On degenerate geometric rank

Derek Wu; Mat\v{e}j Dole\v{z}\'alek; Pawe{\l} Pielasa

REVIEW 2 major objections 5 minor 21 references

On degenerate geometric rank

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Rank-neutral directions reveal when degenerate tensors cannot lift

desk verdict A genuine generalization of Draisma's unliftability criterion to general rank loci, plus a clean classification of rank-1-induced degeneracy; the proofs are mostly sound, with a few spots that need clarification rather than major rework. read the letter →

arxiv 2607.18898 v1 pith:MPZUQCK5 submitted 2026-07-21 math.AG

Matěj Doležálek , Pawe{l} Pielasa , Derek Wu This is my paper · ORCID

classification math.AG MSC 15A6914N0714M12

keywords tensorgeometricranklociliftabilitydegeneratesecantvarietiesrank-neutraldirectionsbordersubrank

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

The paper studies tensors whose geometric rank is lower than expected—tensors in which some locus of matrices of rank at most $r$ has surprisingly large dimension. It proves a sufficient criterion for when such a tensor cannot be extended to a larger tensor without enlarging that rank locus: if the tensor's space of matrices equals its set of rank-neutral directions, it is 'unliftable'. It also characterizes tensors whose degeneracy is caused entirely by the rank-1 locus: either the space contains a linear subspace of the Segre variety, or it is equivalent to $2\times 2$ symmetric matrices, in which case the rank-2 locus also achieves the geometric rank. These results are accompanied by new examples of degenerate geometric rank arising from nonlinear rank loci, including matrix multiplication and the octonions.

What carries the argument

The argument rides on the set of rank-neutral directions $RND_r(E)$: for each maximal component $Y_i$ of the rank-$r$ locus, intersect the spaces $E + \hat{T}_e \sigma_r \mathrm{Seg}$ over all regular points $e \in Y_i$ where $\operatorname{rank}(e)=r$. The tangent-cone formula $\hat{T}_{[e]}\sigma_r \mathrm{Seg} = \{M \mid M(\ker e) \subseteq \operatorname{im} e\}$ turns the criterion into an explicit matrix computation. The proof of Theorem 4 generalizes Draisma's liftability criterion to arbitrary projective varieties $X$, with Lemma 7 as the load-bearing step. Theorem 10 uses Hopf's theorem on the span of an irreducible curve in the Segre variety, plus an induction cutting with a general hyperplane.

What would settle it

Find a triple $(X,E,E')$ with $E' = E + Cv$, where $Y = X \cap PE$ is regular at a point $e$ but $Z = X \cap PE'$ is singular at $e$, and compute $\hat{T}_e Y$ and $\hat{T}_e Z \cap E$. If they differ, Lemma 7 collapses, and with it the general unliftability criterion of Theorem 4.

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Extended reading notes

Core claim

The central results are Theorem 4 and Theorem 10. Theorem 4 states that if $E = RND_r(E)$, then the space $E$ is r-unliftable: it cannot be contained in a larger space $E'$ of matrices with the same codimension of the rank-$r$ locus. The set $RND_r(E)$ collects directions whose addition to $E$ does not force the rank locus to shrink in codimension, computed via tangent cones of the secant variety $\sigma_r \mathrm{Seg}$. Theorem 10 characterizes degenerate geometric rank caused by the rank-1 locus: for a concise degenerate tensor $T$, if $Y^1_T$ achieves $GR(T)$, then either $PT_A(A^*)$ contains a linear subspace of the Segre variety, or some quotient of $T$ is $\operatorname{GL}(B)\times \operatorname{GL}(C)$-equivalent to the space of symmetric $2\times 2$ matrices, and th

Load-bearing premise

The proof of Theorem 4 assumes that, for points $e$ that are regular in the intersection $Y = X \cap PE$ but possibly singular in $Z = X \cap PE'$, the tangent cone of $Y$ equals the intersection of the tangent cone of $Z$ with $E$; this equality is asserted without proof and is not automatic for singular varieties.

Editorial extensions

If this is right

A tensor satisfying E = RND_r(E) cannot be extended to any larger tensor with the same codimension of the rank-r locus, giving a practical unliftability certificate.
Degenerate geometric rank caused solely by the rank-1 locus is severely constrained: only two exceptional configurations exist, which simplifies classification efforts.
The examples of matrix multiplication and octonions show that the criterion works beyond linear spaces of bounded rank, suggesting a new tool for subrank upper bounds.
If the theorem's conclusion holds generally, tensors of degenerate geometric rank must either contain a linear bounded-rank space or exhibit a symmetric 2×2 block; nonlinear rank loci alone cannot be the sole source.
The computationally verified unliftability of several minimal-border-rank tensors provides a new batch of 'maximal' degenerate tensors.

Reading between the lines

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

If the tangent-cone equality in Lemma 7 fails in some case, the Rank-Neutral-Direction criterion would need refinement; a search for such a counterexample could reveal a sharper condition.
The characterization suggests that 'purely nonlinear' degenerate geometric rank may not exist for rank 1, but the paper's own examples show it does for higher ranks; extending Theorem 10 to r>1 is a natural test.
The SageMath implementation could be used to scan known tensor families for unliftability, potentially preparing the ground for new border subrank lower bounds.
The connection to Hopf's theorem hints that span-of-curve inequalities in the Segre variety may yield further bounds for secant varieties, beyond the geometric rank setting.

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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 paper is worth reading. It does two new things: it generalizes Draisma's unliftability criterion from linear spaces of bounded rank to arbitrary rank loci, via the notion of RND_r, and it proves a clean classification of tensors whose rank-1 locus alone achieves geometric rank. The examples—matrix multiplication, octonions, and the SL3-invariant E6—are concrete and should be useful for people hunting for nonlinear flattenings or border subrank bounds.

The main theorems appear correct. Theorem 4 is a natural extension, and the proof follows Draisma's structure. The one spot that needs attention is Lemma 7: the equality \hat T_e Y = \hat T_e Z ∩ E is asserted 'due to the scheme-theoretic intersection.' That is not automatic for tangent cones. But in the situation of the lemma, e is regular in Y, and a standard argument shows that a smooth hyperplane section forces the ambient component to be smooth at e (otherwise the tangent space of the section would be too large). Then the equality holds. The paper does not include that argument, which is an omission, but it is repairable and I don't think the proof is broken.

Theorem 10's induction also skips the detail that the restricted tensor T' may lose conciseness. A genericity choice of the hyperplane should preserve it, but the text doesn't say so. Another minor gap.

The computational claims (octonions unliftable, E6 4-unliftable) are asserted without details in the text. There is a GitHub link, so a referee can check, but as written the reader has to trust the computation. That should be fixed before publication.

Otherwise, the examples are consistent and the Section 5 discussion is honest about the fact that minimal border rank examples always have a linear locus achieving the rank jointly. The paper's claim to have identified nonlinear degeneracy is fine—matrix multiplication and octonions do show that.

Bottom line: this deserves a serious referee. The gaps are expository, not fatal. I would send it to review, and I'd cite it if I worked on geometric rank or tensor flattenings.

Referee Report

2 major / 5 minor

Summary. The paper studies tensors of degenerate geometric rank, i.e., tensors T ∈ A⊗B⊗C for which some rank-r locus Y^r_T has unexpectedly large dimension. The main contributions are: (1) a sufficient criterion for a tensor (equivalently, a linear space E ⊂ Hom(B*,C)) to be r-unliftable, expressed through the set RND_r(E) of rank-r-neutral directions (Theorem 4, generalizing Draisma's criterion for spaces of bounded rank); (2) a structural theorem (Theorem 10) stating that if the rank-1 locus alone achieves degenerate geometric rank, then either the tensor's space of matrices contains a linear subspace of the Segre variety, or a quotient is GL(B)×GL(C)-equivalent to the space of symmetric 2×2 matrices, in which case the rank-2 locus also achieves the geometric rank; (3) several examples of nonlinearly degenerate geometric rank, including matrix multiplication tensors, octonions, minimal-border-rank tensors, and an SL_3-invariant tensor. The paper supplies a SageMath implementation of the liftability criterion.

Significance. If the results hold, they provide a substantial generalization of Draisma's liftability framework to arbitrary rank loci and give the first structural dichotomy for tensors whose degeneracy is caused by the rank-1 locus. The explicit examples and the accompanying code are useful contributions; in particular, the unliftability of matrix multiplication tensors and the octonion structure tensor are concrete and checkable. The paper is written in a clear style and engages honestly with computational verification. The main theorems are significant for the geometric-rank community and for the broader study of tensor degenerations.

major comments (2)

[Section 3, Lemma 7] The proof of U ⊆ ND_X(E) hinges on the equality \hat T_e Y = \hat T_e Z ∩ E, asserted 'due to the scheme-theoretic intersection Y = Z∩PE'. This equality is not a formal consequence of scheme-theoretic intersection for tangent cones when e is singular in Z, and the proof only assumes e ∈ Y_reg, not e ∈ Z_reg. Indeed, the argument explicitly allows dim \hat T_e Z > dim \hat T_e Y. A counterexample to this tangent-cone equality for a singular variety and a hyperplane section would invalidate Lemma 7, and hence Theorem 4. The authors need either to prove this equality under the stated hypotheses (e.g., by showing e is regular in Z or by a transversality statement for tangent cones) or to modify the definition of RND_r(E) so that the argument goes through. As written, the central unliftability criterion is not established.
[Section 4, Theorem 10 proof, induction step] The induction applies the theorem to the restriction T' induced by A ↠ A', where (A')* = T_A^{-1}(T_A(A*)∩H) for a general hyperplane H ⊂ B⊗C. The paper assumes throughout that all tensors are concise, but T' may lose conciseness: the flattenings T'_B and T'_C can fail to be injective even when T is concise. The proof does not show that a general H can be chosen to preserve conciseness, nor does it state a version of Theorem 10 for non-concise tensors. This is a load-bearing omission because the inductive hypothesis requires the theorem to apply to T'. The gap appears repairable by a genericity argument, but it must be supplied.

minor comments (5)

[Title/Abstract] The title line reads 'ON DEGENERA TE GEOMETRIC RANK' due to a missing space; likely a LaTeX typo.
[Section 2.1] The projective dimension of PA* is m−1, but the text sometimes writes dimPA* where the vector-space dimension m is meant. This is a source of the '+' signs in §4; please unify notation.
[Section 3, Definition 3] The set RND_r(E) is a union of intersections; it is not asserted to be a linear subspace. This is fine, but in the remark after Theorem 4 the statement 'RND_r(E) is a linear space of dimension dim E + 1' should be justified, since the union may be nonlinear in general.
[Section 5.1, Example 15] The verification that the chosen permutation matrices A suffice to force the intersection to equal E is sketched rather than proved in full. A short argument for the general n case would improve readability.
[Section 5.3] In the displayed matrices, several entries are written as '0x' without a space (e.g., '0x 0 0' and '0x 0 0 0'); the intended multiplication sign is clear but should be fixed.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: central results derive from external theorems and direct tangent-space arguments; only flagged issue is an unproved tangent-cone equality in Lemma 7, a correctness gap rather than a definitional reduction.

full rationale

I traced the derivation chain and found no step where a claimed prediction or theorem is equivalent to its input by construction. Theorem 4's unliftability criterion is proved through Corollary 8/Lemma 7 using Lemma 9, where Lemma 9 is the external Draisma tangent-space description of sigma_r Seg(PB×PC). The set RND_r(E) in Definition 3 is not secretly the conclusion: it is computed from tangent cones, and Theorem 4 proves a conditional statement about lifts rather than assuming the desired answer. The one real weakness is Lemma 7's assertion: 'Simultaneously, \hat T_e Y = \hat T_e Z ∩ E due to the scheme-theoretic intersection Y = Z ∩ PE.' For tangent cones of a possibly singular Z this equality is not a formal consequence and no proof is supplied. I flag this as an omitted proof/correctness risk: if it fails, the proof of Theorem 4 breaks. It is not circular, however, because the equality is not assumed as the target conclusion and no fitted parameter is renamed as a prediction. Theorem 10 is an independent classification using Hopf's theorem (quoted with proof reference [Smi78]) and elementary span/codimension inequalities; Lemma 14 preserves codimensions by cutting positive-dimensional components with a general hyperplane. The induction's hyperplane restriction may require a genericity argument to preserve conciseness, but that is again a rigor gap rather than a circularity. The self-citations [Dol26] and [DM26] are used only for code and contextual remarks and do not carry the main theorems, so they do not raise the circularity score above 1.

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

Main theorems rest on standard algebraic geometry (projective varieties, tangent cones), the standing conciseness assumption, and two imported results: Hopf's theorem and Draisma's tangent-space description of σ_r Segre. No free parameters or invented entities. The examples rely on external classification results ([KMZ23], [JJ24]) and a GitHub repository without a commit hash.

assumptions (5)

domain assumption All tensors are assumed concise (all three flattening maps injective).
Assumed in Section 2.1; used throughout, and the proof of Theorem 10 implicitly needs it to identify P T_A(A*) with P A*.
standard math Hopf's theorem (Theorem 12) as cited from [Smi78, Prop 1.3].
Used in Lemma 13 to lower-bound the span of a curve in the Segre variety.
standard math Tangent space description of σ_r Seg(PB×PC) (Lemma 9), quoted from [Dra06, Lemma 9].
Used to compute RND_r(E) in Theorem 4 and examples.
domain assumption GR(M⟨n⟩)=⌊3/4 n^2⌋ from [KMZ23, Theorem 6.1].
Used in Example 15 to place the new unliftability result in context.
domain assumption Classification of minimal border rank tensors in [JJ24, Section 4] is correct.
Section 5.3 relies on this list to generate examples.

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Cite this review

Pith. "Pith review of On degenerate geometric rank." pith.science (2026). https://pith.science/paper/MPZUQCK5

@misc{pith2026260718898,
  author       = {Pith},
  title        = {Pith review of: On degenerate geometric rank},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPZUQCK5}},
  note         = {Machine review of arXiv:2607.18898}
}

abstract

A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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2026 , eprint=

Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus , author=. 2026 , eprint=

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work page doi:10.19086/da.73322