The Hesse pencil of plane curves and osculating conics

Ewelina Nawara

REVIEW 3 major objections 5 minor 16 references

The Hesse pencil of plane curves and osculating conics

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every smooth Hesse cubic, the paper writes down explicit equations of the 27 osculating conics and shows which products with the cubic are free or nearly free.

desk verdict Solid explicit computation of osculating conics for the Hesse pencil; the freeness classification is plausible but under-proved for the new parameter values. read the letter →

arxiv 2506.04662 v1 pith:5N5KPFPB submitted 2025-06-05 math.AG

Ewelina Nawara This is my paper

classification math.AG MSC 14H5053A1514N05

keywords osculatingconicssextacticpointsHessepencilsecondHessianfreecurvesnearlyJacobiansyzygiesplanecubics

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 derives, for every smooth curve in the Hesse pencil $F=x^3+y^3+z^3+txyz=0$ with $t^3+27\neq 0$, the explicit coordinates of its 27 sextactic points and the explicit equation of the osculating conic at each point. The central formula, equation (14), gives the conic at the point $(1,1,z_i)$, and all 27 conics follow from a symmetry group action. The paper then checks that products of the cubic with these conics produce free and nearly free curves: for the equianharmonic members $t\in\{0,6,6\varepsilon^2,6\varepsilon^4\}$, single conics give nearly free curves with exponents $(2,3,3)$, certain pairs and triples give free curves, and the remaining pairs are nearly free. For $t\in\{-5,-3(1-\sqrt3),-3(1+\sqrt3)\}$, a computer algebra computation reports nearly free products for single and pair conics and exponents $(5,5,5)$ for triple products. A sympathetic reader cares because explicit sextactic geometry is rare, and these formulas turn a classical invariant-theoretic construction into concrete families of curves with controlled syzygies.

What carries the argument

The load-bearing object is the explicit conic equation (14) and its derivation through a sequence of projective changes of coordinates $A,B,C,D$ that put the cubic into the normal form $F_1(x,y,1)=x+y^2+fx^3+gx^2y+hxy^2+iy^3$, where the corrected osculating-conic formula from Lemma 2.24 of [2] is $-(i^2+h)x^2-ixy+y^2+x$. The factorization of the second Hessian, $H_2(F)=(x^3-y^3)(x^3-z^3)(y^3-z^3)$, locates the 27 sextactic points, and the group generated by $g_0,g_1,g_2$ reduces the computation from 27 conics to three representatives $P_1,P_4,P_7$. The freeness and near-freeness statements are then established by computing minimal Jacobian syzygies and their degrees.

What would settle it

Run the supplied computer algebra scripts for $t=6\varepsilon^2$ and $t=6\varepsilon^4$, compute the minimal Jacobian syzygy degrees of $F\cdot C_i\cdot C_j$ for pairs inside and outside the sets $G_\alpha$, and check whether they are exactly $(3,3)$ and $(3,4,4)$; any deviation would disprove Theorem 4.3. For Theorem 4.4, an independent symbolic derivation of the exponents $(2,3,3)$, $(3,4,4)$, and $(5,5,5)$ for $t=-3(1\pm\sqrt3)$ would confirm what the paper currently reports from computation alone.

Watch

Extended reading notes

Core claim

For $F=x^3+y^3+z^3+txyz$ with $t^3+27\neq 0$, the second Hessian equals $(x^3-y^3)(x^3-z^3)(y^3-z^3)$; intersecting it with $F=0$ and $H\neq 0$ yields the 27 sextactic points. Using Cayley's osculating-conic formula as corrected in equation (7), and then a projective normalization following Lemma 2.24 of [2], the conic at $(1,1,z_i)$ reduces to the compact equation (14): $z_i^2(9-6tz_i)(x^2+y^2)+z_i(15tz_i+18)(xz+yz)-z_i^2(t^2z_i^2+18)xy+(z_i^2t^2-18tz_i-36)z^2=0$. The paper's main freeness classification, Theorem 4.3, states that for $t\in\{0,6,6\varepsilon^2,6\varepsilon^4\}$ every curve $F\cdot C_i$ is nearly free with exponents $(2,3,3)$; the 27 conics split into nine disjoint triples $G_\alpha$ such that inside a triple the product $F\cdot C_i\cdot C_j\cdot C_k$ is free with exponents $(3,5)$, inside pairs are free with exponents $(3,3)$, and outside pairs are nearly free with exponents $(3,4,4)$. Theorem 4.4 reports, from a computer algebra check, that for $t\in\{-5,-3(1-\sqrt3),-3(1+\sqrt3)\}$ single products are nearly free with $(2,3,3)$, pairs are nearly free with $(3,4,4)$, and all triple products have exponents $(5,5,5)$.

Load-bearing premise

The classification for $t\in\{6,6\varepsilon^2,6\varepsilon^4\}$ is not proved in the text but asserted to follow by arguments analogous to the $t=0$ case, and for $t\in\{-5,-3(1\pm\sqrt3)\}$ the exponents rest on a computer algebra computation without an independent mathematical argument.

Editorial extensions

If this is right

Every future computation involving the second Hessian or osculating conics of a Hesse cubic can start from the explicit formulas in equation (14) rather than recomputing them.
The equianharmonic members of the Hesse pencil, including the Fermat cubic, behave uniformly: adding one osculating conic gives a nearly free curve, and the nine triples of conics give free curves with exponents $(3,5)$.
The decomposition of the 27 sextactic points into nine disjoint triples $G_\alpha$ is a concrete combinatorial structure carried by the geometry, not just a computational artifact.
Harmonic cubics do not share the full freeness behavior: their triple products have exponents $(5,5,5)$, which is a genuinely different syzygy pattern.
The supplied computer algebra programs allow readers to verify every claimed exponent for all listed parameter values without reimplementing the geometry.

Reading between the lines

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

The explicit conic equations make it natural to test products of the cubic with four or more conics, extrapolating the freeness pattern beyond what the paper states.
Because the second Hessian factors so cleanly, the 27 sextactic points likely sit inside a classical configuration related to the Hesse arrangement, though the paper does not explore that connection.
The computer-only part of Theorem 4.4 suggests a concrete open task: derive the exponents $(5,5,5)$ for harmonic cubics by an independent symbolic argument, which would clarify why equianharmonic and harmonic members differ.
One can re-run the same osculating-conic construction on other members of the pencil outside the listed special values to see whether the near-freeness of $F\cdot C_i$ is a general phenomenon or is special to equianharmonic and the three named curves.

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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 real value here is the explicit, self-contained computation: equation (14) for the osculating conics at the sextactic points, the corrected sign in the conic formula from [2, Lemma 2.24], and the clean derivation of the second Hessian reproducing Cayley's classical result. The Singular code on GitHub is a genuine asset, and the paper makes the classical machinery concrete for a family that mostly appeared in abstract form before. I would point anyone working on sextactic points or osculating conics to this paper, and the formulas themselves look credible.

The soft spot is the freeness classification in Section 4. Theorem 4.3 asserts a structured partition of the 27 conics and specific exponents for t in {6, 6ε^2, 6ε^4}, but the proof only says "the remaining cases follow by analogous arguments" after citing [8] for t=0. That is not a derivation, even if the Hesse pencil's symmetry makes the claim plausible. The group action is fixed for all t, so the orbit structure should be transferable, but the partition G_α depends on t through the z_i(t), and that dependency is not addressed. Similarly, Theorem 4.4 is a classification statement backed only by Singular output. That is acceptable when the code is available and the claim is computational, but the theorem reads as a mathematical statement, not an experimental report, so the reader should be told exactly what was checked and what remains conjectural.

The stress-test note worries about sign or scaling errors in (14) breaking the freeness results. I did not find evidence of that, and the detailed derivation plus the code make an unnoticed algebraic error less likely. The lack of proof for the new cases is the real issue, not a suspected error. The derivation of the osculating conics themselves is the paper's main contribution, and it seems sound.

Bottom line: this deserves a serious referee. The explicit formulas and the computational infrastructure are worth publishing, but the authors should either fill in the missing arguments for Theorem 4.3 or explicitly rephrase it as a computational conjecture with the verification made precise. As it stands, I would accept the paper with major revision, not a desk reject.

Referee Report

3 major / 5 minor

Summary. The paper studies the Hesse pencil of plane cubics F_t = x^3 + y^3 + z^3 + txyz. It computes the second Hessian, gives explicit coordinates for the 27 sextactic points, and derives explicit equations for the associated osculating conics, with the main formula in equation (14) for the conic at (1,1,z_i(t)) and the remaining conics obtained by a group action. In Section 4 the author uses these conics to form products F*C_i, F*C_i*C_j, and F*C_i*C_j*C_k and states classification results for their freeness: Theorem 4.3 covers t in {0,6,6epsilon^2,6epsilon^4}, and Theorem 4.4 reports exponents for t in {-5,-3(1-sqrt(3)),-3(1+sqrt(3))} based on Singular computations. The paper also provides Singular programs on GitHub.

Significance. If fully established, the paper would give the first detailed explicit treatment of osculating conics and sextactic points for non-Fermat members of the Hesse pencil, together with new families of free and nearly free curves extending the Fermat-case results of [8]. A definite strength is that the derivation is grounded in classical theorems (Cayley, Maugesten-Moe, Balay-Wilson-Brysiewicz) and the main formulas are supported by publicly available Singular scripts. However, the two classification theorems rest in part on omitted arguments or purely computational assertions, so the significance is conditional on completing those verifications.

major comments (3)

[§4, Theorem 4.3] The proof of Theorem 4.3 is incomplete. It cites [8] for t=0 and then says 'The remaining cases follow by analogous arguments and are thus omitted', but the theorem covers t=6, 6epsilon^2, and 6epsilon^4. The points P_i and the conics C_i depend on t through z_i(t), defined by the t-dependent equation (9), and the partition G_alpha into nine triples is asserted to exist separately for each parameter value. None of the freeness or near-freeness claims for these three values is derived in the text, so the central new classification is not supported as written. Please supply the missing arguments or, at minimum, a complete machine-checkable verification that includes the explicit G_alpha for each t.
[§4, Theorem 4.4] Theorem 4.4 is a classification statement whose entire proof is the sentence 'As verified using the Singular software' plus a pointer to a GitHub repository. The paper does not state the algorithm used by the scripts, the exact equations fed into them, or the output tables for the 27 conics and their products. Since the exponents (2,3,3), (3,4,4), and (5,5,5) are the theorem's content, this is load-bearing. Please include the computational details in the paper or give an independent mathematical argument, and report representative outputs (for example, syzygy degrees for one or two indices) so the result can be checked without re-running an external script.
[§3, Eq. (14)] The formula for the osculating conic at (1,1,z_i) is obtained through transformations A, B, C, D and a correction of [2, Lemma 2.24] whose sign error is announced but not demonstrated. Because all later freeness claims use equation (14), I ask for a direct consistency check in the text: verify that (14) vanishes to order at least six at (1,1,z_i) on F_t and that the remaining 26 conics are exactly the transforms of (14) under the group G generated by g0, g1, g2. This would rule out sign or scaling errors before they propagate to Theorems 4.3 and 4.4.

minor comments (5)

[Abstract] The phrase 'extending recent findings the freeness of curves' is missing a preposition; it should read 'extending recent findings on the freeness of curves'.
[§3, after Eq. (10)] The sentence 'perform further reduction of coefficients in (11)' should refer to equation (10), since the reduction described takes place before equation (11) is displayed.
[§3, before Eq. (7)] There is a typo: 'Calyley' should be 'Cayley'.
[§4, after Theorem 4.4] The sentence 'The reader's convenience, we provide Singular programs' is missing the preposition 'For'; it should read 'For the reader's convenience, we provide Singular programs'.
[§3, matrices A, B, C, D] The matrix D contains the expression z_i*sqrt(6 - t z_i), and its inverse appears in equation (13). Please specify the branch of the square root used for each z_i, or note explicitly that the final equation (14) is independent of that choice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the osculating-conic formulas and freeness constructions are derived from stated classical theorems and explicit computations, not from the target conclusions; the omitted and computer-assisted cases are completeness issues, not circularity.

full rationale

The paper's derivation chain is not circular. The second Hessian is computed from Cayley's classical formula as corrected by Maugesten and Moe [11], and the sextactic points are obtained by solving the explicit system H2(F)=0, F=0, H≠0. The osculating conics are computed in two independent ways: first by applying Theorem 2.1 (Cayley's osculating-conic formula) with the explicitly computed Λ(P), and second via the projective-transformation method of Balay-Wilson and Brysiewicz [2, Lemma 2.24]. Both routes lead to equation (14), whose derivation does not assume the freeness results. The freeness statements in Theorem 4.3 use the case t=0 from [8], an external reference by different authors, and assert the remaining equianharmonic cases 'follow by analogous arguments'; this is an omitted proof, not a circular reduction. Theorem 4.4 is supported by Singular computations made available in [16], which is an external computational check rather than a fitting of parameters to the claimed exponents. No parameter is fitted to the target data: t is a fixed member of the Hesse pencil, and the 27 conics are indexed by explicit z_i(t) solutions. The manuscript itself flags the omitted 'analogous' derivations and invites verification via the Singular programs, so those gaps are correctly classified as proof incompleteness or verification risk, not as circularity.

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

The central computations rely on classical formulas for the second Hessian and osculating conics taken from Cayley and Maugesten-Moe, and on a normal-form lemma from Balay-Wilson-Brysiewicz. The freeness results depend on the t=0 case established in [8] and on computer algebra verification for the new parameter values. No free parameters or invented entities are introduced; the parameter t is part of the family definition.

assumptions (5)

standard math Theorem 2.1 (Cayley's osculating conic theorem)
Used in the first method to derive the osculating conic equation for a point P; stated in Section 2 of the paper.
standard math Definition 2.3 of the second Hessian H2(F) from Maugesten and Moe [11, Theorem 1.1]
The paper adopts this corrected formula for H2(F) and uses it to compute the second Hessian for the Hesse pencil.
standard math Lemma 2.24 of [2] giving the osculating conic for curves in normal form
The second method applies this lemma after projective changes of coordinates; the author corrects a sign error in the lemma's statement (equation 12).
domain assumption Freeness result for t=0 (Fermat cubic) from [8]
Theorem 4.3 relies on [8] for the t=0 case and asserts the other t values follow by analogous arguments, which are not presented.
domain assumption Non-singularity assumption t^3+27 != 0
The paper assumes this to exclude singular points, which cannot be sextactic points; stated before the computations in Section 3.

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Cite this review

Pith. "Pith review of The Hesse pencil of plane curves and osculating conics." pith.science (2026). https://pith.science/paper/5N5KPFPB

@misc{pith2026250604662,
  author       = {Pith},
  title        = {Pith review of: The Hesse pencil of plane curves and osculating conics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5N5KPFPB}},
  note         = {Machine review of arXiv:2506.04662}
}

read the original abstract

In this paper, we revisit the classical problem of determining osculating conics and sextactic points for a given algebraic curve. Our focus is on a particular family of plane cubic curves known as the Hesse pencil. By employing classical tools from projective differential geometry, we derive explicit coordinates for these special points. The resulting formulas not only clarify previous approaches but also lead to the construction of new families of free and nearly free curves, extending recent findings the freeness of curves.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[8]

Dimca A., Ilardi G., Malara G., Pokora P.:Construction of free curves by adding osculating conics to a given cubic curve, to appear in: Int. Math. Res. Not.,https://doi.org/10.1093/imrn/rnae273

work page doi:10.1093/imrn/rnae273
[1]

Artebani, M., Dolgachev, I.:The Hesse Pencil of plane cubic curve.Enseign. Math. 55 (2009), no. 3/4, 235–273

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Cayley, A.:On the Conic of Five-Pointic Contact at Any Point of a Plane Curve, Philosophical Transactions of the Royal Society of London149, 371–400 (1859)

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Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.:Singular4-4-0 — A computer algebra system for polynomial computations. https://www.singular.uni-kl.de (2024)

work page 2024

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Computations performed withSingular:https://github.com/EwelinaNaw/Sextactic-Points.git Ewelina Nawara, Department of Mathematics, University of the National Education Com- mission, Podchor¸ ażych 2, 30-084 Kraków, Poland, E-mail address:ewelina.nawara@op.pl 10

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [8]

Dimca A., Ilardi G., Malara G., Pokora P.:Construction of free curves by adding osculating conics to a given cubic curve, to appear in: Int. Math. Res. Not.,https://doi.org/10.1093/imrn/rnae273

work page doi:10.1093/imrn/rnae273

[2] [1]

Artebani, M., Dolgachev, I.:The Hesse Pencil of plane cubic curve.Enseign. Math. 55 (2009), no. 3/4, 235–273

work page 2009

[3] [2]

Rose-Hulman Undergraduate Mathematics Journal 15 (2014), 1–22

Balay-Wilson, L., Brysiewicz, T.:Points of ninth order on cubic curves. Rose-Hulman Undergraduate Mathematics Journal 15 (2014), 1–22

work page 2014

[4] [3]

European Mathematical Society (EMS), Zürich, 2009, xvi+491 pp

Beltrametti C.M., Carletti E., Gallarati D., Bragadin G.M.,:Lectures on Curves, Surfaces and Pro- jective Varieties. European Mathematical Society (EMS), Zürich, 2009, xvi+491 pp

work page 2009

[5] [4]

Cayley, A.:On the Conic of Five-Pointic Contact at Any Point of a Plane Curve, Philosophical Transactions of the Royal Society of London149, 371–400 (1859)

work page

[6] [5]

Cayley, A.:On the Sextactic Points of a Plane Curve, Philosophical Transactions of the Royal Society of London155, 545–578 (1865)

work page

[7] [6]

Coolidge J.L.:A Treatise on Algebraic Plane Curves,Clarendon Press, pp.xxiv+513 (1931)

work page 1931

[8] [7]

https://www.singular.uni-kl.de (2024)

Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.:Singular4-4-0 — A computer algebra system for polynomial computations. https://www.singular.uni-kl.de (2024)

work page 2024