On the existence of maximizing curves of odd degrees

Izabela Le\'sniak; Marek Janasz

REVIEW 4 major objections 5 minor 11 references

On the existence of maximizing curves of odd degrees

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A reduced simply singular plane curve of odd degree whose singularities lie in the stated ADE ranges cannot be maximizing.

desk verdict 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. read the letter →

arxiv 2411.17366 v2 pith:YOH3BMYZ submitted 2024-11-26 math.AG math.AC

Marek Janasz , Izabela Leśniak This is my paper · ORCID

classification math.AGmath.AC MSC 14H5032S2514C20

keywords maximizingcurvesodddegreeplaneADEsingularitieslogcanonicalthresholdfreeTjurinanumberM-curveslinearrangements

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

The argument depends on the accuracy of two black-box facts: the Dimca-Sernesi lower bound $ operatorname{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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

I 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.

This 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.

Recommendation: engage with it seriously, ask for the theorem statement and proof typo to be cleaned up, and accept.

Referee Report

4 major / 5 minor

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.

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 (4)

[Theorem 3.1] 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 3, proof of Theorem 3.1] 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 4, Definitions 4.1 and 4.7] 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.
[Theorem 4.10] 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.

minor comments (5)

[Section 3, proof of Theorem 3.1] 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.'
[Example 4.6] 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 4, introductory paragraph] 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'.
[Introduction] 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.
[Definition 2.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-existence criterion and the M-curve characterizations are derived from cited external theorems and direct computations, not from definitions that force the conclusions.

full rationale

The paper contains no fitted-parameter-then-prediction loop. Theorem 3.1 derives the non-maximality criterion from the Dimca-Sernesi inequality mdr(f) >= alpha_C deg(C) - 2, cited as [5, Theorem 1.2], together with the standard log canonical threshold values for ADE singularities cited to [9]; both are external results and are not supplied by the authors. The M-curve characterizations in Theorems 4.3 and 4.8 are derived from the du Plessis-Wall freeness criterion and the Dimca-Sernesi lower bound forced by a simple elliptic singularity; the defining property is freeness plus a minimal mdr value, while the Tjurina-number formula is an output of the derivation, not an input. The examples in Section 4 are checked with SINGULAR and are consistent with the stated combinatorial counts. There are no load-bearing self-citations and no renaming of a known result as a new derivation. The only notable issues are presentation-level: the phrase 'admitting' in Theorem 3.1 should mean 'admitting only' for the alpha_C-minimum argument to yield the stated criterion, and the proof contains a sign typo where 'not maximizing' should be 'maximizing' in the contradiction setup. These affect clarity and correctness checking, but they do not make any central claim equivalent to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or new physical entities are introduced. The central results are derived from cited theorems, known lct values, and definitions; the new M-curve class is a mathematical definition, not a postulated entity.

assumptions (5)

standard math Dimca-Sernesi bound: mdr(f) ≥ α_C deg(C) − 2 for reduced plane curves with quasi-homogeneous singularities.
Invoked in the proof of Theorem 3.1 as [5, Theorem 1.2]; the whole non-existence criterion is a rearrangement of this inequality.
standard math The du Plessis-Wall freeness criterion and τmax bounds: a curve of degree d with mdr r satisfies τ ≤ τmax(d,r), with equality iff free when r < d/2.
Used in Remark 2.6 and in the characterization of M-curves in Theorems 4.3 and 4.8 as [6, Theorem 4.2].
standard math The listed log canonical threshold values for A_k, D_l, E6, E7, E8.
Quoted in the proof of Theorem 3.1 from [9]; the inequalities k≤4m, l≤2m+1 and the E6/E7/E8 degree thresholds are obtained by algebra from these values.
standard math Simple elliptic singularities X9 and T_{2,3,6} are quasi-homogeneous with log canonical threshold 1/2.
Used in Section 4 to set α_C = 1/2 and derive the minimal possible mdr for M-curves.
domain assumption All curves are reduced plane curves over C with only quasi-homogeneous singularities.
The definitions of maximizing curves and M-curves and the cited theorems are stated in this setting.

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Cite this review

Pith. "Pith review of On the existence of maximizing curves of odd degrees." pith.science (2026). https://pith.science/paper/YOH3BMYZ

@misc{pith2026241117366,
  author       = {Pith},
  title        = {Pith review of: On the existence of maximizing curves of odd degrees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOH3BMYZ}},
  note         = {Machine review of arXiv:2411.17366}
}

abstract

In this paper we provide the non-existence criterion for the so-called maximizing curves of odd degrees. Furthermore, in the light of our criterion, we define a new class of plane curves that generalizes the notion of maximizing curves which we call as $M$-curves.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 9 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

V. I. Arnold, Local normal forms of functions. Invent. Math. 35: 87 – 109 (1976)

1976

[2] [2]

Cuntz, S

M. Cuntz, S. Elia, J.-P. Labb´ e, Congruence normality of simplicial hyperplane arrangements via oriented matroids. Ann. Comb. 26(1): 1 – 85 (2022)

work page 2022

[3] [3]

Dimca, G

A. Dimca, G. Ilardi, P. Pokora, G. Sticlaru, Construction of free curves by adding lines to a given curve. Results Math. 79: Art. Id 11 - 31 pages (2024)

work page 2024

[4] [4]

Dimca and P

A. Dimca and P. Pokora, Maximizing Curves Viewed as Free Curves. Int. Math. Res. Not. IMRN 22: 19156 – 19183 (2023)

work page 2023

[5] [5]

Dimca and E

A. Dimca and E. Sernesi, Syzygies and logarithmic vector ﬁelds alon g plane curves. (Syzygies et champs de vecteurs logarithmiques le long de courbes p lanes.) J. ´Ec. Polytech., Math. 1: 247 – 267 (2014)

work page 2014

[6] [6]

A. A. Du Plessis and C. T. C. Wall, Application of the theory of the dis criminant to highly singular plane curves. Math. Proc. Camb. Philos. Soc. 126(2): 259 – 266 (1999)

work page 1999

[7] [7]

Decker, G.-M

W. Decker, G.-M. Greuel, G. Pﬁster, and H. Sch¨ onemann, Singu - lar 4-1-1 — A computer algebra system for polynomial computations . http://www.singular.uni-kl.de (2018)

work page 2018

[8] [8]

Persson, Horikawa surfaces with maximal Picard numbers

U. Persson, Horikawa surfaces with maximal Picard numbers. Math. Ann. 259: 287 – 312 (1982)

work page 1982