual checks are acceptable, providing the scripts or at least a summary of the commands would improve reproducibility and is in the spirit of the journal's standards.","section":"Computer checks"}],"recommendation":"major_revision","confidential_remarks":"The paper is novel and likely correct, and the first main theorem is well supported. The main obstacle is that the second main theorem depends on a computer verification that is not independently reproducible from the manuscript, and on a Rao-theorem step that needs to be spelled out. If the authors supply code or certificates and expand the derivation of the building block's Hartshorne-Rao module, I would support acceptance. The self-citation pattern is not excessive and the results are not circular."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine contribution. It brings liaison addition into hyperplane arrangement theory, proves a clean Cohen-Macaulay criterion for the top-dimensional part and radical of the Jacobian ideal (Theorem 3.2), shows the CM property of the three natural ideals is independent when the hypothesis fails, and gives a slick inductive construction of arrangements whose Hartshorne-Rao module is one-dimensional in a single degree with arbitrary dimension r. The liaison-addition arguments are standard but cleverly applied, and the examples are informative. The writing is clear about what is proven and what is checked by computer.\n\nThe main results hold up on reading. The proof of Theorem 3.2 uses liaison addition to glue non-reduced components, and the no-plane hypothesis is exactly what makes the required regular sequences. The independence examples (Example 4.3) and the Betti-diagram variation for same combinatorics (Example 4.6) are useful and correct. The second theorem (Theorem 6.2) rests on a building block arrangement F = xyzw(x+y)(y+z)(z+w)(w+x)(w+x+y+z), whose displayed Betti diagram indicates a single copy of k as Hartshorne-Rao module. The stress-test worry about the Rao step is not a real gap: since M(C) is finite length and the presentation has four generators and four linear first syzygies, the linear forms must generate the maximal ideal, so the cokernel is k. The paper does not spell that out, but it follows immediately, so this is a matter of exposition, not correctness.\n\nThe genuine soft spot is reproducibility. The paper uses CoCoA for several Betti diagrams but gives no scripts or output files. The polynomials are explicit, so a reader can rerun the checks, but the field is moving toward sharing code and it would be better to include it. This is a minor issue, not a flaw in the mathematics. I would not hold up acceptance over it; I would ask the authors to add the scripts or at least a precise statement of the computer algebra system and version.\n\nWho is this for? People working on hyperplane arrangements, liaison theory, and CM properties of Jacobian ideals. It is not a Terao-conjecture paper, but it complements that literature. It deserves a serious referee. The results are new, the proofs are mostly clear, and the examples are valuable. I would accept it with minor revisions.","headline":"A solid, genuinely new paper that brings liaison addition into hyperplane arrangement theory; the main theorems hold up, and the only real weakness is the lack of computer-check scripts, which is standard for the field.","tokens_in":18491,"tokens_out":4150,"would_cite":true,"duration_ms":42388,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N20","52C35","14M06","14M07","14M05","13D02","13N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Liaison addition makes the singular-locus schemes of hyperplane arrangements Cohen-Macaulay in most cases — and prescribes their failures otherwise.","keywords":["hyperplane arrangements","Jacobian ideal","Cohen-Macaulay","liaison addition","basic double linkage","Hartshorne-Rao module","Terao conjecture","singular locus"],"falsifier":"Compute the Hartshorne-Rao module of the nine-plane arrangement $F=xyzw(x+y)(y+z)(z+w)(w+x)(w+x+y+z)$ by resolving the saturated ideal of its top-dimensional Jacobian component; if the module is not one-dimensional in exactly one degree, Theorem 6.2 collapses. A negative test for the positive theorem would be any arrangement satisfying the separation hypothesis whose $\\overline J$ or $\\sqrt J$ is not Cohen-Macaulay.","tokens_in":17537,"feed_emoji":"📐","tokens_out":9199,"duration_ms":84977,"temperature":0.7,"pith_summary":"The paper brings two classical tools from liaison theory — liaison addition and basic double linkage — to bear on hyperplane arrangements, and uses them to control the schemes supported on the arrangement's singular locus. Its first theorem says that if no hyperplane of the arrangement contains the supports of two distinct non-reduced components of the Jacobian scheme, then the top-dimensional part of the Jacobian ideal, its radical, and the ideals obtained by fattening each radical component to a suitable power are all Cohen-Macaulay. Its second theorem says that when this hypothesis fails, the failure can be tuned: for every positive integer $r$ there is an arrangement in $\\mathbb P^3$ whose top-dimensional singular curve and whose reduced singular curve each have Hartshorne-Rao module of dimension $r$ concentrated in a single degree — the minimal possible failure — and infinitely many shifts of the same even liaison class are again realised by arrangements. The results matter because they separate the geometry of the singularity from the freeness question of Terao's conjecture, and they show that liaison classes of curves are populated by arrangement-defined schemes in a controlled way.","feed_headline":"For most arrangements, the singular locus is Cohen-Macaulay","feed_subtitle":"Liaison addition also lets one tune the singular curve's failure to be Cohen-Macaulay, to any size, in one degree.","key_machinery":"The central mechanism is liaison addition (Theorem 2.4): for two codimension-two schemes $V_1,V_2$ with $F_i\\in I_{V_i}$ forming a regular sequence $(F_1,F_2)$, the ideal $F_2 I_{V_1}+F_1 I_{V_2}$ defines the union of $V_1,V_2$ and the complete intersection, and the Hartshorne-Rao module splits as a direct sum with degree shifts. Basic double linkage (Proposition 2.5) is the degenerate case where one scheme is empty; it adds a complete intersection while shifting the Hartshorne-Rao module by one degree. Translated to arrangements (Propositions 2.6 and 2.9), these two operations say that for products $F$ and $G$ of linear forms, $M(C_{FG}) \\cong M(C_F)(-p)\\oplus M(C_G)(-m)$ and $M(C_{LF}) \\cong M(C_F)(-1)$, which is exactly what makes the positive theorem and the prescribed-failure construction work.","core_discovery":"On the paper's own terms, the central discovery is that the schemes cut out by the height-two primary components of the Jacobian ideal of a hyperplane arrangement behave like liaison-theoretic curves: they can be assembled piece by piece. Under the hypothesis that no hyperplane contains the support of two distinct non-reduced components, Theorem 3.2 shows in $\\mathbb P^3$ that $\\overline J$ is Cohen-Macaulay by adding non-reduced components one at a time with liaison addition and then adding the reduced lines with basic double linkage; Corollaries 3.5 and 3.6 extend the same conclusion to $\\sqrt J$ and to $\\cap \\mathfrak p_i^{b_i}$, and Corollary 3.7 lifts the result to $\\mathbb P^n$ by a general hyperplane section. When the hypothesis is dropped, Examples 4.1–4.5 show that the three ideals can fail independently, and that a free arrangement can even have a non-Cohen-Macaulay radical. The second main theorem, Theorem 6.2, asserts that for every $r\\ge 1$ there is an arrangement whose top-dimensional part has Hartshorne-Rao module of dimension $r$ supported in exactly one degree, and Corollary 6.5 gives the same for the radical; the proof builds these curves by taking $r$ general copies of a fixed nine-plane building block and combining them with liaison addition, then shifting with basic double linkage.","pith_inferences":["Because liaison addition gives a direct-sum decomposition of Hartshorne-Rao modules, the family of modules arising from arrangement singular loci is closed under shifted direct sums; if this closure is robust, one could conjecture that every even liaison class of curves in $\\mathbb P^3$ contains an arrangement-defined curve, not just the one-dimensional-supported classes treated here.","The one-degree, dimension-$r$ failures are the signature of Buchsbaum curves, so arrangement singular loci provide many new examples of Buchsbaum curves with locally free syzygy bundles; one could test whether the same operations produce modules supported in several degrees with prescribed maps, which would answer the paper's open question about classifying arising liaison classes.","The separation hypothesis looks combinatorial in nature — it concerns which planes contain which triple-line supports — so one could try to turn Theorem 3.2 into a pure intersection-lattice criterion for Cohen-Macaulayness of $\\overline J$ and $\\sqrt J$, and test on the two arrangements of Example 4.6 whether their common lattice already decides the Cohen-Macaulay property."],"forward_implications":["Whenever the separation hypothesis holds, $\\overline V$, $V_{\\mathrm{red}}$ and the fattened ideals $\\cap \\mathfrak p_i^{b_i}$ are arithmetically Cohen-Macaulay, independent of whether the arrangement itself is free.","For graphic arrangements, the hypothesis translates to a graph condition: if no two 3-cycles of a graph share an edge, then $R/\\sqrt J$ and $R/\\overline J$ are Cohen-Macaulay; every bipartite graph satisfies this.","For every $r\\ge 1$, the even liaison class whose Hartshorne-Rao module is one-dimensional in a single degree contains infinitely many curves arising as top-dimensional parts of Jacobian ideals of plane arrangements, in infinitely many degree shifts; the same holds for reduced singular loci.","The failure of Cohen-Macaulayness of the singular curve can be made to occur in exactly one degree with dimension $r$, so arrangement singular loci realise the minimal possible non-ACM behaviour; in these constructions the syzygy bundle of the Jacobian ideal is locally free.","The hypothesis cannot simply be dropped: arrangements with the same intersection lattice can have different Betti diagrams for $\\overline J$ and $\\sqrt J$, so the Cohen-Macaulay question is genuinely finer than pure combinatorics."],"supporting_citations":[{"why":"Supplies the generalized liaison addition theorem used to assemble non-reduced components of the Jacobian scheme into an ACM curve.","marker":"[7]"},{"why":"Provides the liaison-addition statement and the hyperplane-section criterion used to pass from $\\mathbb P^3$ to $\\mathbb P^n$.","marker":"[12]"},{"why":"Contains the basic double G-linkage version of basic double linkage used to add planes while shifting the Hartshorne-Rao module.","marker":"[13]"},{"why":"Introduced basic double linkage for curves in $\\mathbb P^3$, the operation used to shift the failure to higher degrees.","marker":"[11]"},{"why":"Gives the starting example of a plane arrangement whose Jacobian algebra is not Cohen-Macaulay, which the paper modifies as its building block.","marker":"[14]"},{"why":"Supplies the theorem that the Hartshorne-Rao module determines the even liaison class and connects the module's resolution to the curve's resolution.","marker":"[16]"}],"fun_headline_variants":["Liaison addition makes singular locus Cohen-Macaulay","Tuning singular curve failure to any degree via liaison","Hyperplane arrangements: Cohen-Macaulay singular locus via liaison","For arrangements, singular locus Cohen-Macaulay under mild hypothesis","Hartshorne-Rao module: measure singular locus failure in one degree"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The positive results rest on the assumption that no plane of the arrangement contains the supports of two distinct non-reduced (thickened) singular components; the failure-tuning results rest on a computer check that a specific nine-plane arrangement has a one-dimensional Hartshorne-Rao module in exactly one degree.","fun_headline_variants_meta":{"raw":{"variants":["Liaison addition makes singular locus Cohen-Macaulay","Tuning singular curve failure to any degree via liaison","Hyperplane arrangements: Cohen-Macaulay singular locus via liaison","For arrangements, singular locus Cohen-Macaulay under mild hypothesis","Hartshorne-Rao module: measure singular locus failure in one degree"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001018,"raw_usage":{"total_tokens":4477,"prompt_tokens":1303,"completion_tokens":3174,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":919,"completion_tokens_details":{"reasoning_tokens":3086}},"tokens_in":919,"tokens_out":3174,"duration_ms":21268,"temperature":1.0,"reasoning_tokens":3086,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:57:58.112302+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Hartshorne-Rao module of the nine-plane arrangement $F=xyzw(x+y)(y+z)(z+w)(w+x)(w+x+y+z)$ by resolving the saturated ideal of its top-dimensional Jacobian component; if the module is not one-dimensional in exactly one degree, Theorem 6.2 collapses. A negative test for the positive theorem would be any arrangement satisfying the separation hypothesis whose $\\overline J$ or $\\sqrt J$ is not Cohen-Macaulay.","supporting_citations":[{"cited_title":"Geramita and J","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized liaison addition theorem used to assemble non-reduced components of the Jacobian scheme into an ACM curve."},{"cited_title":"Introduction to Liaison Theory and Deficiency Modules,","cited_arxiv_id":null,"evidence_quote":"Provides the liaison-addition statement and the hyperplane-section criterion used to pass from $\\mathbb P^3$ to $\\mathbb P^n$."},{"cited_title":"Migliore and U","cited_arxiv_id":null,"evidence_quote":"Contains the basic double G-linkage version of basic double linkage used to add planes while shifting the Hartshorne-Rao module."},{"cited_title":"Lazarsfeld and A.P","cited_arxiv_id":null,"evidence_quote":"Introduced basic double linkage for curves in $\\mathbb P^3$, the operation used to shift the failure to higher degrees."},{"cited_title":"Mustat ¸ˇ a and H","cited_arxiv_id":null,"evidence_quote":"Gives the starting example of a plane arrangement whose Jacobian algebra is not Cohen-Macaulay, which the paper modifies as its building block."},{"cited_title":"Rao, Liaison among curves in P3, Invent","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that the Hartshorne-Rao module determines the even liaison class and connects the module's resolution to the curve's resolution."}],"review_version":1}