{"id":"0b84302a-8ac4-46d8-898b-72097b26e053","arxiv_id":"2507.04902","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every genus g ≤ 12, the paper identifies exactly which Brill–Noether loci are contained in which, yielding a complete relative-position classification.","lead":"This paper determines all inclusions and exclusions among the Brill–Noether loci in the moduli space of curves of genus up to 12. It combines K3 surfaces and Lazarsfeld–Mukai bundles with explicit plane-curve constructions to complete the low-genus containment picture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of the omitted admissible-assignment enumerations is the load-bearing risk; a missed admissible assignment would directly falsify a claimed non-containment.","rationale":"The paper's strongest claim is a finite classification, and the proofs are largely convincing: the explicit constructions are concrete, the use of established results is appropriate, and spot-checks of the numerical bounds (e.g. Proposition 3.29(ii) and Example 2.9) are consistent with the correct stable-sheaf moduli dimension bound. The single place where a single omission would change the theorem is the exhaustive enumeration used to prove non-containments. The reader's weakest assumption identifies exactly this point. I did not find an internal inconsistency that would make the classification false, but the omitted enumerations are load-bearing for the claimed completeness, so the conditional verdict is appropriate. No new objection beyond the reader's concern is warranted.","tokens_in":41252,"tokens_out":35351,"duration_ms":393602,"concrete_test":"Write an independent exhaustive script that, for each lattice Λ^r_{g,d} and each terminal filtration type used in Propositions 3.4(iii), 3.8, 3.15, and 3.29, enumerates all integer (x,y) satisfying inequalities (4)–(5) and Lemma 2.10, computes the resulting c2 lower bound using the stable-factor inequality c2 ≥ ((r−1)/(2r))c1^2 + (r^2−1)/r, and compares the minimum with the asserted thresholds (e.g. ≥4 for g1_3 in Proposition 3.15(i); ≥7 and ≥8 for g2_6 and g2_7 in Proposition 3.29(ii); ≥9 and ≥10 in Proposition 3.29(iv)). Also independently enumerate the 4x4 admissible fillings for Proposition 3.27. If every minimum matches the paper's bound and no c2 value below the stated threshold appears, the non-containment proofs and the classification stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a complete classification of containments, so every claimed non-containment must be certain. Several such claims—Propositions 3.15, 3.29, and also 3.4(iii), 3.8, and the 4x4 tableau check in Proposition 3.27—rest on assertions of exhaustive enumerations of admissible assignments, with no code, tables, or displayed case analysis. This is not cosmetic: the necessary-condition formalism of Section 2 says a g^s_e on C can exist only if some admissible assignment of the terminal filtration satisfies the slope/Gelfand–Tsetlin and c2 constraints. If one admissible assignment was overlooked, it could yield a c2 value below the asserted threshold and give exactly the linear series the proposition rules out. For instance, Proposition 3.29(ii) requires that on Pic = Λ^2_{10,7} the only admissible assignments for a g^2_e have c2-bounds 7 and 8, and Proposition 3.29(iv) requires the same finite list of four destabilizing classes for two different ranks on Λ^3_{10,9}; these lists are stated without derivation. Spot-checking Proposition 3.29(ii) against the correct moduli-dimensional stability bound c2 ≥ ((r−1)/(2r))c1^2 + (r^2−1)/r gives the claimed 7 and 8, so the argument is plausible; but the classification as a whole is conditional on completeness of all such checks.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relative positions (containments and non-containments) of Brill–Noether loci M^r_{g,d} with ρ(g,r,d)<0 in low genus. The main result, Theorem A, asserts that for g≤6 all containments are trivial, while for 7≤g≤12 the complete poset of non-trivial containments is described by the Hasse-like diagrams in Figures 2–5, 7–8. The proof combines classical results (Castelnuovo–Severi, Max Noether, Mori complete intersections), the refined Brill–Noether theory for fixed gonality, chains of elliptic curves, explicit nodal-plane-curve constructions, and an analysis of admissible assignments of terminal filtrations of Lazarsfeld–Mukai bundles on K3 surfaces. The paper also proposes Conjectures A and B on expected containments in large genus and raises questions on irreducibility of Brill–Noether loci.","tokens_in":41523,"tokens_out":6208,"duration_ms":64758,"significance":"If Theorem A is correct, the paper gives the first complete classification of non-trivial containments among Brill–Noether loci in genera 7 through 12, a finite list that should be a benchmark for refined Brill–Noether theory. A notable strength is that many containments are proved by explicit geometric constructions rather than by a black-box computation, and non-containments are often witnessed by concrete curves (smooth plane curves, bielliptic covers, Castelnuovo curves, K3 sections). The K3-admissible-assignment formalism is a useful heuristic and is clearly connected to the Donagi–Morrison circle of ideas. However, the classification is not independently verifiable as written: several load-bearing non-containment proofs rest on assertions of exhaustive finite enumerations for which no tables, code, or detailed case analysis are supplied. The result is plausible and the methods are appropriate, but the missing evidence for these enumerations is a correctness risk that must be addressed before the classification can be accepted.","major_comments":[{"comment":"The non-containment proofs for genus 9 depend on the assertion that certain lists of admissible assignments are complete. For instance, Proposition 3.15(i) states that for Λ^2_{9,6} the only possible admissible assignments of a g^1_e are those with c1(E1) ∈ {2H−4L, H−L, 2L, −H+4L}, and Proposition 3.15(ii) states that for Λ^2_{9,7} all admissible assignments of a g^2_e are of type 1⊂3 and have destabilizing sub-line bundle H−L or L. The text gives no derivation, no table of the checked cases, and no code. Since by §2.2 every terminal filtration yields an admissible assignment, an overlooked admissible assignment could directly produce a g^s_e below the asserted c2 threshold and invalidate the claimed non-containment. This is load-bearing for Theorem A in genus 9.","section":"§3.4, Proposition 3.15"},{"comment":"The same issue occurs in the genus 10 classification. Proposition 3.29 states that for Λ^2_{10,7} the only admissible assignments for g^1_k and g^2_e are as listed, and for Λ^3_{10,9} that the destabilizing classes for g^2_e are exactly {2H−3L, H−L, L, −H+3L}, that the only types for g^3_e are 1⊂4, 2⊂4, and 1⊂2⊂4, and that there are no admissible assignments of type 1⊂2⊂4. These exhaustive statements are asserted with phrases such as “checking all possibilities” and “one can check,” without displaying the enumeration or the inequalities from Equations (4) and (5) that eliminate the remaining cases. An omitted admissible assignment in any of these lists would falsify one of the non-containments (i)–(iv) and hence the genus 10 part of Theorem A.","section":"§3.5, Proposition 3.29"},{"comment":"The non-containment M^2_{10,7} ⊄ M^3_{10,9} is proved by degenerating to a chain of ten elliptic curves and asserting: “There is no admissible filling with the entries 1,…,10 of a 4×4 square with the same torsion conditions.” This is a finite combinatorial check, but no table of attempted fillings, no explicit obstruction, and no reproducible algorithm is provided. The statement is essential to the proof of the proposition, and the reader cannot verify it from the text. Please provide the complete case analysis or a short script with documented output.","section":"§3.5, Proposition 3.27"},{"comment":"The paper would benefit from a precise description of how the “check all possibilities” enumerations are performed. Lemma 2.10 gives finite ranges for the coefficients (x,y) of c1(E/E_i), so an exhaustive search is in principle possible, but the manuscript does not state the algorithm: which terminal filtration types are considered in each genus, how the Gelfand–Tsetlin inequalities together with Equations (4) and (5) are used to prune the search, and how the 'quotient non-negative' and 'quotient slope-positive' conditions of Remark 2.6 are applied. Supplying this information, together with the resulting tables for each Λ^r_{g,d} used in Propositions 3.4(iii), 3.8, 3.15, and 3.29, would make the central classification checkable and would remove the main correctness risk.","section":"§2.3–§2.4, Lemma 2.10 and the enumeration method"}],"minor_comments":[{"comment":"The statement appears as “M^2_{1,6} ⊈ M^1_{7,3}”; the subscript “1,6” should presumably be “7,6”. Please correct this typo.","section":"§3.2, Proposition 3.4(iii)"},{"comment":"The proof says that a basepoint on the g^2_7 sends C into M^2_{12,6} ⊂ M^3_{12,9} “by Proposition 4.14,” but Proposition 4.14 does not state M^2_{12,6} ⊂ M^3_{12,9}. The intended argument is likely via Remark 4.13 together with κ(12,3,9)=3; please fix the reference.","section":"§4.2, Proposition 4.16"},{"comment":"The phrase “4 dimensional linear system of conics through 2 of the nodes” is ambiguous: the vector space of conics through two fixed points has dimension 4, but the resulting projective linear system has dimension 3, which is what produces the g^3_{10}. Please clarify whether “dimension” refers to the projective system or the vector-space dimension.","section":"§4.2, Proposition 4.16"},{"comment":"The diagrams are difficult to read in the current rendering: many labels and arrows overlap, and in places the arrows are difficult to distinguish from the 'trivial containment' arrows that are intentionally omitted. A larger vector graphic or a separate list of the cover relations would improve usability.","section":"Figures 2–5, 7–8"},{"comment":"There are several typographical errors, e.g. “satsify” in Section 1.1 and “Corolalry” in the proof of Proposition 4.14; also the notation for Brill–Noether loci sometimes omits spaces, as in “M2 1,6.” A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a serious and substantial contribution to refined Brill–Noether theory, and I see no reason to doubt the main geometric constructions. My recommendation of major revision is driven by the reproducibility gap in the finite enumerations: the classification theorem is only as strong as those checks, and the manuscript currently asks the reader to take several of them on faith. The author could realistically fix this by adding an appendix with the full admissible-assignment tables for each lattice Λ^r_{g,d} used, or by posting a short verification script. I would not recommend rejection, because the issue is local and fixable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is a full determination of the relative positions of Brill–Noether loci for g ≤ 12, and as far as I can tell it is probably right. The genuinely new part is the systematic treatment of genera 7 through 12, including several new containments (e.g., M^2_{11,7} ⊂ M^3_{11,10} and M^2_{12,7} ⊂ M^3_{12,10}) that earlier work missed, plus a clean set of non-containments. The paper also does something useful conceptually: it carefully separates what is K3-expected from what is actually proved, and it does not try to sell the heuristic as the argument. The constructions for containments are explicit (nodal plane curves, secant projections, Castelnuovo curves), and the non-containments from Pflueger-style gonality bounds or classical Castelnuovo–Severi are on solid ground.\n\nThe soft spot is exactly where the reader put it: several non-containments rely on “one can check” exhaustive enumerations of admissible assignments, with no code, no tables, and no displayed case analysis. Propositions 3.15, 3.29, and parts of 3.4 and 3.27 are the load-bearing ones. If even one admissible assignment is missed, a claimed non-containment could collapse, and because the theorem is a complete classification, the risk is not cosmetic. I spot-checked Proposition 3.29(ii) against the moduli-dimension stability bound, and the numbers 7 and 8 come out correctly, which makes me think the author did run the checks properly. But I cannot verify completeness from the printed text, and the paper would be much easier to trust if the enumerations were supplied as ancillary tables or a short script.\n\nThere are also minor presentation issues: some figures are not rendered in the text version, and there are typos (e.g., “M^2_{1,6}” for the genus-7 non-containment). These don't affect the mathematics. The circularity worry raised in the report is a non-issue: K3 expectations are used only as heuristics, and the actual containments are proven by explicit constructions.\n\nWho is this for? Anyone working on Brill–Noether loci, moduli of curves, or K3-surface methods, especially those interested in low-genus classification and conjectures in higher genus. It deserves a serious referee: the result is substantial, the strategy is sound, and the main gap is fixable by adding the missing enumerations to an appendix or a repository. I would accept it for peer review, with the request that the author make the enumerations independently checkable.","headline":"A likely-correct complete classification of Brill–Noether containments through genus 12, with the main gap being unshown exhaustive enumerations that are load-bearing for the non-containments.","tokens_in":42087,"tokens_out":1180,"would_cite":true,"duration_ms":16258,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H51","14J28","14H60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every genus up to 12, the complete containment pattern of Brill–Noether loci is now known, with explicit witnesses for every non-containment.","keywords":["Brill–Noether loci","relative positions of moduli subvarieties","K3 surfaces","Lazarsfeld–Mukai bundles","admissible assignments","gonality stratification","linear series on curves","moduli of curves"],"falsifier":"Re-run the finite enumeration of quotient non-negative, quotient slope-positive admissible assignments for Lazarsfeld–Mukai bundles on K3 surfaces with Picard lattice $\\Lambda^r_{g,d}$ for each case in Propositions 3.15, 3.29, and 4.14; for instance, in genus 9 with $\\Lambda^2_{9,7}$, an admissible assignment with destabilizing subsheaf other than $H-L$ or $L$ giving $c_2(E_{C,g^2_e}) \\leq 6$ would disprove the claimed non-containment $M^2_{9,7} \\not\\subset M^2_{9,6}$.","tokens_in":41015,"feed_emoji":"🗺️","tokens_out":7952,"duration_ms":81455,"temperature":0.7,"pith_summary":"The paper aims to determine, for each genus $g \\leq 12$, exactly which Brill–Noether loci $M^r_{g,d}$ (moduli of curves carrying a linear series of degree $d$ and dimension $r$) contain which others, when the Brill–Noether number $\\rho(g,r,d)$ is negative. It proves that in genus $3 \\leq g \\leq 6$ only trivial containments occur, and it gives complete figures for the non-trivial containments in genera 7 through 12. This matters because Brill–Noether loci are usually reducible, so the coarse structure of their inclusions is the first answer refined Brill–Noether theory can give before a component-by-component description exists. The same K3-surface machinery also yields explicit conjectural thresholds in large genus.","feed_headline":"Every Brill–Noether inclusion identified through genus 12","feed_subtitle":"Complete figures show which curve families force which linear series, settling the low-genus containment order.","key_machinery":"The load-bearing object is the Lazarsfeld–Mukai bundle $E_{C,(A,V)}$ attached to a linear series on a smooth curve $C$ inside a K3 surface with Picard lattice $\\mathbb{Z}[H] \\oplus \\mathbb{Z}[L]$; when $\\rho < 0$ this bundle is non-simple. Its possible destabilizing subsheaves are organized into terminal filtrations, and the numerical data of such a filtration form an admissible assignment, a Gelfand–Tsetlin pattern of slopes together with the conditions that each quotient is nef and slope-positive. An admissible assignment gives a lower bound on $c_2(E_{C,g^s_e})$, hence a non-containment $M^r_{g,d} \\not\\subset M^s_{g,e}$; conversely, when a bundle realizing an assignment can be constructed, one obtains a containment. The secondary machinery consists of the gonality stratification function $\\kappa(g,r,d)$ from the refined Brill–Noether theory of $k$-gonal curves, secant expected containments via the determinantal cycles $V^{r-s}_{d-e}(g^r_d)$, and classical plane-curve and Castelnuovo-curve constructions.","core_discovery":"The central claim, Theorem A, is that the relative positions of Brill–Noether loci in genus $g \\leq 6$ are governed solely by the trivial containments obtained by adding base points and subtracting non-base points, while for $7 \\leq g \\leq 12$ the complete set of non-trivial containments is displayed in Figures 2–5, 7, and 8. In particular, every containment among loci with $\\rho < 0$ and $d \\leq g-1$ is a sequence of the displayed covers, trivial containments, or Serre duality, and every non-containment is witnessed by a specific family of curves. The argument combines the refined Brill–Noether theory for curves of fixed gonality, classical bounds (Castelnuovo, Castelnuovo–Severi, Martens, Coppens), explicit constructions with nodal plane curves and Castelnuovo curves, and an analysis of unstable Lazarsfeld–Mukai bundles on K3 surfaces. On K3 surfaces, the existence of a $g^s_e$ forces an admissible assignment for a terminal filtration of the associated Lazarsfeld–Mukai bundle, and the absence of such assignments yields non-containments; when assignments exist, the bundles can sometimes be built as direct sums, yielding containments such as $M^2_{11,7} \\subset M^3_{11,10}$.","pith_inferences":["The same admissible-assignment calculus should be run in genus 13–15; even a partial run could show where the low-genus pattern breaks and how quickly the Conjecture A threshold becomes accurate.","The equalities in low Clifford index, e.g. $M^{e-1}_{g,2e}$ all equal for $g \\geq 11, e \\geq 4$ and $M^4_{g,11}=M^4_{g,10}=M^3_{g,8}$ for $g \\geq 13$, suggest a general Clifford-coalescence principle: loci of a fixed small Clifford index eventually coincide, a principle that could be tested by computing $\\kappa$ in higher genus.","The explicit constructions through linear systems on nodal plane curves indicate that many K3-expected containments can be realized without K3 surfaces; this raises the possibility of a purely classical proof of Conjecture A's containment direction.","If the hand-checked enumerations are replaced by a certified computer enumeration, the non-containment half of Theorem A would be independently verifiable and the method could be extended mechanically to bounded genus."],"forward_implications":["For $g \\leq 12$, the inclusion partial order among Brill–Noether loci with $\\rho < 0$ is finite and explicit: any containment question reduces to reading the appropriate figure together with trivial containments and Serre duality.","Families realizing the non-containments (smooth plane sextics, bielliptic covers, trigonal curves, Castelnuovo curves, nodal plane curves) certify that the omissions in the figures are genuine.","The realized K3-expected containments, such as $M^2_{11,7} \\subset M^3_{11,10}$ and $M^2_{12,7} \\subset M^3_{12,10}$, provide explicit decompositions of Lazarsfeld–Mukai bundles as sums of line bundles and smaller Lazarsfeld–Mukai bundles.","Conjecture A gives a concrete numerical threshold, $e \\geq d - 2r + s + \\frac{g-d+r+1}{2} + \\frac{(s-2)(r-1)-1}{s-1}$, predicting exactly when $M^r_{g,d} \\subset M^s_{g,e}$ in large genus, and Conjecture B predicts that no containments into $M^2_{g,e}$ occur outside the secant range for $r \\geq 3$."],"supporting_citations":[{"why":"Supplies the Lazarsfeld–Mukai bundle construction and the non-simplicity criterion when the Brill–Noether number is negative.","marker":"[43]"},{"why":"Establishes the K3-surface admissible-assignment method for distinguishing Brill–Noether loci, which the low-genus non-containment proofs rely on.","marker":"[6]"},{"why":"Provides the gonality function $\\kappa(g,r,d)$ and the criterion $M^r_{g,d} \\not\\subset M^s_{g,e}$ when $\\kappa$ decreases, used throughout.","marker":"[7]"},{"why":"Proves the refined Brill–Noether theorem for curves of fixed gonality, giving the dimension and existence statements behind the trigonal containments.","marker":"[36]"},{"why":"Contributes stability results and terminal-filtration bounds for rank-3 Lazarsfeld–Mukai bundles, giving non-containments of the form $M^r \\not\\subset M^1$ and $M^r \\not\\subset M^2$.","marker":"[46]"},{"why":"Introduces generalized Lazarsfeld–Mukai bundles and the $c_2$ bounds from terminal filtrations, used to rule out admissible assignments.","marker":"[47]"},{"why":"Gives the refined Brill–Noether theory for curves on Hirzebruch surfaces, used for plane-curve and scroll containments in genera 10 and 12.","marker":"[41]"},{"why":"Supplies irreducibility statements and chains-of-elliptic-curves limits for $M^2_{10,7}$ and related loci, plus the reducible examples motivating the component question.","marker":"[30]"},{"why":"Provides the Castelnuovo-curve bounds on linear series used in the genus-12 non-containments.","marker":"[11]"},{"why":"Supplies classical formulas for secant lines and Castelnuovo bounds, used in the genus-11 4-secant argument and plane-curve arguments.","marker":"[4]"}],"fun_headline_variants":["Complete Brill–Noether containment map for genus ≤12","All Brill–Noether locus relations settled up to genus 12","Low-genus Brill–Noether lattice fully worked out","Genus-12-and-below Brill–Noether inclusions fully listed","Relative positions of Brill–Noether loci nailed for low genus"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exhaustive enumerations of admissible assignments in the non-containment proofs are assumed complete; if a single admissible assignment was missed, one of the claimed non-containments could fail.","fun_headline_variants_meta":{"raw":{"variants":["Complete Brill–Noether containment map for genus ≤12","All Brill–Noether locus relations settled up to genus 12","Low-genus Brill–Noether lattice fully worked out","Genus-12-and-below Brill–Noether inclusions fully listed","Relative positions of Brill–Noether loci nailed for low genus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1516,"prompt_tokens":949,"completion_tokens":567,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":481}},"tokens_in":565,"tokens_out":567,"duration_ms":5951,"temperature":1.0,"reasoning_tokens":481,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:36:29.389932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the finite enumeration of quotient non-negative, quotient slope-positive admissible assignments for Lazarsfeld–Mukai bundles on K3 surfaces with Picard lattice $\\Lambda^r_{g,d}$ for each case in Propositions 3.15, 3.29, and 4.14; for instance, in genus 9 with $\\Lambda^2_{9,7}$, an admissible assignment with destabilizing subsheaf other than $H-L$ or $L$ giving $c_2(E_{C,g^2_e}) \\leq 6$ would disprove the claimed non-containment $M^2_{9,7} \\not\\subset M^2_{9,6}$.","supporting_citations":[{"cited_title":"Brill–Noether–Petri without dege nerations","cited_arxiv_id":null,"evidence_quote":"Supplies the Lazarsfeld–Mukai bundle construction and the non-simplicity criterion when the Brill–Noether number is negative."},{"cited_title":"Distinguishing Brill-Noether loci","cited_arxiv_id":"2406.19993","evidence_quote":"Establishes the K3-surface admissible-assignment method for distinguishing Brill–Noether loci, which the low-genus non-containment proofs rely on."},{"cited_title":"Maximal Brill--Noether loci via the gonality stratification","cited_arxiv_id":"2310.09954","evidence_quote":"Provides the gonality function $\\kappa(g,r,d)$ and the criterion $M^r_{g,d} \\not\\subset M^s_{g,e}$ when $\\kappa$ decreases, used throughout."},{"cited_title":"Brill–Noether the ory for curves of a fixed gonality","cited_arxiv_id":null,"evidence_quote":"Proves the refined Brill–Noether theorem for curves of fixed gonality, giving the dimension and existence statements behind the trigonal containments."},{"cited_title":"Stability of rank-3 Lazarsf eld–Mukai bundles on K3 surfaces","cited_arxiv_id":null,"evidence_quote":"Contributes stability results and terminal-filtration bounds for rank-3 Lazarsfeld–Mukai bundles, giving non-containments of the form $M^r \\not\\subset M^1$ and $M^r \\not\\subset M^2$."},{"cited_title":"Generalized Lazarsfeld–Mu kai bundles and a conjecture of Donagi and Morrison","cited_arxiv_id":null,"evidence_quote":"Introduces generalized Lazarsfeld–Mukai bundles and the $c_2$ bounds from terminal filtrations, used to rule out admissible assignments."},{"cited_title":"Some reducible and irreducible Brill–Noether loci, 2025","cited_arxiv_id":null,"evidence_quote":"Supplies irreducibility statements and chains-of-elliptic-curves limits for $M^2_{10,7}$ and related loci, plus the reducible examples motivating the component question."},{"cited_title":"Towards a Halphen th eory of linear series on curves","cited_arxiv_id":null,"evidence_quote":"Provides the Castelnuovo-curve bounds on linear series used in the genus-12 non-containments."},{"cited_title":"Griffith s, and Joe Harris","cited_arxiv_id":null,"evidence_quote":"Supplies classical formulas for secant lines and Castelnuovo bounds, used in the genus-11 4-secant argument and plane-curve arguments."}],"review_version":1}