{"id":"c514dc53-c73e-451a-b916-344d52670d09","arxiv_id":"2411.17366","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new criterion shows that odd-degree plane curves whose singularities are all of certain mild types cannot be maximizing, and introduces a broader class called M-curves.","lead":"Mathematicians found a rule that rules out a special kind of rare curve, called a maximizing curve, whenever its singular points come only from a restricted list. The rule explains why such curves are hard to find and introduces a broader, easier-to-construct family called M-curves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's 'admitting' must mean 'admitting only'; as literally written the lct-minimum argument does not follow, and the proof contains a negation typo. With 'only' inserted, the argument is sound.","rationale":"The reader's verdict CONDITIONAL is correct. Their weakest_assumption concerned external black boxes (the Dimca-Sernesi inequality and the lct table), which I believe are standard and correct; the threshold algebra in the proof checks out. The real soft spot is internal: Theorem 3.1's hypothesis is quantified ambiguously and the proof only establishes the 'admitting only' version; without that reading the lct-minimum argument does not go through. I also noticed the 'is not maximizing' typo in the proof. None of this invalidates the intended theorem, so the verdict should remain CONDITIONAL until the statement is tightened and the typo is fixed.","tokens_in":49,"tokens_out":24247,"duration_ms":356811,"concrete_test":"Test the intended quantification by amending Theorem 3.1 to 'admitting only singularities of the listed types' and re-running the proof's threshold check: for every allowed A_k (k <= 4m), D_l (l <= 2m+1), E6, E7, E8, verify lct > (m+1)/(2m+1), hence alpha_C > threshold, contradicting alpha_C <= (m+1)/(2m+1) from mdr(f) = m-1. As a control, compare with Example 3.2: if the first bullet alone were intended as a hypothesis, the degree-9 maximizing curve (seven A1 nodes) would be forbidden; it is only excluded from the theorem's scope because it also has E7, which the stated list excludes for m=4. This makes the 'admitting only' reading the only one consistent with the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 is the paper's central claim, and it states a sufficient condition for non-maximality in terms of allowed singularity types. The proof works by showing that a maximizing curve has alpha_C <= (m+1)/(2m+1), then checking that every listed singularity type has lct strictly above this threshold. This only yields alpha_C > threshold if every singular point of C lies in the listed classes, i.e. if 'admitting' means 'admitting only'. The bullet list in the theorem is not explicit about this: read as 'C has at least one A_k, one D_l, and one E6', the theorem is not what is proved; read as 'C has an A_k' alone, even the degree-9 maximizing example C'' admits A1 nodes. The proof also contains a typographical inversion ('Assume that C ... is not maximizing' where the contradiction setup requires the contrary). These are presentation-level flaws, not failures of the underlying algebra: the lct inequalities for A_k (k <= 4m), D_l (l <= 2m+1), E6/E7/E8 are correct, and the Dimca-Sernesi bound together with freeness gives the stated contradiction. The conditional verdict is appropriate: the theorem should be restated with 'admitting only' and the proof should be corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies maximizing plane curves of odd degree, i.e. reduced simply singular curves of degree 2m+1 with total Tjurina number 3m^2+1. The main result, Theorem 3.1, asserts that no such curve can be maximizing if its singularities are all of the listed ADE types: A_k with k≤4m, D_l with l≤2m+1, E6, and for larger m also E7 and E8. The proof combines the Dimca–Sernesi lower bound mdr(f) ≥ α_C deg(C) − 2 with the freeness property of maximizing curves to force α_C ≤ (m+1)/(2m+1), then checks that each listed singularity type has log canonical threshold strictly above this value. The paper also introduces M-curves, which are free curves with ADE and simple elliptic singularities and prescribed Tjurina numbers, and gives criteria and examples for even and odd degrees.","tokens_in":8577,"tokens_out":9187,"duration_ms":78942,"significance":"If correctly stated, Theorem 3.1 is a useful non-existence criterion that explains the scarcity of maximizing curves of odd degree, and the proposed M-curves generalize the notion with several worked examples. The algebraic core is a sound rearrangement of known results (Dimca–Sernesi, du Plessis–Wall, standard lct tables), the arithmetic of the inequalities is correct, and the examples are checked with SINGULAR. The paper does not rely on fitted parameters or circular reasoning; it is a concise contribution to the study of free and maximizing plane curves.","major_comments":[{"comment":"The statement of Theorem 3.1 is ambiguously quantified. The proof only works if every singular point of C is of one of the listed types, i.e. if 'admitting' is read as 'admitting only'. As written, 'C admits singularities of type A_k, D_l and E6' can be read as 'C has at least one singularity of each of these types', which is not what is proved. The bullet list should be rephrased to say that the singularities of C are only of the types A_k (k≤4m), D_l (l≤2m+1) and E6 (with E7 for m≥5, E8 for m≥8). Without this correction the central argument fails, since a single singular point outside the list could lower α_C below the threshold. The same ambiguity appears in the abstract and the introduction, and should be fixed consistently.","section":"Theorem 3.1"},{"comment":"The sentence 'Assume that C of a fixed degree d = 2m+1 ≥ 7 is not maximizing' is logically inverted. The proof has just established that a maximizing curve must satisfy α_C ≤ (m+1)/(2m+1), so to use the lct computations one should either assume C is maximizing and derive a contradiction, or directly observe that if all singularities have lct > (m+1)/(2m+1), then α_C > (m+1)/(2m+1) and the earlier rephrasing shows C is not maximizing. As written, the assumption 'is not maximizing' is incompatible with the subsequent inequalities, which are about the condition lct_p(C) > (m+1)/(2m+1). This is a typographical/logical inversion in a central proof text, not a substantive algebraic error, but it must be corrected.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The Definitions 4.1 and 4.7 of M-curves use the phrase 'admitting ADE and SE singularities, meaning here that C admits at least one SE singularity of any type'. This does not clarify whether C may have other singularities outside ADE and SE. The proofs of Theorems 4.3 and 4.8 rely on the bound mdr(f) ≥ m−2 (even case) and mdr(f) ≥ m−1 (odd case), which in the paper is derived from α_C = 1/2, valid only if all singularities have lct ≥ 1/2. If non-ADE/non-SE singularities (e.g. an ordinary point of multiplicity at least 5) are allowed, the bound may fail and the characterizations τ(C) = 3m^2−3m+3 and τ(C) = 3m^2+1 need not be equivalent to freeness with the stated mdr. The definitions and theorems should explicitly require that C has only ADE and SE singularities (with at least one SE), so that α_C = 1/2 is justified.","section":"Section 4, Definitions 4.1 and 4.7"},{"comment":"In Theorem 4.10, the derivation passes from the general formula sum_{k≥2}(k−1)t_k = d1 d2 + d1 + d2 to the special formula n2 + 2n3 + 3n4 = d1 d2 + d1 + d2 without stating the assumption that all intersection points have multiplicity at most 4. This is true for line arrangements whose only singularities are A1, D4 and X9, and the examples listed satisfy it, but the theorem is stated for all M-line arrangements. If an M-line arrangement is defined as in the corrected Definitions 4.1/4.7, then no point can have multiplicity ≥5, and the formula follows; this point should be made explicit. Otherwise the general formula with all t_k should be kept, and the simplified formula should be stated as a special case under the multiplicity≤4 assumption.","section":"Theorem 4.10"}],"minor_comments":[{"comment":"Apart from the inversion mentioned above, the line 'Assume that C of a fixed degree d = 2m+1 ≥ 7 is not maximizing' should be replaced by a sentence that clearly introduces the case distinction. For instance, 'Now we determine, for each singularity type, when lct_p(C) > (m+1)/(2m+1); if all singular points are of these types, then α_C > (m+1)/(2m+1) and C is not maximizing.'","section":"Section 3, proof of Theorem 3.1"},{"comment":"The text refers to 'τ(CL16) = 93' and 'mdr(Q16) = 4', but the curve is named CL earlier and the labels CL16 and Q16 are not defined. These should be replaced by CL or a consistent notation.","section":"Example 4.6"},{"comment":"The phrase 'admitting ADE and SE singularities, meaning here that C admits at least one SE singularity' is confusing and should be rewritten, for example as 'whose singularities are only of ADE and simple elliptic type, with at least one simple elliptic singularity'.","section":"Section 4, introductory paragraph"},{"comment":"The abstract and introduction use the same ambiguous 'admitting' wording as Theorem 3.1. Since this wording is the basis of the main claim, these statements should be aligned with the corrected 'admitting only' formulation.","section":"Introduction"},{"comment":"In the displayed definitions of μ_p and τ_p, the notation 'dim C' is ambiguous (the subscript C should be the ring of convergent power series). This is a minor typesetting issue, but it would improve readability.","section":"Definition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short note within the scope of the journal. The central non-existence criterion is defensible after the quantifier in Theorem 3.1 is corrected to 'admitting only'. The M-curves section also needs a precise statement of the allowed singularity types. None of the issues appear to be fatal, but they affect load-bearing statements, so a major revision is appropriate. The authors should also propagate the corrected wording through the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful thing here is a new obstruction for maximizing curves of odd degree: if a simply singular curve of degree 2m+1 admits only A_k (k≤4m), D_l (l≤2m+1), and E6 (plus E7 for m≥5, E8 for m≥8), then it cannot be maximizing. The derivation is a clean rearrangement of the Dimca–Sernesi syzygy bound and known log canonical threshold values, and the arithmetic checks out. That is a legitimate, if modest, new statement: it tells you where not to look for odd-degree maximizing curves, which are rare and hard to construct. The paper also introduces M-curves, a broader class of free curves with ADE and simple elliptic singularities, and gives a very usable combinatorial equation (n2 + 2n3 + 3n4 = ...) for testing whether a line arrangement can be an M-curve. The examples, including the degree-9 maximizing curve and several line arrangements, are concrete and reportedly checked with SINGULAR; I have no reason to doubt them.\n\nThe soft spots are real but mostly presentational. Theorem 3.1 as written says C \"admits\" the listed singularity types, which can be read as \"has at least one of each\". The proof requires that these are the only types present, because it bounds the log canonical threshold by taking the minimum over all singular points. The intended reading is clear from the context and from Example 3.2, but the statement should say \"admitting only\". The proof also contains an apparent typo: \"Assume that C ... is not maximizing\" appears where the computation simply needs the assumption that all singularities belong to the listed classes; as written it is a non sequitur. Theorem 4.8's proof is omitted as \"analogous\" to Theorem 4.3; that is annoying but harmless, since the analogy is exact. These are all fixable in a revision.\n\nI would not call the main theorem deep, but it is correct under the intended reading and it does something the cited papers did not do: it turns known bounds into an explicit non-existence criterion with a simple combinatorial test for arrangements. The bibliography is appropriate and the reliance on Dimca–Sernesi and Paemurru–Viswanathan is standard. There is no fitted data, no circular reasoning, and the computations are reproducible.\n\nThis paper is for people working on free plane curves, line arrangements, and singularity theory. For them it is a useful tool and a source of new examples. I would be comfortable sending it to a serious referee; it deserves a regular peer review rather than a desk rejection, and it only needs minor revision before publication.\n\nRecommendation: engage with it seriously, ask for the theorem statement and proof typo to be cleaned up, and accept.","headline":"Main theorem is correct once 'admitting' is read as 'admitting only'; the paper needs a revised statement and a proof typo fix, but the underlying mathematics is sound and the examples are concrete.","tokens_in":9152,"tokens_out":2571,"would_cite":true,"duration_ms":24383,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H50","32S25","14C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A reduced simply singular plane curve of odd degree whose singularities lie in the stated ADE ranges cannot be maximizing.","keywords":["maximizing curves","odd degree plane curves","ADE singularities","log canonical threshold","free curves","Tjurina number","M-curves","line arrangements"],"falsifier":"A SINGULAR computation that produces a reduced simply singular plane curve of degree 11 with only $A_k$ ($k \\le 44$), $D_l$ ($l \\le 23$), and $E_6,E_7$ singularities, total Tjurina number $\\tau=76$, and $\\operatorname{mdr}(f)=4$ would disprove Theorem 3.1, since such a curve would be free and maximizing.","tokens_in":8123,"feed_emoji":"","tokens_out":6894,"duration_ms":62962,"temperature":0.7,"pith_summary":"This paper establishes a non-existence criterion for maximizing plane curves of odd degree. A curve of degree $2m+1$ with only $A_k$ ($k \\le 4m$), $D_l$ ($l \\le 2m+1$), and $E_6$ singularities, with $E_7$ added when $m \\ge 5$ and $E_8$ when $m \\ge 8$, cannot be maximizing. The proof compares the Arnold exponent of the singularities with the Dimca-Sernesi lower bound on the minimal degree of Jacobian relations. If true, this explains why no maximizing curve of odd degree $\\ge 11$ is known: any such curve would need at least one singularity outside these ranges. The paper also introduces M-curves, free curves with ADE and simple elliptic singularities, and characterizes them by Tjurina numbers and, for line arrangements, by a combinatorial condition.","feed_headline":"Odd-degree maximizing curves need wilder singularities","feed_subtitle":"The paper proves that curves with only A_k (k≤4m), D_l (l≤2m+1), and E6 singularities cannot be maximizing.","key_machinery":"The argument runs on a threshold comparison: the Arnold exponent $\\alpha_C$ is the minimum log canonical threshold among the singular points of $C$, and the Dimca-Sernesi inequality $\\operatorname{mdr}(f) \\ge \\alpha_C \\deg(C)-2$ turns that exponent into a lower bound on the minimal degree of Jacobian relations. For a maximizing odd-degree curve this bound would have to give $m-1$, but whenever $\\alpha_C > (m+1)/(2m+1)$ it gives at least $m$. The paper combines this with the freeness criterion of du Plessis and Wall, which identifies a free curve of degree $d$ by the equality $\\tau(C)=(d-1)^2-r(d-r-1)$ with $r=\\operatorname{mdr}(f)$, and with the log canonical threshold table for ADE singularities.","core_discovery":"The paper's main result is that a reduced simply singular plane curve of odd degree $d=2m+1$ ($m \\ge 3$) whose singular points are only of types $A_k$ with $k \\le 4m$, $D_l$ with $l \\le 2m+1$, and $E_6$, with $E_7$ allowed when $m \\ge 5$ and both $E_7,E_8$ when $m \\ge 8$, cannot be maximizing. A maximizing curve of odd degree would be free with $\\operatorname{mdr}(f)=m-1$ and $\\tau(C)=3m^2+1$; the proof shows that under these singularity restrictions the Arnold exponent $\\alpha_C$ exceeds $(m+1)/(2m+1)$, so the Dimca-Sernesi bound $\\operatorname{mdr}(f) \\ge \\alpha_C(2m+1)-2$ forces $\\operatorname{mdr}(f) \\ge m$, a contradiction. In the same paper the authors define M-curves, free curves with ADE and simple elliptic singularities that attain the minimal possible mdr, and give Tjurina-number and combinatorial characterizations.","pith_inferences":["One consequence the authors leave implicit is that the search for odd-degree maximizing curves reduces to looking for singularities whose log canonical threshold is no larger than $(m+1)/(2m+1)$; the paper's list covers the common ADE types but not, for example, very high index $A_k$ singularities.","The M-curve formalism suggests a systematic construction strategy: because the M-curve condition is fixed by degree alone, any free curve of degree $2m$ with $\\tau=3m^2-3m+3$ is automatically an M-curve, so scanning known arrangements against this single numeric invariant may be productive.","The sharpness at degree 9 hints that exceptional maximizing curves may appear only when some singularity sits exactly on the threshold, generalizing the role played by $E_7$ in the known example.","If M-curves are as abundant as the examples indicate, they give a wider testing ground for freeness and syzygy questions than maximizing curves, whose odd-degree cases now appear largely obstructed."],"forward_implications":["No maximizing curve of odd degree $2m+1 \\ge 7$ can have only the ADE types listed in Theorem 3.1, so any future example must contain a singularity outside those ranges.","A maximizing odd-degree curve must satisfy $\\alpha_C \\le (m+1)/(2m+1)$, meaning at least one singular point has a log canonical threshold at or below that bound.","The known degree-9 maximizing curve, with $A_1$ and $E_7$ singularities, sits exactly at the edge of the criterion, which the paper presents as evidence that the bound is sharp.","The new M-curve class includes the Hesse arrangement, several conic-line arrangements, simplicial arrangements, and the Klein arrangement, and for these curves the total Tjurina number is determined solely by the degree.","For M-line arrangements, the weak combinatorics satisfy $n_2+2n_3+3n_4=m^2+2m-1$ in odd degree and $m^2+m-3$ in even degree, giving a quick numerical test for whether a line arrangement can be an M-curve."],"supporting_citations":[{"why":"Introduces maximizing curves of odd degree and supplies the freeness and mdr properties that the non-existence proof targets.","marker":"[4]"},{"why":"Provides the inequality mdr(f) ≥ α_C deg(C) − 2 that is the engine of the threshold comparison in Theorem 3.1.","marker":"[5]"},{"why":"Supplies the log canonical threshold values for A_k, D_l, E6, E7, and E8 used to derive the singularity restrictions.","marker":"[9]"},{"why":"Gives the freeness criterion and the maximal Tjurina number bound used to prove the M-curve characterizations.","marker":"[6]"},{"why":"Provides the explicit degree-9 maximizing curve that shows the non-existence criterion stops exactly at the known example.","marker":"[3]"},{"why":"Supplies the conic-line and simplicial arrangements that serve as M-curve examples and as tests for the combinatorial condition.","marker":"[10]"}],"fun_headline_variants":["Maximizing odd curves need singularities beyond ADE","No maximizing curves with only mild singularities","M-curves: new class from non-maximizing proof","Odd-degree maximizers require exotic singularities","Maximizing curves exist only with unusual singularities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the accuracy of two black-box facts: the Dimca-Sernesi lower bound $\noperatorname{mdr}(f) \\ge \\alpha_C \\deg(C) - 2$ for reduced plane curves with quasi-homogeneous singularities, and the listed log canonical threshold values for ADE singularities; if either had a different constant or value, the threshold comparison would break.","fun_headline_variants_meta":{"raw":{"variants":["Maximizing odd curves need singularities beyond ADE","No maximizing curves with only mild singularities","M-curves: new class from non-maximizing proof","Odd-degree maximizers require exotic singularities","Maximizing curves exist only with unusual singularities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000391,"raw_usage":{"total_tokens":1997,"prompt_tokens":823,"completion_tokens":1174,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":1099}},"tokens_in":439,"tokens_out":1174,"duration_ms":8833,"temperature":1.0,"reasoning_tokens":1099,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:12:19.161551+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A SINGULAR computation that produces a reduced simply singular plane curve of degree 11 with only $A_k$ ($k \\le 44$), $D_l$ ($l \\le 23$), and $E_6,E_7$ singularities, total Tjurina number $\\tau=76$, and $\\operatorname{mdr}(f)=4$ would disprove Theorem 3.1, since such a curve would be free and maximizing.","supporting_citations":[{"cited_title":"Dimca and P","cited_arxiv_id":null,"evidence_quote":"Introduces maximizing curves of odd degree and supplies the freeness and mdr properties that the non-existence proof targets."},{"cited_title":"Dimca and E","cited_arxiv_id":null,"evidence_quote":"Provides the inequality mdr(f) ≥ α_C deg(C) − 2 that is the engine of the threshold comparison in Theorem 3.1."},{"cited_title":"Paemurru and N","cited_arxiv_id":null,"evidence_quote":"Supplies the log canonical threshold values for A_k, D_l, E6, E7, and E8 used to derive the singularity restrictions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the freeness criterion and the maximal Tjurina number bound used to prove the M-curve characterizations."},{"cited_title":"Dimca, G","cited_arxiv_id":null,"evidence_quote":"Provides the explicit degree-9 maximizing curve that shows the non-existence criterion stops exactly at the known example."}],"review_version":1}