Algebraically Skew Embeddings of Curves

Andy B. Day

REVIEW 4 major objections 4 minor 26 references

Algebraically Skew Embeddings of Curves

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves that in P4 the only algebraically skew curves are the rational normal curve and the elliptic normal curves, and that in P3 the only such curve is the twisted cubic.

desk verdict A plausible classification of skew curves in P3 and P4 with a newly developed blowup method, but the load-bearing P4 count is under-verified and one stated lemma is false; it deserves serious refereeing. read the letter →

arxiv 2501.18132 v4 pith:RLQLZXMO submitted 2025-01-30 math.AG

Andy B. Day This is my paper · ORCID

classification math.AG MSC 14H5014N0514N1014C17

keywords algebraicallyskewembeddingcurvesTerracinilociGaussmapexcessintersectionblowuprationalnormalcurveelliptic

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 studies algebraically skew embeddings: smooth complex projective varieties whose embedded tangent spaces at any two distinct points are disjoint, with an extra condition ruling out infinitesimal tangencies. It proves that any n-dimensional variety admits such an embedding into $P^{{4n+1}}$ and none into a projective space of dimension below 3n. For curves the classification is complete: in P3 the only algebraically skew curve is the twisted cubic, and in P4 the only ones, up to linear equivalence, are the rational normal curve and the elliptic normal curves. A sympathetic reader should care because this settles the projective analogue of the classical skew-loop question for curves in the two lowest nontrivial ambient dimensions, and the counting formula it produces gives a concrete enumerative invariant for every degree and genus.

What carries the argument

The main tool is the degeneracy locus D_1 = {(U,W) : dim(U∩W) ≥ 1} inside the product Gr(n,N)×Gr(n,N), together with the product Γ = C×C of a curve's Gauss image with itself; skewness is exactly the statement that D_1 meets Γ only along the diagonal. Because D_1 is singular along the diagonal, the paper blows up the diagonal of the Grassmannian product, writes the Chow-ring classes of the proper transforms, and for P4 performs a second blowup along the diagonal curve to resolve tangency. The auxiliary incidence variety \hat D_1 ≅ P(Q) ×_{P4} P(Q) (with Q the universal quotient bundle on P4) makes the Chow ring of the blown-up degeneracy locus computable, and the symmetric excess-intersection formula converts the class computation into the numerical count of non-skew pairs.

What would settle it

Compute the number of non-skew ordered pairs of tangent lines for the canonical genus 5 curve in P4 by an independent method: the paper predicts 240. Alternatively, test Lemma 3.1(ii) on Gr(2,4): the class i_*(σ21) must lie in codimension 7, while the stated formula places it in codimension 3, so the formula as written is false and the count built on it needs correction or a new justification.

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Extended reading notes

Core claim

The central claim is that algebraic skewness of a curve can be detected by an excess-intersection count of its tangent lines. For a smooth degree d genus g curve in P4 whose Gauss map is an isomorphism onto its image, the number of ordered pairs of distinct points whose embedded tangent lines meet is (2d+2g−2)^2 − 20d − 44g + 44, counting degenerate third osculating planes. Algebraic skewness means this number is zero, and solving the resulting equation with the constraints on degree and genus leaves exactly the rational normal curve (d=4, g=0) and the elliptic normal curves (d=5, g=1) in P4, together with the twisted cubic in P3. The proof also establishes the general bounds 3n ≤ msdim X ≤ 4n+1 for an n-dimensional smooth projective variety X.

Load-bearing premise

The load-bearing step is the paper's formula for pushing a Schubert class forward along the diagonal inclusion of a Grassmannian; as stated that formula gives a class of the wrong codimension in a case used later, and every subsequent intersection count inherits it.

Editorial extensions

If this is right

In P3, a smooth spatial curve with pairwise disjoint embedded tangent lines must be the twisted cubic; no other degree or genus is possible.
In P4, algebraic skewness forces either the degree 4 rational normal curve or a degree 5 elliptic normal curve, up to linear equivalence.
Every n-dimensional smooth projective variety has an algebraically skew embedding into P^{4n+1}, and none exists in P^N with N < 3n.
For any smooth degree d genus g curve in P4 with nondegenerate Gauss map, the number of non-skew ordered pairs of tangent lines is (2d+2g−2)^2 − 20d − 44g + 44; for the canonical genus 5 curve this is 240.
Generic skew scrolls in P3 and P4 must be rational normal scrolls, so non-rational scrolls always contain intersecting rulings.

Reading between the lines

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

If Lemma 3.1(ii) is corrected rather than discarded, the classification may survive, but the enumerative constant in Lemma 4.11 would likely change; the paper's numerical predictions are the place to test the repair.
The same double-blowup strategy should apply to curves in P5 and beyond, where the relevant degeneracy loci are D_r with r ≥ 1 and the expected answers may be families rather than single curves.
The coincidence of the genus 5 count with a Steiner complex cardinality suggests a hidden theta-characteristic interpretation of non-skew tangent pairs for canonical curves, which the paper leaves open.
The real totally skew surface constructed from an affine elliptic normal curve in R8 may be a test case for whether the algebraic classification has smooth counterparts in low-dimensional real Euclidean spaces.

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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

Hi — short take: this is a real research paper with a plausible and interesting classification and a genuinely new technique, but the key P4 enumeration is not yet demonstrated as written, and one stated lemma is false. I would send it to a referee; I would not yet rely on the P4 theorem.

What is new: the 3n lower bound for algebraically skew embeddings, the blowup-of-Gauss-images method, and the classifications — twisted cubic in P3, rational normal and elliptic normal curves in P4. The paper is also straight about the overlap with Ciliberto: Lemma 4.11 is the same count as the Terracini-locus computation in [Cil24], and the author reports a Macaulay2 check for one genus-5 canonical curve giving 240. That is real evidence and the right level of honesty.

The soft spots are equally real. Lemma 3.1(ii), as stated, cannot be right: diagonal pushforwards have codimension |p| + dim G, not |p|, and the displayed sum is dimensionally inconsistent. The paper then uses this lemma in the P3 computation. However, the particular pushforward actually used for Gr(2,4), σ21 ↦ σ21⊗σ22 + σ22⊗σ21, is correct and can be checked directly, so the P3 theorem may survive a repaired lemma. The bigger problem is the P4 count: Lemma 4.11 is the output of a long Chow/excess-intersection computation, and the only independent check reported is a single (d,g)=(8,5) example. One Macaulay2 point is not enough to support a classification statement. Theorem 4.13 also has a scrambled and under-explained exclusion of the remaining degree-genus pairs; the GH page reference is a pointer, not a proof of nonexistence. Lemma 3.10's GKZ quote is sloppy, though the formula itself is standard.

If Ciliberto's independent computation is sound, the enumeration is probably right; if it is not, the paper's central theorem has no adequate verification yet. The author should repair Lemma 3.1, expand the case exclusion in Theorem 4.13, and either get an independent check of Lemma 4.11 or write out the full computation in enough detail to be checked. A referee with Terracini-loci and Schubert calculus background can do that. Worth sending out.

Referee Report

4 major / 4 minor

Summary. The paper defines algebraically skew embeddings of smooth projective varieties and studies the minimal skew embedding dimension msdim X. It proves general bounds 3n ≤ msdim X ≤ 4n+1, then specializes to curves. The main technical tool is a blowup of G×G along the diagonal and an excess-intersection computation on the proper transforms of the degeneracy locus D1 and of Γ = γ(X)×γ(X). The claimed results are that the only algebraically skew curve in P3 is the twisted cubic (Theorem 3.15), and that the only algebraically skew curves in P4 are the rational normal curve and the elliptic normal curves (Theorem 4.13). The paper also applies the method to scrolls and includes a Macaulay2 verification of one numerical case, Example 4.12.

Significance. If the central enumeration Lemma 4.11 is correct, the classification of algebraically skew curves in P3 and P4 is a clean and natural result, and the blowup/excess-intersection approach offers a genuinely different method from the Terracini-locus and Porteous computations of [Cil24]. The paper is self-contained, has no fitted parameters, and provides an independent machine check in Example 4.12; the general bounds in Section 2 are also a useful contribution. However, the written derivation contains a demonstrably false lemma in the intersection calculus and several unclear steps in the classification argument, so the main classification is not yet established as written.

major comments (4)

[§3.1, Lemma 3.1(ii)] Lemma 3.1(ii) is false as stated. Since the diagonal inclusion i: ΔG → G×G has codimension dim G, the pushforward i_*(σ_p) must land in codimension |p| + dim G, not |p|. For G = Gr(2,4) the displayed rule would place i_*(σ_{21}) in codimension 3, whereas the correct class has codimension 7; the identity actually used in Lemma 3.9 and Lemma 3.11, i_*(σ_{21}) = σ_{21}⊗σ_{22} + σ_{22}⊗σ_{21}, has total codimension 7. The proof given for the lemma confuses the condition σ_aσ_b = σ_p with the Künneth decomposition of the diagonal class. Because the same pushforward rule is used in the blowup ring structure underlying the P4 computation leading to Lemma 4.11, this is a load-bearing error and must be replaced by the correct diagonal-class computation.
[§3.4, Lemma 3.10] The dual-degree formula d∨ = 2d + 2g − 2 is a load-bearing input, used in Lemmas 3.9, 4.2, and 4.11, but the proof is not acceptable as written: it cites [GKZ94, p.61] and simultaneously asserts that the formula printed there is a typo, without reproducing the correction. Moreover, the proof writes the point class of Gr(1,N) as σ_{22}, which is correct only for N = 3; for N = 4 the point class is σ_{33}. Please supply a direct proof of the dual-degree formula (for instance via the Gauss map or Plücker-type class formulas) or a correct, unambiguous citation.
[§4.8, Theorem 4.13] The final classification contains a scrambled and insufficient exclusion of impossible pairs. After the algebra, the author obtains (g,d) ∈ {(0,4),(1,5),(2,5),(5,4)}. The next sentence says that 'the cases (5,2),(4,5) are in fact not possible', which does not match this list; presumably the intended exclusions are (g,d) = (2,5) and (g,d) = (5,4). Even with that correction, the appeal to [GH94, p.253] is not a proof: the nonexistence of nondegenerate smooth genus-2 degree-5 and genus-5 degree-4 curves in P4 should be shown directly, for example by Riemann–Roch and Clifford's theorem. The derivation of the bounds 6 ≤ d∨ ≤ 16 and d ≤ 5 should also be written out explicitly. As it stands, Theorem 4.13 is not fully proven.
[§4.3–4.8, Lemma 4.11] The numerical heart of the paper is Lemma 4.11, which converts the Chow-ring computation into the count (2d+2g−2)^2 − 20d − 44g + 44. The proof is a long sequence of blowup and Chern-class calculations whose only reported check is the single case (d,g) = (8,5) in Example 4.12. Given that Lemma 3.1(ii) is false and that the later classification step is scrambled, this computation is not verifiable as written. In addition, the step from the excess-intersection identity to 'the number of nonskew pairs' implicitly assumes that all off-diagonal contributions are reduced, transverse points; this should be stated and justified. I ask for a fully displayed calculation, cross-checked against [Cil24] or verified in additional examples, before the classification can be accepted.

minor comments (4)

[§1, Definition 1.2] There are several typos: 'sekw' should be 'skew', and the abstract contains 'minim al'.
[§4.2, Theorem 4.2] The sentence 'Since there is no elliptic curves of degree ≤ 2 in P4, the case of g = 1 is impossible' is unclear: the preceding equation gives d∨ = 5 for g = 1, not d∨ ≤ 2, and an elliptic scroll of degree 5 in P4 is not obviously ruled out. Please rewrite the exclusion.
[§2.1, Notation] The angle-bracket notation for numerical classes, written with '/llbracket' and '/rrbracket', is rendered inconsistently and is not defined cleanly; please define it once and use it uniformly.
[§3.4, Lemma 3.10] In the last sentence of the proof, 'To concludes the proof' should be 'To conclude the proof'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification and enumeration are derived from independent excess-intersection computations, with external checks rather than fitted inputs.

full rationale

The paper's central claims, Lemma 4.11 and Theorem 4.13, are obtained from explicit Chow-ring and excess-intersection computations in Sections 4.4 through 4.7. The inputs are the definition of algebraically skew embeddings (Definitions 1.1 and 1.2), standard Schubert calculus, and blowup constructions; no parameter is fitted to the target count. The single Macaulay2 check in Example 4.12 is an independent verification of one case, not an input to the formulas. The acknowledgment that Ciliberto's work [Cil24] gives the same count as Lemma 4.11 is a consistency note, not the basis for the proof; the derivation does not rely on that citation. The paper contains no load-bearing self-citations and no uniqueness theorem imported from the author's own prior work. The questionable Lemma 3.1(ii) is a mathematical correctness concern rather than a circularity: if the lemma is false, the computation is unsupported, but the paper does not define its objects in terms of its conclusions and does not rename a fitted quantity as a prediction. The derivation is therefore self-contained with respect to circularity, and the correct score is 0.

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

The paper introduces no fitted parameters and no new physical or geometric entities. Its computations rest on standard intersection-theoretic tools plus several domain assumptions about curves, dual varieties, and genericity of scrolls. The shakiest input is the dual-degree formula d∨ = 2d+2g-2, cited to [GKZ94] while claiming a typo in that source, and the Gauss-map diffeomorphism assumption stated in Lemma 4.11.

assumptions (5)

standard math Porteous formula and standard degeneracy locus computations over Grassmannians
Used in Lemma 2.7 and Lemma 3.5 to compute the class of the degeneracy locus D1 and its proper transform.
standard math Blowup Chow ring structure and the excess intersection formula from [EH16] and [Ful98]
Used throughout Sections 3 and 4, most explicitly in Theorem 4.4, to compute classes in the blowup spaces.
domain assumption The degree of the dual hypersurface of a smooth curve in P^N is 2d+2g-2
Stated in Lemma 3.10 with a reference that the paper claims contains a typo; this value is substituted into every subsequent intersection formula.
domain assumption The Gauss map of the curves under study is a diffeomorphism to its image
Assumed in Lemma 4.11 and used to identify diagonal classes; for algebraically skew curves it is implied by the definitions, but the lemma states it as a hypothesis.
domain assumption Generic conditions on scrolls in Lemma 4.1 and Theorem 4.2
Used to rule out non-generic excess components; asserted without a proof of genericity.

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Cite this review

Pith. "Pith review of Algebraically Skew Embeddings of Curves." pith.science (2026). https://pith.science/paper/RLQLZXMO

@misc{pith2026250118132,
  author       = {Pith},
  title        = {Pith review of: Algebraically Skew Embeddings of Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLQLZXMO}},
  note         = {Machine review of arXiv:2501.18132}
}

abstract

Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not intersect. In this work, we establish an upper bound and a lower bound of the minimal dimension $N$ such that there exists an algebraically skew embedding into $\mathbb{P}^N$ in terms of the dimension of the given smooth variety $X$. Then we further classify the algebraic curves in terms of their minimal skew embedding dimensions, and apply the same technique to other one-parameter family of lines.

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Ballico, A

E. Ballico, A. Bernardi, and P. Santarsiero. Terracini locus for three points on a segre variety. Asian J. Math. , 27:375--404, 2023. https://doi.org/10.4310/AJM.2023.v27.n3.a3 doi:10.4310/AJM.2023.v27.n3.a3

work page doi:10.4310/ajm.2023.v27.n3.a3 2023

[2] [2]

Ballico and L

E. Ballico and L. Chiantini. On the terracini locus of projective varieties. Milan J. Math. , 89:1--17, 2021. https://doi.org/10.1007/s00032-020-00324-5 doi:10.1007/s00032-020-00324-5

doi:10.1007/s00032-020-00324-5 2021

[3] [3]

Terracini loci of curves

E. Ballico and L. Chiantini. Terracini loci of curves, 2023. https://arxiv.org/abs/2305.02035 arXiv:2305.02035

work page Pith review arXiv 2023

[4] [4]

M. C. Beltrametti and A. J Sommese. On k-jet ampleness. In Complex Analysis and Geometry , pages 355--376. Springer US, 1993. https://doi.org/10.1007/978-1-4757-9771-8_15 doi:10.1007/978-1-4757-9771-8_15

work page doi:10.1007/978-1-4757-9771-8_15 1993

[5] [5]

Bansal, S

A. Bansal, S. Sarkar, and S. Vats. Isomorphism of multiprojective bundles and projective towers, 2023. https://arxiv.org/abs/2311.00999 arXiv:2311.00999

arXiv 2023

[6] [6]

Ballico and E

E. Ballico and E. Ventura. A note on very ample terracini loci. Rend. Circ. Mat. Palermo (2) , 72:4115--4118, 2023. https://doi.org/10.1007/s12215-023-00886-4 doi:10.1007/s12215-023-00886-4

work page doi:10.1007/s12215-023-00886-4 2023

[7] [7]

Enumeration of Terracini schemes

C. Ciliberto. Enumeration of terracini schemes, 2024. URL: https://arxiv.org/abs/2406.16514, https://arxiv.org/abs/2406.16514 arXiv:2406.16514

work page Pith review arXiv 2024

[8] [8]

I. V. Dolgachev. Classical Algebraic Geometry: A Modern View . Cambridge University Press, 2012

work page 2012