{"id":"afec259b-99a6-44a9-a4c3-51ad9c23d915","arxiv_id":"1908.03648","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under explicit numerical inequalities, the cokernel of a matrix whose maximal minors define a codimension-three ideal has the Weak Lefschetz Property, and its non-Lefschetz locus is concentrated in one degree.","lead":"This paper studies the Weak Lefschetz Property for certain finite-length graded modules over a polynomial ring in three variables, and it defines a non-Lefschetz locus for graded modules. A generalist might read it because it extends a classical property of complete intersections to non-cyclic modules and introduces a module-level analogue of Gorenstein symmetry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.10, on which the omitted proof of Theorem 4.3 explicitly relies, is false: for n=1, (a1; b1,b2,b3)=(0;2,3,5) it predicts strict unimodality, but the Hilbert function is (1,3,5,6,6,5,3,1).","rationale":"I read the paper as aiming to prove WLP for cokernels via semistability and splitting type, with Sections 2-3 supplying numerical control of the Hilbert function. The codimension-3 Buchsbaum-Rim framework is standard and is not my main concern. The missing proof of Theorem 4.3 is the obvious weakness, and the reader already made the verdict CONDITIONAL. My stress-test found a sharper problem: Proposition 3.10, which the author explicitly invokes as the purpose of the unimodality analysis, is false as stated. The n=1 complete intersection of degrees (2,3,5) satisfies assumption (a) but has a flat middle Hilbert function. More structurally, for d even, c = d-3 is odd and symmetry from Proposition 3.9 forces h_{(c-1)/2} = h_{(c+1)/2}, so strict unimodality cannot hold. The proof's error is treating binomial coefficients as polynomials on intervals where they vanish combinatorially, making the differentiation argument invalid. This does not prove Theorem 4.3 false, and the Section 5 results on non-Lefschetz loci appear independent, but the central theorem's proof cannot be the asserted 'mutatis mutandis' adaptation of [9]. I would keep the verdict CONDITIONAL, with the explicit condition that Proposition 3.10 be corrected or replaced and a full proof of Theorem 4.3 be supplied that does not rely on strict unimodality. If those are not supplied, the WLP theorem should be treated as unproved.","tokens_in":17741,"tokens_out":15534,"duration_ms":152921,"concrete_test":"Compute the Hilbert function of M = R/(f1,f2,f3) with deg f = (2,3,5), i.e. n=1, a1=0, b=(2,3,5). The hypotheses of Proposition 3.10(a) hold, since d=10, d'=2, and d'+b2+2 = 7 > 5 = b3. The complete-intersection Hilbert series (1-t^2)(1-t^3)(1-t^5)/(1-t)^3 gives h = (1,3,5,6,6,5,3,1), which is not strictly unimodal. This directly falsifies Proposition 3.10 and shows the omitted argument for Theorem 4.3 cannot be the claimed adaptation of [9].","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing support for Theorem 4.3 is not the Buchsbaum-Rim resolution setup but Proposition 3.10, which the text says is 'precisely the purpose' of the unimodality analysis before invoking the mechanics of [9]. Proposition 3.10 is false as stated. Take n=1, a1=0, b1=2, b2=3, b3=5, with K of characteristic zero. Then M = R/(f1,f2,f3) is a complete intersection, so the ideal of maximal minors has codimension 3. Here d=10 is even, d'=2, and d' + 2 + b2 = 2+2+3 = 7 > 5 = b3, so hypothesis (a) holds. The Hilbert function is h = (1,3,5,6,6,5,3,1), which is symmetric but not strictly unimodal. In fact, whenever d is even, c = d-3 is odd and Proposition 3.9's symmetry forces the two middle values to be equal, so strict unimodality is impossible. The proof's error is that the binomial coefficients in (⋆⋆) are treated as polynomials on intervals where they vanish combinatorially, so differentiation is not justified. Since the proof of Theorem 4.3 is omitted and the reader is told it follows [9] using this strict-unimodality input, the central WLP theorem is not established by the manuscript. The theorem may still be true, and Section 5 may stand independently, but the stated proof strategy collapses.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies finite-length graded modules M = coker(φ) over R = K[x,y,z], where φ: ⊕_{j=1}^{n+2} R(-b_j) → ⊕_{i=1}^n R(-a_i) is a graded map whose ideal of maximal minors has codimension three. It computes the minimal free resolution of M via the Buchsbaum-Rim complex, derives formulas for the Betti degrees, and then analyzes the Hilbert function for symmetry and strict unimodality. The main theorem (Theorem 4.3) asserts that M has the Weak Lefschetz Property under the numerical conditions that a1 = 0 and either (a) d even and d' + 2 + b_{n+1} > b_{n+2}, or (b) d odd and d' + 1 + b_{n+1} > b_{n+2}, where d = Σb_j - Σa_i and d' = Σ(b_i - a_i). The second half defines the non-Lefschetz locus for finite-length graded modules over S = K[x_1,...,x_r], proves a containment result for the ideals defining the locus (Proposition 5.7), and derives that for level modules the non-Lefschetz locus is contained in two degrees, while for symmetrically Gorenstein modules it is supported in a single degree.","tokens_in":18055,"tokens_out":10030,"duration_ms":88615,"significance":"If Theorem 4.3 were established, it would provide a substantial family of non-cyclic finite-length modules over K[x,y,z] with the Weak Lefschetz Property, generalizing the complete intersection case. The non-Lefschetz locus results extend the framework of Boij, Migliore, Miró-Roig, and Nagel from cyclic algebras to arbitrary finite-length graded modules, and the connection with Artin level modules is potentially useful. The explicit Buchsbaum-Rim resolution computations in Section 2 and the symmetric Gorenstein criterion in Section 3 are also of independent interest. However, the central WLP claim rests on an omitted proof and on a false strict-unimodality statement, so these contributions are contingent on a successful revision.","major_comments":[{"comment":"Proposition 3.10 is false as stated. Take n=1, a1=0, b1=2, b2=3, b3=5 over K of characteristic zero. The maximal minors of the 1×3 matrix are the three generators of a complete intersection of degrees 2, 3, and 5 in R=K[x,y,z], so the codimension-three hypothesis holds. Here d=10, d'=2, and d' + b_{n+1} + 2 = 2+3+2 = 7 > 5 = b_{n+2}, so hypothesis (a) holds. The Hilbert function of R/(f1,f2,f3) is (1,3,5,6,6,5,3,1), which is symmetric but not strictly unimodal because the two middle values coincide. The proof's error is that the binomial-coefficient expressions in (⋆⋆) are differentiated on intervals where the binomial coefficients vanish combinatorially, so the polynomial treatment is not justified. Since the proof of Theorem 4.3 explicitly states that Proposition 3.10 supplies the strict-unimodality input, this false proposition undermines the proof of the main theorem.","section":"§3, Proposition 3.10"},{"comment":"Theorem 4.3 is the central claim of the paper but its proof is omitted, with only the sentence 'The proof of Theorem 4.3 works entirely in the same way as the proof ([9], Theorem 2.3), changing only what is necessary, so we omit the details.' This is not adequate for a substantial generalization to non-cyclic modules. Moreover, the omitted proof depends on Proposition 3.10, which is false (see the preceding comment). The author must supply a complete proof, or at least a fully detailed explanation of why the arguments of [9] carry over, and must address how the failure of strict unimodality (e.g., the plateau in the complete intersection example) is handled in the Lefschetz argument.","section":"§4, Theorem 4.3"},{"comment":"Proposition 5.7 is load-bearing for Corollary 5.8 and Proposition 5.11, the main non-Lefschetz locus results, yet its proof is omitted with only 'the proof ... mutatis mutandis'. The statement relates the containment of ideals of maximal minors to a socle condition and to growth of the Hilbert function; this is a nontrivial matrix-ideal statement, and the cyclic case in [2] does not automatically transfer to modules. The details should be written out or a precise reference supplied. Without this proof, Corollary 5.8 and Proposition 5.11 are not established.","section":"§5, Proposition 5.7"}],"minor_comments":[{"comment":"The manuscript contains many typographical errors, including 'Kosuzl' in the abstract, 'Gorenstien' in Proposition 3.9, 'neessary' in Section 2, 'disucssion', 'ony', and a duplicated 'want to know' in Section 5. The paper requires careful proofreading before resubmission.","section":"Throughout"},{"comment":"In formula (⋆), the binomial coefficients such as binom(t+2-a_i,2) should be defined explicitly to be zero when the upper argument is smaller than 2. The proof later differentiates these expressions as polynomials on intervals where the combinatorial values vanish, which is the source of the error; a precise convention and a corrected argument are needed.","section":"§3, Proposition 3.10"},{"comment":"In the proof of Corollary 5.8, the inclusion chain is written for i = 0, ..., j-1, but the initial inclusion I(L_{N,j-1}) ⊆ I(L_{N,j-2}) is not justified by the displayed chain; the indexing should be checked and clarified.","section":"§5, Corollary 5.8"},{"comment":"Reference [5] is cited in the text as 'Peterson Z.', but the arXiv listing for arXiv:1803.10337 gives 'Peterson C.'; the author's name should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The false Proposition 3.10 is a serious issue, but the counterexample does not disprove Theorem 4.3 itself (indeed the example is a complete intersection with WLP). The paper may be salvageable if the author corrects the unimodality statement, supplies full proofs of Theorem 4.3 and Proposition 5.7, and clarifies the plateau case. I would be willing to consider a revised version; however, the omissions and the false proposition currently preclude acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has a genuine new strand—the non-Lefschetz locus for finite-length graded modules and its connection to level modules—but the proof of the main WLP theorem runs through Proposition 3.10, which is false. Both Theorem 4.3 and Proposition 5.7 have their proofs omitted. The paper deserves a serious referee, but not in its current form.\n\nWhat is actually good: Sections 2 and 3 cleanly compute the degrees in the Buchsbaum-Rim resolution of M, and Proposition 3.9, using Kunte's symmetrically Gorenstein condition to get symmetric Hilbert functions, is a nice observation. Section 5 is the real news: Definition 5.2, Corollary 5.8, and Proposition 5.11 generalize the Boij-Migliore-Miro-Roig-Nagel non-Lefschetz locus from Gorenstein algebras to level and symmetrically Gorenstein modules. That is a reasonable step beyond the cyclic case, and it does not depend on the false unimodality claim.\n\nWhere it falls down: Proposition 3.10 is wrong as stated. Take n=1 and (a1; b1, b2, b3) = (0; 2, 3, 5). The map is a complete intersection, so the ideal of maximal minors has codimension three. The hypotheses of part (a) hold: d=10 is even, d'=2, and d' + 2 + b2 = 7 > 5 = b3. But the Hilbert function is (1,3,5,6,6,5,3,1), symmetric and unimodal yet not strictly unimodal. The proof's error is that binomial coefficients are treated as polynomials on intervals where they vanish combinatorially, so differentiation is not justified. Worse, for even d the socle degree is odd, so symmetry already forces the two middle Hilbert function values to be equal; strict unimodality is impossible. Since the proof of Theorem 4.3 is omitted and explicitly leans on Proposition 3.10, the main theorem is not established here. It may still be true—the complete intersection case is known—but it is also weaker than the companion paper [5], so even without this bug it would not be the main contribution.\n\nProposition 5.7 is also stated without proof ('mutatis mutandis'), even though it drives Proposition 5.11 and Corollary 5.12. That omission is less damaging because the proof pattern exists in the cited literature, but it should be written out. The citation pattern elsewhere is honest: the author tells you the restrictions were removed in [5], so the novelty claim is correctly located in Section 5.\n\nBottom line: send this to a referee. The referee should be asked to check Proposition 3.10, require a repaired proof of Theorem 4.3 or its demotion to a remark, and demand full proofs for Section 5. The Section 5 material deserves a home; the current manuscript does not yet deserve acceptance.","headline":"Section 5 on the non-Lefschetz locus for graded modules is the real contribution; the main WLP proof is unsupported because Proposition 3.10 is false.","tokens_in":18639,"tokens_out":6964,"would_cite":false,"duration_ms":65841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","13D40","13E10","14F05","14M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit degree inequalities give non-cyclic modules over $K[x,y,z]$ the Weak Lefschetz Property.","keywords":["Weak Lefschetz Property","graded modules","Buchsbaum-Rim complex","Hilbert function","symmetrically Gorenstein","non-Lefschetz locus","Artinian modules","vector bundles on P^2"],"falsifier":"Exhibit a single matrix $\\phi$ satisfying the hypotheses of Theorem 4.3 (codimension-three maximal minors, positive-degree entries, $a_1=0$) for which multiplication by a general linear form is not of maximal rank in some degree—or, equivalently, for which the vector bundle $E$ of Lemma 4.1 is not semistable; computing $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}}(-1))$ or $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}})$ via the exact sequences $(\\star\\star)$ and $(\\star\\star\\star)$ would settle it. If every such map has semistable $E$, the numerical inequalities are unnecessary; if one map with the inequalities has an unstable $E$, Theorem 4.3 is false.","tokens_in":17492,"feed_emoji":"📐","tokens_out":8377,"duration_ms":74019,"temperature":0.7,"pith_summary":"Over the polynomial ring $R = K[x,y,z]$ with $K$ algebraically closed of characteristic zero, the paper studies modules $M$ that are cokernels of graded maps $\\phi : \\bigoplus_{j=1}^{n+2} R(-b_j) \\to \\bigoplus_{i=1}^{n} R(-a_i)$ whose ideal of maximal minors has codimension three. Its central goal is to show that a large family of these non-cyclic finite-length modules satisfies the Weak Lefschetz Property—the condition that multiplication by a general linear form has maximal rank in every degree. The main theorem gives explicit numerical inequalities on the shifts $a_i, b_j$ under which this holds, and the proof passes through three structurally interesting results: the Hilbert function of $M$ is symmetric, it is strictly unimodal, and the associated rank-two vector bundle on $\\mathbb{P}^2$ is semistable. A second thread defines the non-Lefschetz locus for arbitrary graded finite-length modules and shows that for level modules it is supported in at most two degrees, and for symmetrically Gorenstein modules in a single degree. A sympathetic reader would care because these are the first general WLP results for non-cyclic modules in three variables beyond isolated examples, connecting commutative algebra to vector-bundle stability.","feed_headline":"New family of modules obeys the weak Lefschetz property","feed_subtitle":"Explicit degree inequalities guarantee WLP for non-cyclic cokernels with codimension-three minors.","key_machinery":"The load-bearing machinery is the Buchsbaum–Rim complex, which supplies the minimal free resolution $0 \\to \\bigoplus_{i=1}^n R(-d_i) \\to \\bigoplus_{j=1}^{n+2} R(-c_j) \\to \\bigoplus_{j=1}^{n+2} R(-b_j) \\xrightarrow{\\phi} \\bigoplus_{i=1}^n R(-a_i) \\to M \\to 0$ when the ideal of maximal minors has codimension three; its graded twists compute to $c_j = d - b_j$ and $d_i = d - a_i$. From those shifts, the paper extracts the Hilbert function via a known syzygy formula, and establishes symmetry through Kunte's notion of a Symmetrically Gorenstein module—an Artinian module whose minimal resolution is self-dual with an antisymmetric middle map. Unimodality is then checked interval by interval on the piecewise-quadratic Hilbert function. Finally, the sheafified kernel $E$ of $\\phi$ is a rank-two bundle on $\\mathbb{P}^2$, and the same inequalities guarantee $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}}(-1)) = 0$ (or $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}}) = 0$ in the odd case), which is the semistability condition that Grauert–Mülich converts into a splitting type and hence into the Weak Lefschetz Property.","core_discovery":"On the paper's own terms, the central discovery is Theorem 4.3: if $a_1 = 0$ and either $d$ is even with $d' + 2 + b_{n+1} > b_{n+2}$, or $d$ is odd with $d' + 1 + b_{n+1} > b_{n+2}$, where $d = \\sum_j b_j - \\sum_i a_i$ and $d' = \\sum_{i=1}^n (b_i - a_i)$, then $M = \\operatorname{coker}(\\phi)$ has the Weak Lefschetz Property. The theorem applies to modules that are usually not cyclic (in fact the minimal number of generators is $n$), so it genuinely extends the classical result that codimension-three complete intersections have WLP. The proof establishes along the way that $M$ is Symmetrically Gorenstein in the sense of Kunte when $a_1=0$, giving a symmetric Hilbert function, and that under the same inequalities the Hilbert function is strictly unimodal. It then identifies, via the Buchsbaum–Rim resolution, the first syzygy sheaf $E$ as a rank-two vector bundle on $\\mathbb{P}^2$ with $c_1(E) = -d$; the numerical inequalities are exactly what forces $E$ to be semistable, and Grauert–Mülich then fixes its splitting type, which is the input that yields maximal rank for multiplication by a general linear form.","pith_inferences":["The numerical inequalities are presented as sufficient; a natural next question is whether they are also necessary for semistability of $E$ when the zero pattern of $\\phi$ is generic, which would characterize WLP for this family by degree data alone.","The non-Lefschetz locus results do not require a unimodal Hilbert function: the proof suggests the locus is concentrated around any degree where the Hilbert function changes direction, so a 'peak degree' version of Proposition 5.11 might hold for non-level modules with a symmetric socle.","Example 4.5's circulant-like matrices resemble discretizations of differential operators; one could test WLP numerically for larger $n$ and for rings in more variables, where the Buchsbaum–Rim resolution is not minimal but the non-Lefschetz locus is still defined by minors of linear-form matrices.","The connection between semistability and WLP may transfer to other Artinian modules whose first syzygy sheaf is a vector bundle of rank $>2$; Grauert–Mülich has higher-rank analogues that could fix the splitting type up to a bounded spread."],"forward_implications":["Every cokernel satisfying the inequalities has a symmetric, strictly unimodal Hilbert function, not just the modules that are complete intersections; this gives a large pool of test modules for numerical Hilbert-function questions.","The explicit family in Example 4.5—banded matrices whose nonzero entries are a regular sequence $f_1,f_2,f_3$ of equal degree—yields non-cyclic Artinian modules with WLP for every $q \\ge 3$ and $n > 1$.","The proof template (self-dual resolution, symmetric Hilbert function, semistability of the syzygy bundle, WLP) applies whenever the Buchsbaum–Rim resolution is minimal, so any future improvement of the inequalities in Theorem 4.3 would automatically enlarge the WLP family.","For level modules, the non-Lefschetz locus is contained in the union of two consecutive degrees, and for symmetrically Gorenstein modules it is a single degree; this recovers and extends the known single-degree result for Gorenstein algebras.","Complete intersections in $K[x,y,z]$ obtain a new proof of WLP as the $n=1$ case of Theorem 4.3, with the remaining range covered by a known bound."],"supporting_citations":[{"why":"Supplies the Buchsbaum–Rim complex as the minimal free resolution when the maximal minors have codimension three.","marker":"[3]"},{"why":"The theorem being generalized: complete intersections in three variables have WLP; its proof template and vector-bundle semistability argument are adapted.","marker":"[9]"},{"why":"Defines Symmetrically Gorenstein modules and characterizes them by self-dual resolutions, used to prove symmetry of the Hilbert function.","marker":"[10]"},{"why":"Provides the Hilbert-function formula (⋆) for modules with a three-term resolution, the basis for the unimodality analysis.","marker":"[4]"},{"why":"Gives the Grauert–Mülich theorem and semistability criteria for rank-two bundles on P^2, used to find the splitting type of E.","marker":"[14]"},{"why":"Introduces the non-Lefschetz locus scheme and the containment result that the paper generalizes to modules.","marker":"[2]"},{"why":"Shows the dual of a level module is level, used in proving the two-degree non-Lefschetz locus bound.","marker":"[1]"},{"why":"Bounds the codimension of the ideal of maximal minors in terms of a zero submatrix, used to prove Lemma 2.1.","marker":"[7]"},{"why":"Supplies the missing bound for complete intersections when the inequality fails, completing Corollary 4.4.","marker":"[15]"}],"fun_headline_variants":["Explicit degree inequalities guarantee weak Lefschetz property","New module family obeys weak Lefschetz via simple bounds","Graded modules with weak Lefschetz from codim-three minors","Numerical conditions force weak Lefschetz in graded modules","Extending WLP to non-cyclic modules via degree inequalities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the imported fact that the Buchsbaum–Rim complex is the minimal free resolution of $M$ whenever the ideal of maximal minors has codimension three and every entry of $\\phi$ is zero or a positive-degree form; if that resolution is not minimal, all the degree-shift computations and the Hilbert-function formula collapse.","fun_headline_variants_meta":{"raw":{"variants":["Explicit degree inequalities guarantee weak Lefschetz property","New module family obeys weak Lefschetz via simple bounds","Graded modules with weak Lefschetz from codim-three minors","Numerical conditions force weak Lefschetz in graded modules","Extending WLP to non-cyclic modules via degree inequalities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000311,"raw_usage":{"total_tokens":1776,"prompt_tokens":953,"completion_tokens":823,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":753}},"tokens_in":569,"tokens_out":823,"duration_ms":7711,"temperature":1.0,"reasoning_tokens":753,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:08:30.152485+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a single matrix $\\phi$ satisfying the hypotheses of Theorem 4.3 (codimension-three maximal minors, positive-degree entries, $a_1=0$) for which multiplication by a general linear form is not of maximal rank in some degree—or, equivalently, for which the vector bundle $E$ of Lemma 4.1 is not semistable; computing $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}}(-1))$ or $H^0(\\mathbb{P}^2, E_{\\mathrm{norm}})$ via the exact sequences $(\\star\\star)$ and $(\\star\\star\\star)$ would settle it. If every such map has semistable $E$, the numerical inequalities are unnecessary; if one map with the inequalities has an unstable $E$, Theorem 4.3 is false.","supporting_citations":[{"cited_title":"Springer-Verlag New York, Inc","cited_arxiv_id":null,"evidence_quote":"Supplies the Buchsbaum–Rim complex as the minimal free resolution when the maximal minors have codimension three."},{"cited_title":"Journal of Algebra 262(1), pp","cited_arxiv_id":null,"evidence_quote":"The theorem being generalized: complete intersections in three variables have WLP; its proof template and vector-bundle semistability argument are adapted."},{"cited_title":"Mathematische Nachrichten 284(7), pp","cited_arxiv_id":null,"evidence_quote":"Defines Symmetrically Gorenstein modules and characterizes them by self-dual resolutions, used to prove symmetry of the Hilbert function."},{"cited_title":"Springer Science+Business Media, Inc","cited_arxiv_id":null,"evidence_quote":"Provides the Hilbert-function formula (⋆) for modules with a three-term resolution, the basis for the unimodality analysis."},{"cited_title":"Birkh¨ auser Boston","cited_arxiv_id":null,"evidence_quote":"Gives the Grauert–Mülich theorem and semistability criteria for rank-two bundles on P^2, used to find the splitting type of E."},{"cited_title":"The non- Lefschetz locus, Journal of Algebra 505: 288-320 (2018)","cited_arxiv_id":null,"evidence_quote":"Introduces the non-Lefschetz locus scheme and the containment result that the paper generalizes to modules."},{"cited_title":"361-374, 2000","cited_arxiv_id":null,"evidence_quote":"Shows the dual of a level module is level, used in proving the two-degree non-Lefschetz locus bound."},{"cited_title":"and Merle M., Singularit´ es isol´ ees et sections planes de vari´ et´ es d´eterminantielles","cited_arxiv_id":null,"evidence_quote":"Bounds the codimension of the ideal of maximal minors in terms of a zero submatrix, used to prove Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the missing bound for complete intersections when the inequality fails, completing Corollary 4.4."}],"review_version":1}