{"id":"c14070fe-e11b-4eed-897e-da08d0509ea7","arxiv_id":"2608.06019","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For several large classes of licci ideals, the number of generators is bounded by the largest shift in the last step of the graded free resolution, confirming part of three new conjectures.","lead":"This paper introduces three conjectures restricting the shape of graded free resolutions of licci ideals, most centrally that the number of generators never exceeds the largest degree shift in the final syzygy module. It proves the conjectures for several large classes, including codimension two ideals, codimension three Gorenstein ideals, and ideals containing a full regular sequence of quadrics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems rely on two cited regularity bounds ([23, Cor 5.13] and [10, Cor 4.2]); if either has unstated hypotheses or fails for some licci ideal, the deductions in Theorems 3.1–3.3 and 5.2 collapse.","rationale":"The reader's weakest_assumption identifies exactly the two external bounds as the load-bearing point, and my reading agrees. The internal arguments—Boij–Söderberg decomposition in Theorem 3.1, the combination of the two bounds in Theorem 3.2, the Gorenstein duality in Theorem 3.3, and the iterative linking in Theorem 5.2—are coherent and appear correct once those inequalities are granted. The dependence on [23, Cor 5.13] is especially strong because it supplies the final comparison b_1 ≤ T_c in Theorems 3.1, 3.2, 3.3, and it drives the socle-degree bounds in Theorem 5.2. The computational tables in Section 6 are corroborating evidence rather than proof; the lack of a reproducible AI protocol is a secondary concern, but the tables list deviation and socle degree, so the conjecture is checkable by inspection. The textual slips (e.g., the 'norm 2' phrase in Theorem 2.2 and the compressed justification around s = n−1 in Theorem 5.2) do not appear to affect correctness. Therefore, the manuscript should be accepted conditionally, pending verification that the cited bounds are accurately stated and applicable to the full class of licci ideals considered.","tokens_in":15697,"tokens_out":45840,"duration_ms":403816,"concrete_test":"Check the exact statements of [23, Cor 5.13] and [10, Cor 4.2] in the original papers, confirming that their hypotheses cover all homogeneous licci ideals in polynomial rings over a field and that the inequalities use the same shifts (t_1, T_1, T_k, T_c) as in the manuscript. As a computational cross-check, select a non-Gorenstein licci ideal with a nearly pure resolution (e.g., a codimension-3 ideal from Table 2) and compute its Betti table in Macaulay2 to verify T_c > (c−1)t_1 and T_c ≤ T_k + (c−k)T_1 for all 2 ≤ k ≤ c.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central positive results are conditional on two external inequalities that are cited but not proved or stated in detail. Theorem 3.1 uses the licci bound T_c(S/I) > (c−1)t_1(S/I) from [23, Cor 5.13] to conclude b_1 ≤ (c−1)t_1+1 ≤ T_c. Theorem 3.2 additionally uses T_c ≤ T_k + (c−k)T_1 from [10, Cor 4.2] to propagate the bound to intermediate homological degrees. Theorem 3.3 and Theorem 5.2 also invoke [23, Cor 5.13]. If either inequality has hypotheses not met by all homogeneous licci ideals in polynomial rings (for instance, if the first requires Gorensteinness, equigeneration, or a different shift variable), or if the inequalities are misstated, then the proofs do not go through, even though Conjecture 1.2 might still be true. No internal inconsistency was found: the Boij–Söderberg decomposition, the mapping-cone computations, and the induction in Theorem 5.2 appear correct under these assumptions. The concern is therefore not a detected flaw but a genuine contingency: the main theorems are only as secure as the cited bounds.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Let S be a polynomial ring over a field. For a homogeneous licci ideal I of codimension c, the paper introduces Conjecture 1.2, asserting that μ(I) ≤ T_c(S/I), where T_c(S/I) is the maximal shift in the last module of the minimal free resolution of S/I; it also gives the square-free monomial Conjecture 1.1 and the local m-primary Conjecture 1.3. The authors prove these conjectures in several cases: codimension-two Cohen-Macaulay ideals (Theorem 2.1), codimension-three Gorenstein ideals (Theorem 2.2), licci ideals with nearly pure resolutions (Theorem 3.2), equigenerated Gorenstein monomial licci ideals (Corollary 3.4), and licci ideals containing a maximal regular sequence of quadrics (Theorem 5.2). They also prove the stronger bound b_1 ≤ (c−1)t_1 + 1 in these cases. Section 4 relates the invariants Δ(I), Δ(J), and Δ(L) for an ideal, its link, and their sum, and shows that the Gorenstein case of the conjecture implies the conjecture for at least one of I or J.","tokens_in":15943,"tokens_out":15258,"duration_ms":135834,"significance":"If correct, these conjectures express a new numerical restriction on the Betti tables of licci ideals, complementing the known Cohen-Macaulay and strongly Cohen-Macaulay properties. The partial results cover a wide range of known licci classes, and the proofs use Boij–Söderberg decompositions, mapping-cone resolutions of links, and Golod/Gorenstein structure theory in a transparent way. The authors are explicit about what is conditional: Section 4 isolates the Gorenstein case as a hypothesis, and the tables in Section 6 provide concrete checks on classified families. The arguments are reproducible from the cited literature, and I found no internal inconsistency in the main derivations.","major_comments":[{"comment":"The proofs of Theorems 3.1, 3.2, 3.3, and 5.2 depend on two external numerical inequalities that are cited but never stated precisely: the licci bound T_c(S/I) > (c−1)t_1(S/I) from [23, Corollary 5.13] and the bound T_c ≤ T_k + (c−k)T_1 from [10, Corollary 4.2]. These inequalities are load-bearing for deriving (c−1)t_1 + 1 ≤ T_c and for propagating the bound through intermediate homological degrees, so the manuscript should state their exact hypotheses and conclusions in the graded setting and confirm that they apply to every homogeneous licci ideal, not only to equigenerated or Gorenstein ideals. If either bound has an unstated hypothesis or is misstated, the deductions in these theorems do not survive; the conjectures themselves might still be true, but the present proofs would need repair.","section":"§3, Theorems 3.1–3.3; §5, Theorem 5.2"}],"minor_comments":[{"comment":"The phrase 'I is neither the unit ideal norm 2' appears to contain a typo; it should presumably read 'nor m^2'.","section":"Theorem 2.2"},{"comment":"In the final paragraph of the proof, 'K contains no linear forms' should read 'K′ contains no linear forms,' since K contains the linear forms x_{n−a−b+1},...,x_n by construction.","section":"Theorem 5.2"},{"comment":"The sentence 'Hence ... for all 0 ≤ k ≤ c−1' should either be restricted to 2 ≤ k ≤ c−1 or reconciled with the fixed integer k in the preceding sentence, since the displayed inequality was derived for 2 ≤ k ≤ c−1.","section":"Theorem 3.3"},{"comment":"The sentence 'using AI prepared a table' would benefit from a brief explanation of how the table was generated and rechecked, for reproducibility; the tables themselves are a useful supplement.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper is a solid contribution if the cited inequalities are exactly as stated. I found no internal inconsistency; the main risk is the unverified applicability of [23, Corollary 5.13] and [10, Corollary 4.2] to the full class of homogeneous licci ideals. I recommend asking the authors to state these results explicitly and to verify their hypotheses in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read of Huneke–Polini–Ulrich, arXiv:2608.06019.\n\nThe paper is a genuine contribution, not a repackaging. It introduces three conjectures around the bound mu(I) <= T_c(S/I) for licci ideals, and proves them in several nontrivial classes: codimension two Cohen–Macaulay ideals, codimension three Gorenstein ideals, licci ideals with nearly pure resolutions, equigenerated Gorenstein monomial ideals, and ideals containing a maximal regular sequence of quadrics. The Boij–Soderberg argument in Theorem 3.1 is clean, and the sum-of-links formula in Theorem 4.5 is a nice structural result. The authors are also explicit about where they assume the Gorenstein case: Section 4 states it as a hypothesis, so there's no sleight of hand.\n\nThe main thing to scrutinize is the dependence on two cited bounds: [23, Cor 5.13] gives T_c(S/I) > (c-1)t_1(S/I) for licci ideals, and [10, Cor 4.2] gives T_c <= T_k + (c-k)T_1. These are used to verify the hypotheses of Theorem 3.1 in Theorems 3.2, 3.3, and 5.2. They are published results, so citing them is legitimate, but the paper doesn't restate their exact hypotheses. If either bound fails or has an unstated condition for some licci ideal, the deductions collapse. I don't have evidence that they fail, and the usage looks correct, but a referee should ask the authors to spell out the statements.\n\nThere are a couple of genuine typos: \"norm 2\" in the proof of Theorem 2.2 looks like it should be \"order 2\" or \"unit ideal,\" and \"K contains no linear forms\" in Theorem 5.2 should be \"K'.\" These are minor but worth fixing.\n\nThe computational section is less solid: the tables are useful, but the AI-assisted verification has no code or protocol, so the examples aren't independently reproducible. That's a weakness in presentation, not in the math.\n\nOverall: the central conjectures are plausible and the partial results are real. The paper deserves a serious referee. I'd send it to review with a request to verify the cited bounds' hypotheses and to clean up the typos. I'd likely cite it in my own work on liaison.","headline":"Solid new conjectures and partial proofs for Betti-table bounds on licci ideals; the main theorems lean on two cited bounds that a referee should check.","tokens_in":16503,"tokens_out":2970,"would_cite":true,"duration_ms":25207,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C20","13A30","14M10","14J17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for many homogeneous licci ideals the minimal number of generators is at most the largest shift in the last module of the minimal free resolution, and conjectures the bound always holds.","keywords":["licci ideals","linkage","Betti tables","minimal free resolution","number of generators","deviation","regularity","Boij-Söderberg theory"],"falsifier":"Find a licci ideal whose minimal free resolution has last module concentrated in degrees strictly below the number of minimal generators; the conjecture would be false. This can be checked by computing the Betti table of any licci ideal, for instance by linking a complete intersection, and comparing $\\mu(I)$ with $T_c(S/I)$.","tokens_in":15473,"feed_emoji":"🔗","tokens_out":16461,"duration_ms":122705,"temperature":0.7,"pith_summary":"Licci ideals—ideals obtained from complete intersections by repeated linkage—form a central class in commutative algebra, but their Betti tables are not fully understood. This paper introduces several conjectures comparing the number of minimal generators of a homogeneous licci ideal to the largest shift appearing in the last module of its minimal free resolution. The main conjecture states that $\\mu(I) \\le T_c(S/I)$ for a licci ideal of codimension $c$, equivalently that the deviation of $I$ is at most its regularity; the paper proves this in a large number of cases, including codimension two Cohen–Macaulay ideals, codimension three Gorenstein ideals, licci ideals with nearly pure resolutions, equigenerated Gorenstein monomial ideals, and ideals containing a maximal regular sequence of quadrics. A section on sums of links shows that the Gorenstein case would imply strong evidence for the general conjecture. If true, these restrictions would confirm the earlier expectation that square-free monomial licci ideals are of linear type with Cohen–Macaulay Rees algebras.","feed_headline":"In five big families, licci ideals satisfy a tight Betti bound","feed_subtitle":"New proofs cap the generator count by the resolution's top degree in several large licci families.","key_machinery":"The argument is carried by the interaction between two numerical bounds on the shifts of the minimal free resolution of a licci ideal $I$ of codimension $c$. The first, cited from the structure of linkage, gives $T_c(S/I) > (c-1)t_1(S/I)$; the second, cited from the regularity of Tor, gives $T_c(S/I) \\le T_k(S/I)+(c-k)T_1(S/I)$ for each $k$. Together these force the inequalities $t_k > (k-1)T_1$ needed in the pure-diagram argument. The Boij–Söderberg decomposition expresses the Betti table of $S/I$ as a positive rational combination of pure diagrams bounded between the minimal and maximal shift sequences; reducing the generator bound to a pointwise inequality on pure diagrams yields $\\mu(I) \\le (c-1)t_1+1$, which the licci shift bound then upgrades to $\\mu(I) \\le T_c(S/I)$. In low codimension, Hilbert–Burch and Buchsbaum–Eisenbud supply the structural facts, and the sum-of-links construction $L=I+J$ (always Gorenstein of codimension $c+1$) connects the deviations of $I$, its link $J$, and $L$ through the identity $\\Delta(I)+\\Delta(J)-\\Delta(L)=T_c(S/I)-t_1(S/I)-c$.","core_discovery":"The central assertion of the paper is Conjecture 1.2: for every homogeneous licci ideal $I$ of codimension $c$ in a polynomial ring over a field, the minimal number of generators $\\mu(I)$ is at most $T_c(S/I)$, the maximal degree occurring in the last syzygy module of $S/I$. This is equivalent—after cutting down by a regular sequence of linear forms—to the local statement that $m^{d(I)} \\not\\subset I$ for an $m$-primary licci ideal, where $d(I)=\\mu(I)-c$ is the deviation. The authors prove this conjecture for several families: in codimension two for all Cohen–Macaulay ideals via Hilbert–Burch; in codimension three for Gorenstein ideals via the Buchsbaum–Eisenbud structure theorem together with the fact that non-hypersurface Golod Gorenstein rings cannot occur; for licci ideals with nearly pure resolutions and for equigenerated Gorenstein monomial ideals via the Boij–Söderberg decomposition combined with two bounds on Betti shifts; and for ideals containing a maximal regular sequence of quadrics by induction using links and sums of links. They also prove that if the conjecture holds for the Gorenstein ideal obtained as the sum of links of an ideal and its link, then it holds for at least one of the two linked ideals.","pith_inferences":["The two shift bounds used here are not proved in the paper; a direct proof or counterexample for them would sharpen the whole program, since the Boij–Söderberg reduction in Theorem 3.1 works for any ideal satisfying $t_k > (k-1)T_1$.","The sum-of-links identity suggests an inductive strategy on codimension: proving the Gorenstein case might let one pass to arbitrary licci ideals by linking, reducing the conjecture to checking a $t_1$-to-deviation inequality.","Conjecture 1.2, if true, would imply the square-free monomial case Conjecture 1.1, so the linear-type and Cohen–Macaulay Rees algebra statements for those ideals would follow from a purely numerical bound on resolutions.","A systematic computer search over licci ideals of small codimension could directly test Conjecture 1.2 beyond the families proved here, since the Betti tables of linked ideals are computable."],"forward_implications":["For every codimension-two Cohen–Macaulay ideal and every codimension-three Gorenstein ideal, the local form of the conjecture holds: $m^{d(I)} \\not\\subset I$, bounding the deviation by the regularity.","Licci ideals with nearly pure resolutions and equigenerated Gorenstein monomial licci ideals satisfy the stronger inequality $\\mu(I) \\le (c-1)t_1(S/I)+1$, which implies $\\mu(I) \\le T_c(S/I)$.","If the conjecture holds for Gorenstein licci ideals, then for any licci ideal $I$ with link $J$ and Gorenstein sum $L$, at least one of $I$ or $J$ satisfies the conjecture; moreover the identity relating the three deviations shows that the Gorenstein case controls the general case.","Every licci ideal containing a maximal regular sequence of quadrics has deviation at most its socle degree, so the conjecture holds for that whole family.","The tables assembled from the cited classifications of licci monomial and binomial edge ideals confirm the conjecture for every licci ideal in those lists, while the non-licci examples in the lists fail it."],"supporting_citations":[{"why":"Proves the licci bound $T_c(S/I) > (c-1)t_1(S/I)$ used in Theorems 3.1-3.3 and 5.2.","marker":"[23]"},{"why":"Gives the complementary bound $T_c(S/I) \\le T_k(S/I)+(c-k)T_1(S/I)$ used in the nearly-pure case.","marker":"[10]"},{"why":"Establishes the Boij–Söderberg decomposition of Betti tables into pure diagrams that carries the reduction in Theorem 3.1.","marker":"[11]"},{"why":"Provides the structure theorem for codimension three Gorenstein ideals used in Theorem 2.2.","marker":"[3]"},{"why":"Supplies the mapping-cone description of links and the canonical module identification used throughout Sections 4 and 5.","marker":"[30]"},{"why":"Shows sums of linked ideals are Gorenstein of codimension $c+1$, the key object in Section 4.","marker":"[36]"},{"why":"Gives the licci property for links after cutting down by linear forms, used in the quadrics induction.","marker":"[24]"},{"why":"Identifies $K/(\\gamma)$ with $I/(\\alpha)$ up to shifts, used to transfer deviations in the quadrics proof.","marker":"[22]"},{"why":"Records that Golod Gorenstein rings with $I\\subset m^2$ are hypersurfaces, the contradiction in Theorem 2.2.","marker":"[1]"}],"fun_headline_variants":["Five licci families satisfy the Betti bound conjecture","Generator bound for licci ideals proven in several families","Licci ideals: generator count tied to top resolution degree","New proofs verify licci ideal generator bound across families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorems lean on two cited inequalities about the degrees appearing in the minimal free resolution of licci ideals; if either of those inequalities fails for some licci ideal, the proofs in this paper would not carry through, although the conjectures might still be true.","fun_headline_variants_meta":{"raw":{"variants":["Five licci families satisfy the Betti bound conjecture","Generator bound for licci ideals proven in several families","Licci ideals: generator count tied to top resolution degree","New proofs verify licci ideal generator bound across families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":1927,"prompt_tokens":863,"completion_tokens":1064,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":1000}},"tokens_in":479,"tokens_out":1064,"duration_ms":9615,"temperature":1.0,"reasoning_tokens":1000,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:22:37.238757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a licci ideal whose minimal free resolution has last module concentrated in degrees strictly below the number of minimal generators; the conjecture would be false. This can be checked by computing the Betti table of any licci ideal, for instance by linking a complete intersection, and comparing $\\mu(I)$ with $T_c(S/I)$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the licci bound $T_c(S/I) > (c-1)t_1(S/I)$ used in Theorems 3.1-3.3 and 5.2."},{"cited_title":"Eisenbud, C","cited_arxiv_id":null,"evidence_quote":"Gives the complementary bound $T_c(S/I) \\le T_k(S/I)+(c-k)T_1(S/I)$ used in the nearly-pure case."},{"cited_title":"Eisenbud and F","cited_arxiv_id":null,"evidence_quote":"Establishes the Boij–Söderberg decomposition of Betti tables into pure diagrams that carries the reduction in Theorem 3.1."},{"cited_title":"Buchsbaum and D","cited_arxiv_id":null,"evidence_quote":"Provides the structure theorem for codimension three Gorenstein ideals used in Theorem 2.2."},{"cited_title":"Peskine and L","cited_arxiv_id":null,"evidence_quote":"Supplies the mapping-cone description of links and the canonical module identification used throughout Sections 4 and 5."},{"cited_title":"Ulrich,Sums of linked ideals, Trans","cited_arxiv_id":null,"evidence_quote":"Shows sums of linked ideals are Gorenstein of codimension $c+1$, the key object in Section 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the licci property for links after cutting down by linear forms, used in the quadrics induction."},{"cited_title":"Huneke and B","cited_arxiv_id":null,"evidence_quote":"Identifies $K/(\\gamma)$ with $I/(\\alpha)$ up to shifts, used to transfer deviations in the quadrics proof."},{"cited_title":"Avramov,Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), Progr","cited_arxiv_id":null,"evidence_quote":"Records that Golod Gorenstein rings with $I\\subset m^2$ are hypersurfaces, the contradiction in Theorem 2.2."}],"review_version":1}