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REVIEW 3 major objections 3 minor 17 references

Maximal minors of $1$-generic matrices have rational singularities

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

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Pith's one-line read A quotient of a polynomial ring by the maximal minors of a $1$-generic matrix has rational singularities whenever the matrix has at least as many columns as rows.

desk verdict A fixable but real flaw: the base change to an algebraically closed field changes the 1-generic hypothesis, so the theorem as stated is false, though the intended application of a known method looks right. read the letter →

arxiv 2506.08385 v1 pith:PQA5H263 submitted 2025-06-10 math.AC math.AG

Trung Chau , Manoj Kummini This is my paper · ORCID
classification math.ACmath.AG MSC 13C4014M1213D02
keywords determinantalrings1-genericmatricesmaximalminorsrationalsingularitiesnormalityEagon-NorthcottcomplexKoszulresolutionlinearsections
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 claims that for any field $k$ and any $m\times n$ matrix $M$ of linear forms satisfying the $1$-generic condition—no nonzero row vector $\lambda$ and column vector $\mu$ make $\lambda M\mu^{\mathsf T}$ identically zero—the quotient ring $k[M]/I_m(M)$ by the maximal minors has rational singularities, meaning its singularities are cohomologically as mild as smooth points. If the claim holds, every such ring is normal, confirming a 1988 conjecture for this class, and the result extends a 2018 theorem about Hankel determinantal rings from characteristic zero to arbitrary fields. The route is geometric: a smooth zero scheme in a product of projective spaces is shown to give a rational resolution of the variety cut out by the maximal minors, and a graded blow-up argument carries the conclusion from the projective variety to its affine cone.

What carries the argument

The carrying object is the Koszul complex on $\mathbb{P}^r\times\mathbb{P}^{m-1}$ associated to the $n$ bilinear forms $\sum_i y_i\ell_{i,j}$; the proof's crucial step is that this complex is a locally free resolution of its zero scheme $Z$ precisely when $M$ is $1$-generic. Pushing this resolution forward along the first projection gives an explicit locally free resolution of $\pi_{1*}\mathcal{O}_Z$ on $\mathbb{P}^{m-1}$, which supplies the cohomology vanishing that makes $\pi_1\colon Z\to X$ a rational resolution. The remaining machinery is the Cohen-Macaulay criterion for rational singularities and a Rees-algebra blow-up argument that transfers the property from $X$ to $\operatorname{Spec} k[M]/I_m(M)$.

What would settle it

Over $\mathbb{Q}$, the matrix $\begin{pmatrix}x_0&x_1\\-x_1&x_0\end{pmatrix}$ has no nonzero $\lambda,\mu$ with $\lambda M\mu^{\mathsf T}=0$, so it is $1$-generic by the paper's definition; over $\mathbb{C}$, taking $\lambda=\mu=(1,i)$ makes the bilinear form vanish. Checking whether the key rank condition in the proof still holds for this example would test whether the argument covers it.

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Extended reading notes

Core claim

The central claim is that rational singularities hold for the maximal-minor quotient of any $1$-generic matrix, not just for generic, symmetric, or Hankel matrices. Equivalently, the paper asserts that the projective scheme $X=\operatorname{Proj} k[M]/I_m(M)$ admits a rational resolution whose higher direct images vanish, and that $k[M]/I_m(M)$ itself is a normal Cohen-Macaulay domain. The author would state it as Theorem 1.1: with $m\le n$, $k$ arbitrary, and $M$ 1-generic, the ring $k[M]/I_m(M)$ has rational singularities.

Load-bearing premise

The proof assumes that a matrix which is $1$-generic over a field stays $1$-generic when the field is extended to its algebraic closure, because that extension is used to check that the defining Koszul complex is a resolution.

Editorial extensions

If this is right

  • Every generic, generic symmetric, and generic Hankel matrix is $1$-generic, so their maximal-minor quotient rings have rational singularities in arbitrary characteristic, not only in characteristic zero.
  • The ring $k[M]/I_m(M)$ is normal, settling the 1988 normality conjecture for this class.
  • The dimension estimate $\dim R\ge 2m-2$ together with regularity $m-1$ forces a negative $a$-invariant, so the Rees algebra of the graded maximal ideal is well behaved.
  • The Eagon-Northcott resolution gives the minimal free resolution of $R$, so the rationality result comes with full information about syzygies.

Reading between the lines

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

  • If the base-change gap is patched, the natural repair is to define 1-genericity after passing to the algebraic closure; that stronger hypothesis would make the proof's key rank statement true and might be the intended meaning.
  • The same pushforward-by-Koszul method could be tried for submaximal minors where the ideal is prime, even though the paper notes that submaximal minors of 1-generic matrices can behave badly.
  • In positive characteristic, the explicit resolution suggests testing F-rationality or F-regularity of these quotient rings, which would give an arithmetic analogue of rational singularities.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper claims that for any field k and any m×n 1-generic matrix M with m≤n, the quotient ring k[M]/I_m(M) has rational singularities. The proof proceeds by constructing a bilinear Koszul complex on P^r × P^{m-1}, showing that its direct image identifies O_X with O_Z, proving normality of R, and then applying Lipman's criterion and Hyry's blow-up theorem to obtain rational singularities. The main theorem would partially answer an Eisenbud normality conjecture and generalize the rational-singularity part of the CMSV18 Hankel determinantal result. The core argument is the Kempf-Lascoux-Weyman geometric method, combined with external results of Eisenbud, Lipman, and Hyry.

Significance. If the theorem were true, it would be a significant contribution: it would settle Eisenbud's 1988 normality conjecture for 1-generic matrices and extend known rational-singularity results to a wide class of determinantal rings. The proof is conceptually clean and does not rely on fitted parameters or circular reasoning. However, the theorem as stated is false, and the proof's very first reduction step (base change to an algebraically closed field) is invalid because 1-genericity is not preserved by flat scalar extension. This failure is not cosmetic: it produces an explicit counterexample to the main theorem. The paper's intended approach is sound in spirit and would likely prove a correct theorem under a base-change-stable hypothesis, but the central claim in Theorem 1.1 cannot stand.

major comments (3)
  1. [Lemma 3.4] The first paragraph of the proof asserts: 'Since extending the base field is faithfully flat, we may assume that k is algebraically closed.' This reduction is invalid because 1-genericity is not preserved by base field extension. For example, over k=Q, the 2×2 matrix M=[[x0,x1],[-x1,x0]] is 1-generic: for nonzero λ=(a,b) and μ=(c,d), λ^T M μ = (ac+bd)x0 + (ad-bc)x1, and the system ac+bd=0, ad-bc=0 has no nonzero rational solution. But over C, choosing λ=(1,i) and μ=(1,i) gives λ^T M μ = (1+i^2)x0 + (i-i)x1 = 0, so the base-changed matrix is not 1-generic. Consequently, inside the proof, the claimed rank rk A_b = n fails at non-rational closed points such as b=[1:i]; at that point the matrix A_b has rank 1, not 2. The Koszul complex is therefore not a locally free resolution of O_Z after base change, and all subsequent statements depending on this lemma—Proposition 3.5, Corollary 3.6, Proposition 3.7, and Theorem 1.1—rest on an invalid reduction.
  2. [Observation 3.2] The argument that r+1 ≥ m+n-1 uses the statement that the solution set of λ^T M μ = 0 inside P^{m-1} × P^{n-1} is empty. For k not algebraically closed, the solution set may have no k-rational points while still being nonempty over the algebraic closure, so 'empty' does not imply the claimed codimension bound. For the same matrix M=[[x0,x1],[-x1,x0]] over Q, the inequality r+1 ≥ m+n-1 would give 2 ≥ 3, which is false. This is not a minor edge case: the bound r+1 ≥ m+n-1 is used in the proof of Theorem 1.1 to conclude dim R ≥ 2m-2 and hence that the a-invariant is negative. Without a base-change-stable version of 1-genericity, the observation fails for non-algebraically closed fields.
  3. [Theorem 1.1] The main theorem is false as stated. For k=Q and M=[[x0,x1],[-x1,x0]], the ring R=Q[x0,x1]/(x0^2+x1^2) is not normal: the element t=x0/x1 in the fraction field satisfies t^2+1=0, so it is integral over R, but t is not in R. Rational singularities imply normality, so R cannot have rational singularities. The theorem would become correct if k were assumed algebraically closed or if M were assumed to remain 1-generic after algebraic closure; without such a hypothesis, the present formulation is unsound. Because this is the central claim of the paper, the error cannot be fixed by local edits within the manuscript's stated scope.
minor comments (3)
  1. [Lemma 3.4] There is a typo in the proof: the text says 'this proves the assertion made above about π^{-1}_1 U', but the assertion concerns π^{-1}_2 U; also 'integerdivide' should be 'not in the set of indices'.
  2. [Title and abstract] The typesetting has obvious rendering artifacts, e.g., 'MA TRICES HA VE RA TIONAL SINGULARITIES' and 'an d' in the abstract; these should be corrected before any resubmission.
  3. [General] Since the base-change issue is the core defect, the paper would benefit from an explicit statement of which hypotheses are required for each reduction, e.g., a remark that 1-genericity over k does not imply 1-genericity over \bar{k}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives the main theorem from external results (Eagon–Northcott, Lipman, Hyry) and does not fit parameters or rename inputs as predictions.

full rationale

The paper's derivation is self-contained in the sense relevant to circularity: the main theorem is proved by applying external, independently established results (Eisenbud's structure theorem via the Eagon–Northcott complex, Lipman's duality criterion, Hyry's blow-up theorem, and standard flat base change) to the 1-generic hypothesis. No parameter is fitted, no quantity called a prediction is built from the data it claims to predict, and the authors do not invoke their own prior work as load-bearing evidence. The identified weakness, that extending the base field need not preserve 1-genericity and that the stated theorem is false as written over non-algebraically closed fields, is a mathematical counterexample to an assumption in the proof, not a circular reduction. The argument does not define or justify its conclusion through its own conclusion; it simply has an invalid base-change step. Accordingly, the circularity score is 0.

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

No free parameters or invented entities appear. The proof rests on standard commutative algebra tools plus one false ad hoc assumption: that 1-genericity survives base field extension. That assumption, not circularity, is the defect that breaks the stated theorem.

assumptions (5)
  • ad hoc to paper Base field extension preserves 1-genericity
    Invoked at the start of Lemma 3.4: faithful flatness is claimed to allow assuming k algebraically closed. This is false: 1-generic over k does not imply 1-generic over the algebraic closure. A counterexample is M = [[x0,x1],[-x1,x0]] over Q. The dimension bound in Observation 3.2 depends on the same invalid step.
  • domain assumption Eisenbud's structure theorem for maximal minors
    Remark 3.1 uses [Eis05, Theorem 6.4]: I_m(M) is prime of height n-m+1, R is Cohen-Macaulay with Eagon-Northcott resolution and regularity m-1. These facts underlie the Cohen-Macaulay and a-invariant steps.
  • standard math Lipman's dualizing criterion [Lip94, (4.2)]
    Used to reduce condition (d) of Definition 2.1 to conditions (b) and (c) when the variety is Cohen-Macaulay. This is a standard derived-category statement.
  • standard math Hyry's blow-up theorem [Hyr99, Theorem 1.5]
    Used to pass from rational singularities of Proj R to rational singularities of Spec R via the Rees algebra. The authors sketch a characteristic-free verification with a Leray spectral sequence.
  • standard math Rees algebra of R is a line bundle over Proj R
    Invoked in the proof of Theorem 1.1 with reference to EGA II 8.7.8, allowing Proposition 2.3 to transport a resolution from X to Y.

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Cite this review

Pith. "Pith review of Maximal minors of $1$-generic matrices have rational singularities." pith.science (2026). https://pith.science/paper/PQA5H263

@misc{pith2026250608385,
  author       = {Pith},
  title        = {Pith review of: Maximal minors of $1$-generic matrices have rational singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQA5H263}},
  note         = {Machine review of arXiv:2506.08385}
}
abstract

We show that the quotient ring by the ideal of maximal minors of a $1$-generic matrix has rational singularities. This answers a conjecture of Eisenbud (1988) that such rings are normal, and generalizes a result of Conca, Mostafazadehfard, Singh and Varbaro (2018) that generic Hankel determinantal rings have rational singularities in characteristic zero.

Discussion (0). Continue with ORCID to comment.

Reference graph

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