{"id":"69bdabb6-ffdc-4e77-90f7-ac4cf78982d7","arxiv_id":"1908.00676","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each n at least 4, the paper builds grade 3 ideals with a specified Betti table that are homogeneously licci yet not sequentially bounded licci, settling Chong's question negatively.","lead":"This paper constructs grade-three homogeneous ideals that are homogeneously licci but fail a stronger 'sequentially bounded' property, answering a question posed by E. Chong. The result clarifies the hierarchy of licci properties and provides monomial ideals with matching Hilbert functions in the three-variable case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-SBL proof rests on the explicitly omitted proof in Prop. 3.10 that λ(L) ≤ 1 for ideals with Betti table (3.2); without it, direct links to complete intersections are not ruled out.","rationale":"The reader's weakest-assumption analysis identifies the omitted proof inside Proposition 3.10 and the asserted existence of certain complete intersections. My stress-test agrees with the first of these as the most load-bearing issue. The paper's final non-SBL argument in Theorem 3.12.ii reduces every sequentially bounded double link to the classes (⋆), (⋆⋆), or (♦), and then asserts that no ideal in these classes can be directly linked to a complete intersection. The only stated reason for that impossibility is the unproved bound λ ≤ 1 (and the stronger 'λ would have to be ≥ 3' claim). Because the sequentially bounded property only constrains the types of the complete intersections used, a single direct link from any ideal in these classes to a complete intersection would give a valid sequentially bounded sequence and destroy the theorem. The manuscript explicitly flags the missing proof, so this is not an invented objection. It is a genuine gap, but a plausibly repairable one: if an independent computation verifies λ ≤ 1 and the unique Koszul class, the rest of the case analysis can go through. The existence of ideals satisfying (⋆) is less concerning because Section 5 gives an explicit ideal and the complete intersection listed there is indeed contained in I: I contains x^n, y^(n+3), and z^(2n+4), so x^(n+1), y^(2n+4), and z^(2n+4) lie in I, making the named regular sequence plausible. For these reasons I do not recommend changing the reader's CONDITIONAL verdict; the concern confirms that the proof needs a filling-in before the main theorem is fully established.","tokens_in":36438,"tokens_out":18081,"duration_ms":172805,"concrete_test":"Run Macaulay2 for n = 4 and n = 5 on the explicit ideal L from the Section 5 construction: L = (y*z, x^(n+1), y^(2n+4) - z^(2n+4)) : (z*(z^(2n+3), x, y) + (x^n, y^(n+3))). Compute the minimal free resolution, then decompose the submodule of Syz1 generated by Koszul relations among minimal generators of degrees 2, n, n+2, and 2n+4 modulo m*Syz1. If the dimension is > 1, or if the unique class is not the pair of degrees (n, n+2), Proposition 3.10's opening assertion fails and the non-SBL proof collapses. Also repeat on an ideal with Betti table (3.2) produced by a different construction to test whether λ ≤ 1 is forced by the Betti table alone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.10 opens with an assertion on which the main theorem depends: for an ideal L with Betti table (3.2), λ(L) ≤ 1, the only possible minimal Koszul relation being between the degree n and n+2 generators. The paper states: 'in fact, although the proof is omitted, one can show this must be a minimal Koszul relation.' This bound is then used to conclude that no direct link of L is a complete intersection, and the same λ ≤ 1 is assumed for the related classes (⋆) and (⋆⋆) in Proposition 3.15 and in the final paragraph of Theorem 3.12.ii, where the argument says J′ 'cannot be directly linked to a complete intersection (otherwise, λ(J′) ≥ 3).' If λ(L) were 2, or if the unique minimal Koszul relation had a different pair of degrees, a direct link to a complete intersection would not be excluded, and a sequentially bounded sequence from J to a complete intersection could exist. The promised proof of the λ-bound is not supplied anywhere in the manuscript, so the central non-SBL conclusion is not fully established. This is a fixable gap rather than a demonstrated error, but it is load-bearing: the entire case analysis in Sections 3 and 4 assumes that one can identify all minimal Koszul relations from the Betti table.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sequentially bounded licci (SBL) property for homogeneous grade-3 ideals in polynomial rings. It first shows that the non-minimally licci ideals of Huneke–Migliore–Nagel–Ulrich, and their minimal links, are SBL. It then defines a class of ideals with a fixed Betti table, condition (⋆), and claims in Theorem 3.12 that every such ideal is homogeneously licci but not sequentially bounded licci, giving a negative answer to Chong's Question 1.1. For the three-variable case, Theorem 5.4 constructs, for each ideal satisfying (⋆), a zero-dimensional licci monomial ideal with the same Hilbert function. The proof is a long case analysis based on Ferrand's mapping cone, minimal Koszul relations, and grade-jump lemmas.","tokens_in":36739,"tokens_out":11771,"duration_ms":114959,"significance":"If the main theorem is correct, it provides the first infinite family separating 'homogeneously licci' from 'sequentially bounded licci,' a distinction that is directly relevant to the Eisenbud–Green–Harris conjecture through Chong's Theorem 1.3. The paper's strengths include a systematic use of Ferrand's mapping cone, explicit Betti-table invariants, and a concrete monomial-ideal construction in Section 5 that shows the SBL property is not detected by the Hilbert function. The overall strategy is coherent and does not rely on circular definitions or fitted parameters.","major_comments":[{"comment":"The proof opens with the assertion that every ideal L with Betti table (3.2) satisfies λ(L) ≤ 1, the only possible minimal Koszul relation being between the degree n and n+2 generators, and it explicitly states that the proof is omitted. This assertion is load-bearing: Claim 1 uses it to rule out direct links of L to a complete intersection, Proposition 3.15 and the lemmas in Section 3 assume the analogous λ ≤ 1 for classes (⋆) and (⋆⋆), and the final paragraph of Theorem 3.12.ii concludes that a sequentially bounded double link J′ satisfies λ(J′) ≤ 1 and therefore cannot be directly linked to a complete intersection. If λ(L) were 2, or if the unique minimal Koszul relation had a different pair of degrees, the case analysis would not exclude a sequentially bounded sequence ending at a complete intersection. The manuscript must supply the missing proof, or a reference containing it, before the non-SBL conclusion is established.","section":"Proposition 3.10"},{"comment":"Theorem 3.12 is stated for all N ≥ 3, but from Remark 3.14 onward the proof assumes N = 3. The remark says that for N > 3 one may 'extend to an infinite base field and study the Artinian reduction of the ideals via linear forms,' but no argument is given that the SBL property, the classes (⋆), (⋆⋆), (♦), the existence of the relevant homogeneous complete intersections, or the Betti-table invariants are preserved under this reduction. As written, the proof of Proposition 3.15 and the later propositions covers only the three-variable case. The reduction must be made precise, or the statement of Theorem 3.12 must be restricted to N = 3.","section":"Remark 3.14 and Theorem 3.12"},{"comment":"The text asserts, without proof, that an ideal I with resolution (3.1) contains a complete intersection D of type (2, n+1, 2n+4). This is not an immediate consequence of the displayed Betti table or of the cited grade-jump lemmas, which determine only the minimal type (2, n, 2n+4). The only explicit verification in the paper is for the concrete three-variable ideal in Section 5. Since the construction of the class (⋆) as direct links of the ideals of Theorem 3.4 depends on the existence of D, the paper should either prove this existence in general or clarify that the construction is only carried out for the specific ideals for which D is exhibited.","section":"Section 3.1, paragraph after (3.1)"}],"minor_comments":[{"comment":"There are several typographical errors, including 'Defintion' in Definition 2.7, 'the the main result' in the Introduction, 'Propositon' before Proposition 5.3, and 'impossibe' in Proposition B.4. These should be corrected.","section":"Throughout"},{"comment":"In the proof of part (i), 'set I′ = C : I' should read 'set I′ = C : J', since C is a complete intersection inside J.","section":"Theorem 3.12.i"},{"comment":"Several Betti-table computations are summarized as 'easily checked' or justified by phrases such as 'there are not enough ghost terms.' Given the case-based nature of the proof, these computations should be expanded or placed in an appendix so that the reader can verify them without redoing the entire Ferrand mapping cone analysis.","section":"Proposition 3.10 and Lemma 3.16"},{"comment":"The existence of a complete intersection D of type (2, n+1, 2n+4) inside an arbitrary ideal with resolution (3.1) is asserted rather than proved; even if this is not needed for the main theorem's internal logic, the construction of the family should be unambiguous about which existence statements are being used.","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a commutative algebra journal and the main idea is promising. The most serious issue is the explicitly admitted omission in Proposition 3.10; because the non-SBL conclusion uses λ ≤ 1 as a structural input, the proof is not complete as it stands. The N ≥ 3 reduction in Remark 3.14 is also currently a handwave. I would not recommend rejection: the gaps appear fillable and the explicit Section 5 examples give strong evidence that the main theorem is true for N = 3."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll give it to you straight. The paper does something worthwhile: it constructs a class of grade 3 homogeneous ideals with prescribed Betti tables that are homogeneously licci but not sequentially bounded licci, answering Chong's question in the negative. The companion result—same Hilbert function as a licci monomial ideal in three variables—is a nice addition and gives evidence on Huneke–Ulrich's monomial question.\n\nThe main technique is to use Ferrand's mapping cone and the λ-invariant (minimal Koszul relations) to control what can happen in a sequentially bounded double link. The Betti table analysis is involved and, as far as I can tell, mostly careful. The appendices do real work, and the author is transparent about the limits of what is shown.\n\nNow the soft spots, in proportion.\n\nFirst, the reader's stress-test is on target. Proposition 3.10 begins with the claim that any ideal satisfying (♦) has λ(L) ≤ 1, and says the proof is omitted. This is load-bearing: the whole argument that no direct link of L is a complete intersection, and hence the non-SBL conclusion, depends on it. If λ(L) were 2, a direct link to a CI would not be excluded. The author says 'one can show' but doesn't, anywhere in the manuscript. This is fixable, and probably true, but a referee needs to see the proof.\n\nSecond, the construction of the class (⋆) in Section 3.1 asserts that an ideal with resolution (3.1) contains a complete intersection of type (2, n+1, 2n+4). This is not proved at that point. Section 5 shows it for one explicit ideal, but the general statement is needed. If existence fails for some ideals, the class (⋆) could be smaller than advertised.\n\nThird, the proof of Theorem 3.12.ii after Remark 3.14 assumes N=3, with a remark that larger N reduces to this case by Artinian reduction. That reduction is not shown, and the theorem is stated for all N≥3. The gap is probably patchable, but it should be explicit.\n\nThere are also many 'easily checked' Betti-table computations. They're probably right, but they make verification slow.\n\nWho is this for? Commutative algebraists working in liaison theory, especially those interested in the hierarchy of licci properties and the EGH conjecture. They'll get a genuine new construction and a useful set of tools.\n\nMy recommendation: send it to a serious referee, but a conditional one. The central idea is sound and the answer to Chong's question is likely correct. The missing proof of Proposition 3.10 must be supplied, and the N=3 reduction and CI existence need to be addressed. This is serious work that deserves to be completed, not a desk reject.","headline":"A promising negative answer to Chong's question, but the proof has a self-acknowledged gap in Proposition 3.10 and some loose ends around the N=3 reduction.","tokens_in":37206,"tokens_out":4574,"would_cite":true,"duration_ms":43675,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C40","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"Some licci ideals can never be linked by degree-bounded chains","keywords":["homogeneous liaison","licci ideals","sequentially bounded licci","minimal Koszul relations","Betti tables","Eisenbud-Green-Harris conjecture","monomial ideals","Hilbert functions"],"falsifier":"For n = 4, use the explicit ideal constructed in the proof of Theorem 5.4 that realizes the resolution (3.4), and run a computer algebra search through all homogeneous complete intersections with type bounded above by (2, 5, 12), computing each link with the mapping cone and iterating. The theorem predicts the search never reaches a complete intersection or an ideal of CM type 2; a single sequentially bounded chain that does would refute the main theorem. A second check: compute all minimal Koszul relations of an ideal with Betti table (3.2); any minimal relation other than the one between degrees n and n+2 would disprove the unproved claim inside Proposition 3.10.","tokens_in":36260,"feed_emoji":"🔗","tokens_out":9334,"duration_ms":90000,"temperature":0.7,"pith_summary":"In a polynomial ring, an ideal is homogeneously licci when a finite chain of homogeneous complete-intersection links leads it to a complete intersection. A stronger, numerically controlled version, the sequentially bounded licci (SBL) property, requires the degrees of the regular sequences used for successive links never to increase; a 2015 thesis question asked whether every homogeneously licci ideal is SBL. This paper answers no by constructing, for each n at least 4, a grade-3 ideal with an explicit Betti table that is homogeneously licci but not SBL. In three variables, it also shows each of these ideals has the same Hilbert function as a zero-dimensional monomial licci ideal, so the obstruction cannot be seen from Hilbert functions alone.","feed_headline":"Some licci ideals resist every degree-bounded link","feed_subtitle":"New grade-3 examples answer the question of whether homogeneous licci must be sequentially bounded licci.","key_machinery":"The workhorse is the mapping-cone resolution for a homogeneous link, together with the invariant lambda(I) that counts the minimal Koszul relations among the minimal generators of a grade-3 ideal. When a complete intersection C sits inside an ideal I, the mapping cone writes down a (not necessarily minimal) resolution of the link C : I; the unit entries in the cone differentials occur precisely when generators of C are weak associates of minimal generators of I, or when two generators of C yield a minimal Koszul relation in I. Those unit entries decide which ghost summands trim from the resolution after a double link, which is what lets the paper control the Betti table after repeated or slightly reduced complete-intersection types. The proof packages this control into three Betti-table classes, (diamond), (star), and (star-star), and shows sequentially bounded double links stay inside them.","core_discovery":"The central claim is Theorem 3.12: fix n at least 4 and let J be a grade-3 homogeneous ideal of a polynomial ring whose minimal graded free resolution has the shifts displayed in the paper as (3.4). Then J is homogeneously licci, yet no sequence of sequentially bounded homogeneous links takes J to a complete intersection. The proof is a Betti-table analysis of all possible double links: each sequentially bounded double link of J is shown to be another ideal of type (star), or of the related types (star-star) and (diamond), and all three types have CM type 3 and at most one minimal Koszul relation. Since a direct link to a complete intersection would require enough minimal Koszul relations, none of these ideals can be directly linked to a complete intersection, so the chain can never terminate. The same machinery shows that the previously known non-minimally licci ideals and their minimal links are SBL, which isolates the new class as the genuine obstruction.","pith_inferences":["The paper leaves open whether ideals of type (star-star) are themselves SBL; if any is, that would sharpen exactly which portion of the Betti table, rather than the whole table, forces the obstruction.","The restriction n at least 4 arises because smaller n would let the double-link chain reach a complete intersection of type (1, n-1, 2n+3); testing the same construction for n = 3 with a modified type could reveal whether the phenomenon is unique to large shifts.","The Hilbert-function matching between a non-SBL ideal and a monomial licci ideal suggests that any search for Eisenbud-Green-Harris counterexamples should focus on graded Betti numbers and minimal Koszul relations rather than on Hilbert functions.","A computer algebra search over the n = 4 case could produce a fully explicit certificate of the linkage chain, independent of the paper's case analysis."],"forward_implications":["Question 1.1 is settled in the negative: homogeneously licci does not imply sequentially bounded licci, even for grade-3 ideals in polynomial rings.","The SBL property is not a Hilbert-function invariant: in three variables each counterexample shares its Hilbert function with a zero-dimensional licci monomial ideal (Theorem 5.4).","The Betti table, not the Hilbert function, carries the obstruction: the counterexamples are defined by the resolution (3.4), and every sequentially bounded double link lands in one of the three Betti classes.","The mechanism gives a sufficient condition for failure of SBL: a homogeneously licci ideal of CM type 3 with at most one minimal Koszul relation cannot be sequentially bounded licci, because no direct link to a complete intersection is possible."],"supporting_citations":[{"why":"Supplies the non-minimally licci ideals and the resolution (3.1) that the new class is built from by a direct link.","marker":"[18]"},{"why":"Raises the question of whether every homogeneously licci ideal is sequentially bounded licci, which Theorem 3.12 answers.","marker":"[3]"},{"why":"Introduces the sequentially bounded condition and shows SBL ideals satisfy the Eisenbud-Green-Harris conjecture, motivating the property.","marker":"[4]"},{"why":"Provides the mapping-cone resolution that computes the Betti tables of direct and double links throughout the proof.","marker":"[25]"},{"why":"Defines the invariant lambda of minimal Koszul relations and the trimming criterion for grade-3 links used to control double links.","marker":"[1]"},{"why":"Shows grade-3 Gorenstein ideals are minimally licci, a base case used in the auxiliary SBL results.","marker":"[24]"},{"why":"Gives that grade-3 ideals of CM type 1 are licci, the base case in the SBL reduction.","marker":"[29]"},{"why":"Provides the zero-dimensional licci monomial ideals used as the Hilbert-function comparison in Theorem 5.4.","marker":"[17]"}],"fun_headline_variants":["Homogeneous licci ideals need not be sequentially bounded","Licci but not sequentially bounded: a counterexample","Sequential bounds fail for some licci ideals","Homogeneous licci can avoid sequential boundedness","New examples: licci ideals outside SBL class"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the Betti-table classification of sequential double links: in particular, that an ideal with Betti table (3.2) has at most one minimal Koszul relation (the one between the degree n and n+2 generators), and that an ideal with resolution (3.1) contains a complete intersection of type (2, n+1, 2n+4); both are asserted without proof, and if either enumeration is wrong, a sequentially bounded link sequence might exist.","fun_headline_variants_meta":{"raw":{"variants":["Homogeneous licci ideals need not be sequentially bounded","Licci but not sequentially bounded: a counterexample","Sequential bounds fail for some licci ideals","Homogeneous licci can avoid sequential boundedness","New examples: licci ideals outside SBL class"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1384,"prompt_tokens":877,"completion_tokens":507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":493,"tokens_out":507,"duration_ms":5253,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:38:57.523105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For n = 4, use the explicit ideal constructed in the proof of Theorem 5.4 that realizes the resolution (3.4), and run a computer algebra search through all homogeneous complete intersections with type bounded above by (2, 5, 12), computing each link with the mapping cone and iterating. The theorem predicts the search never reaches a complete intersection or an ideal of CM type 2; a single sequentially bounded chain that does would refute the main theorem. A second check: compute all minimal Koszul relations of an ideal with Betti table (3.2); any minimal relation other than the one between degrees n and n+2 would disprove the unproved claim inside Proposition 3.10.","supporting_citations":[{"cited_title":"Algebra, Geometry and their Interactions (Notre Dame 2 005),","cited_arxiv_id":null,"evidence_quote":"Supplies the non-minimally licci ideals and the resolution (3.1) that the new class is built from by a direct link."},{"cited_title":"Chong, Face vectors and Hilbert functions , Thesis, Cornell Univ., 2015","cited_arxiv_id":null,"evidence_quote":"Raises the question of whether every homogeneously licci ideal is sequentially bounded licci, which Theorem 3.12 answers."},{"cited_title":"Chong, An application of liaison theory to to Eisenbud-Green-Harris c onjecture, J","cited_arxiv_id":null,"evidence_quote":"Introduces the sequentially bounded condition and shows SBL ideals satisfy the Eisenbud-Green-Harris conjecture, motivating the property."},{"cited_title":"Peskine and L","cited_arxiv_id":null,"evidence_quote":"Provides the mapping-cone resolution that computes the Betti tables of direct and double links throughout the proof."},{"cited_title":"Brown, A structure theorem for a class of grade three perfect ideals , J","cited_arxiv_id":null,"evidence_quote":"Defines the invariant lambda of minimal Koszul relations and the trimming criterion for grade-3 links used to control double links."},{"cited_title":"Migliore and U","cited_arxiv_id":null,"evidence_quote":"Shows grade-3 Gorenstein ideals are minimally licci, a base case used in the auxiliary SBL results."},{"cited_title":"Watanabe, A note on gorenstein rings of embedding codimension three , Nagoya Math","cited_arxiv_id":null,"evidence_quote":"Gives that grade-3 ideals of CM type 1 are licci, the base case in the SBL reduction."},{"cited_title":"Huneke and B","cited_arxiv_id":null,"evidence_quote":"Provides the zero-dimensional licci monomial ideals used as the Hilbert-function comparison in Theorem 5.4."}],"review_version":1}