On jet schemes of determinantal varieties

Huaiqing Zuo; Yifan Chen

REVIEW 5 major objections 4 minor 1 cited by

On jet schemes of determinantal varieties

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Pith's one-line read The paper proves that for the first jet scheme of the variety of 3-by-n matrices of rank at most 2, the Hilbert series equals the square of the Hilbert series of the base variety, and that the associated Stanley-Reisner complex is…

desk verdict New Hilbert series computations for jet schemes of 3-by-3 minors and explicit component polynomials, but the central proof rests on an unreported Singular computation and an under-justified reduction from finite cases to all n. read the letter →

arxiv 2506.22898 v1 pith:7CPYABBW submitted 2025-06-28 math.AG

Yifan Chen , Huaiqing Zuo This is my paper · ORCID

classification math.AG MSC 14M1214E1805E40

keywords determinantalvarietiesjetschemesshellabilityHilbertseriesGröbnerbasesStanley-Reisnercomplexirreduciblecomponentsprincipalcomponent

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 jet schemes of determinantal varieties---spaces of truncated power-series arcs through a variety---and looks for a hidden regularity in their Hilbert series. Its main concrete result is that for the first-order jet scheme of the variety of 3-by-n matrices of rank at most 2, the Hilbert series equals the square of the Hilbert series of the underlying determinantal variety. That square relation was previously known for the 2-by-2 minor case, and the paper presents the 3-by-3 case as evidence for a general conjecture: the principal component of any determinantal jet scheme should have Hilbert series equal to the (k+1)-th power of the base series. The proof uses Gröbner bases and a shellable Stanley-Reisner complex, and the paper also writes down explicit polynomial families defining the irreducible components of the general jet schemes $L^{m,n}_{r,p-1}$.

What carries the argument

The carrying mechanism is the correspondence between the leading ideal of the jet ideal and an abstract simplicial complex: squarefree monomials outside the leading ideal are exactly the non-faces, so facets are maximal sets of variables whose product is not killed. The paper shows those facets are unions of non-intersecting paths in a 3-by-n grid, falling into four families $\hat B$, $\hat C$, $\hat D$, $\hat E$, and orders the facets by a path order; the shelling condition reduces to tracking 'NE turns' in the paths. Hilbert series then follows from the standard formula for shellable Stanley-Reisner rings, and the explicit Gröbner basis of the five determinant families is what makes the leading ideal and its simplicial complex visible.

What would settle it

An independent exact Gröbner basis computation for $I^{3,n}_{3,1}$ at $n=10$ in the paper's term order would settle the claim: if the leading ideal contains a monomial not divisible by the leading monomials of the five determinant families, the reduction from finite cases to all $n$ fails and Theorem B is false. A cheaper check is to compare both sides of the Hilbert series identity at $z=1/2$ for $n=5$.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is a factorization: the coordinate ring of $L^{3,n}_{3,1}$ has Hilbert series exactly equal to the square of the coordinate ring of $L^{3,n}_3$, namely $$\left(\frac{1+(n-2)z+\frac{(n-1)(n-2)}{2}$z^{2}$}{(1-z)^{2n+2}}\right)^2.$$ The proof chooses a term order, exhibits five explicit determinant families $a_{p,q,r}$, $b_{p,q,r}$, $c_{p,q,r,s}$, $d_{l,p,q,r,s}$, $e_{l,p,q,r,s}$ whose leading monomials generate the leading ideal, shows the leading ideal is radical, and identifies its Stanley-Reisner complex with a shellable complex whose facets are four families of grid paths. Counting the relevant turns in a shelling recovers the $h$-vector $(1,2n-4,(n-2)(2n-3),(n-1)(n-2)^2,(n-1)^2(n-2)^2/4)$, which sums to the square formula. The same circle of ideas yields explicit defining polynomials for the irreducible components of the general jet scheme $L^{m,n}_{r,p-1}$.

Load-bearing premise

The load-bearing premise is that an unreported computer check for $n=4$ through $9$ correctly computes the leading ideal, and that a lemma then carries those finitely many cases to all $n$; if either step is wrong, Theorem B collapses.

Editorial extensions

If this is right

For every $n$, the coordinate ring of $L^{3,n}_{3,1}$ is Cohen-Macaulay, because a shellable Stanley-Reisner ring is Cohen-Macaulay.
The closed-form Hilbert series makes the dimension and multiplicity of $L^{3,n}_{3,1}$ directly computable from those of the base variety and exhibits the ring as a 'square' in Hilbert-series sense.
The 2-by-2 and 3-by-3 cases give two independent data points for the conjecture that the principal component of every determinantal jet scheme $L^{m,n}_{r,k}$ has Hilbert series equal to the $(k+1)$-th power of the base's Hilbert series.
The explicit polynomial families of Theorem C reduce membership in the irreducible components of $L^{m,n}_{r,p-1}$ to evaluating determinants, rather than to orbit-closure geometry.

Reading between the lines

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

If the square relation is a shadow of a graded algebra factorization or a monomial-order basis bijection, one would expect the jet scheme's $h$-vector to be the convolution square of the base's $h$-vector; the paper does not claim this stronger algebraic statement.
A natural stress test of the general conjecture is the next open case $L^{4,n}_{4,1}$ or the higher-order $L^{3,n}_{3,2}$; if the Hilbert series factors as the predicted power there, the conjecture gains real support.
The delegated Gröbner check for $n=4,\ldots,9$ could in principle be replaced by a uniform row-and-column exchange argument, which would make Theorem B independent of finite computation.
If the principal component's simplicial complex is shellable for all parameters, its Hilbert series would be computable purely from lattice-path counts in an $m$-by-$n$ grid, connecting the conjecture to path enumeration.

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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 a referee's time, but not because the proofs are airtight as written. The genuinely new content: Theorems A and B compute Hilbert series for jet schemes of determinantal varieties in new cases, and Theorem C gives explicit polynomials defining irreducible components, refining Docampo's orbit-closure description. The combinatorial shellability approach is carried out in real detail: four facet families, a shelling order, and h-vector counts. If the Hilbert series formula in Theorem B is right, the square relation for 3-by-3 minors is a nice piece of evidence for Conjecture 8.8.

The soft spot is in the middle of the proof. Section 4 asserts the five families a-e form a Grobner basis for I^{3,n}_{3,1} for all n. The verification is delegated to a Singular run for n=4,...,9 with no code, logs, or certificates. The reduction from those finite cases to all n uses Lemma 4.2 of [2], but as quoted the lemma does not say the replacement preserves the row/column order that defines the term order. Without that hypothesis, relabeling can change leading monomials and the reduction does not follow. Everything after, including the facet description of Delta_0, the shelling, and the Hilbert series, depends on this leading ideal, so Theorem B is not established as written. This is an addressable gap rather than a demonstrated error; the formula is plausible and the facet shapes look coherent.

Theorem A has a smaller but similar hole: the one-paragraph proof assumes the supports D_0,...,D_k of the leading monomials are pairwise disjoint, with no argument given. The claim may be true, but it is load-bearing for the shelling.

Section 8, Proposition 8.10, is garbled: the determinant expansion and the inequality chain near the end need rewriting, with variables misused. The statement might be correct, but I could not verify it from the text. There are also surface typos, e.g., "f rac2" in the h-vector section, which a referee should ask to be cleaned up.

The citation pattern looks fine; the authors build on [2], [5], [12], [13], [14] appropriately. The paper is for specialists in jet schemes and determinantal varieties. The Hilbert series formulas and the explicit component polynomials are concrete enough to be useful, and Conjecture 8.8 gives a clear target. I would not cite it in my own work until the Grobner basis check is reproducible and the reduction to all n is justified; the result is promising but unverified. A serious referee should engage; the gaps are fixable and the paper has real content.

Referee Report

5 major / 4 minor

Summary. The paper studies jet schemes of determinantal varieties. Theorem A treats the square case r=m=n, proving that the Stanley-Reisner complex associated to the leading ideal of I^{m,m}_{m,k} is shellable and computing its Hilbert series as (1-z^m)^{k+1}/(1-z)^{m^2(k+1)}. Theorem B treats the first jet scheme of the 3×n determinantal variety: it asserts that five explicit polynomial families form a Gröbner basis for I^{3,n}_{3,1}, describes the facets of the associated simplicial complex, proves shellability, and derives the Hilbert series as the square of the Hilbert series of the base determinantal variety L^{3,n}_3. Theorem C proposes explicit polynomial families defining the irreducible components of L^{m,n}_{r,p-1}, based on the orbit description in [12]. The paper also formulates Conjecture 1.2/8.8 that the principal component of a general jet scheme has shellable initial complex and Hilbert series equal to the (k+1)-st power of the Hilbert series of L^{m,n}_r.

Significance. If Theorem B is correct, it provides the first square relation for jet schemes defined by 3×3 minors and gives concrete evidence for the general conjecture about principal components. The explicit Gröbner basis and the combinatorial facet description would also be useful tools for further study of determinantal jet schemes. The paper contains no fitted free parameters and its main formulas are explicit and falsifiable. However, the proofs as written are not fully verifiable: the Gröbner basis assertion depends on an unreported computer computation, the shelling proof is garbled in places, and at least one key combinatorial assertion in Section 3 is unproved. The significance of the results is therefore real but conditional on closing these verification gaps.

major comments (5)

[Section 4] The Gröbner basis claim for all n is not established. The paper states that Singular was used to verify the cases of matrices of size 3×4 up to 3×9 and that this suffices, but no code, logs, outputs, or certificates are provided. More importantly, the reduction from these finite cases to all n uses Lemma 4.2 (Lemma 2.5 of [2]), which as quoted only says that replacing a submatrix T by another submatrix T' of the same size preserves the S-polynomial reduction. It does not state that the row/column order defining the monomial order of Section 4 is preserved under the relabeling, so the leading monomials can change and the reduction need not follow. Since Sections 5–7 build the facet description and the Hilbert series of Theorem B entirely from this claimed leading ideal, this is a load-bearing gap. The authors should include a reproducible Singular script and a proof that Lemma 4.2 applies with the fixed monomial order, or give a direct argument for all n.
[Section 3, Proposition 3.6] The proof of shellability for the complex in Theorem A asserts that the sets D_0,...,D_k of variables appearing in the leading monomials lm(f^(0)),...,lm(f^(k)) are pairwise disjoint, but no proof or reference is supplied. This assertion is used to conclude that every facet complement has the form {v_0,...,v_k} with v_i ∈ D_i, and the Hilbert series computation in Theorem 3.9 relies on the resulting counts of elements in each D_i. Without a proof of disjointness, the shelling order and the formula (1-z^m)^{k+1}/(1-z)^{m^2(k+1)} are unsupported.
[Section 5] The paper says: 'Since I^{3,n}_{3,1} is a radical ideal, it corresponds to an abstract simplicial complex ∆_{LM(I^{3,n}_{3,1})}'. This implication is not valid in general: the initial ideal of a radical ideal need not be radical. To define ∆0 as the Stanley-Reisner complex of LM(I^{3,n}_{3,1}) one must prove that LM(I^{3,n}_{3,1}) is squarefree (equivalently, that the leading monomials of the Gröbner basis are squarefree). This is a prerequisite for the entire facet characterization in Section 5 and for the shelling and Hilbert series arguments that follow.
[Section 6, especially case (2) for Q ∈ D̂] The shelling verification is not currently readable or verifiable. The proof contains garbled sentences, for example: 'when x2,r+1 is notin Q, it is sufficient to replace it with x1,q+1. 2,r+1 ∈ Q, it means that x2,r+1 = x2,l2' and similar fragments in the case analysis. The descriptions of the sets c(Q) and the argument that every P < Q contains an element of c(Q) are asserted in a dense and error-prone manner. Since Section 7 computes the h-vector and the Hilbert series from these c(Q) descriptions, the proof of Theorem B is not established as written. The authors need to rewrite the case analysis completely and carefully, or replace it with a machine-checkable shelling verification.
[Section 8, Proposition 8.10] The proof that the smaller family Ω'_λ generates Ω_λ is not coherent. The displayed inequality argument concludes with '(q0−1)⌈i_{q0−1}/(q0−1)⌉ ≥ i_{q0−1}+q0−1', which is not a contradiction; it is an identity-like claim. Moreover, the final sentence of the induction says 'Ω'_{λ,q} can be generated by Ω_{λ,q}', which is the reverse of what must be shown. Consequently Theorem C and the subsequent assertion that the projected family Ω'_λ defines the irreducible component are not proved.

minor comments (4)

[Section 7] Several displayed formulas are corrupted: for example, the count for |c(Q)|=3 in family B is rendered as 'f rac2(n − 1)(n − 2)(n − 3)3' instead of a proper fraction. Please proofread all formulas and ensure they compile correctly.
[Throughout] The paper uses 'Gröbner bases' both as a singular and as a plural noun. Please use 'Gröbner basis' for a single generating set and 'Gröbner bases' only for plural.
[Section 3, Corollary 3.10] The Hilbert polynomial formula contains binomial coefficients that may have negative upper or lower entries; please state the convention used for such binomial coefficients.
[Section 6] The diagrams that illustrate the facets are difficult to parse because the paths are not consistently labelled with the parameters (g1,g2,f1,f2, etc.). Adding explicit labels or a table of the facet families would substantially improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem B is computed from external Gröbner-basis results and a finite computer check, with no fitted parameter or self-referential definition.

full rationale

The derivation chain is not circular. Theorem A uses Proposition 3.1 of [14] for the Gröbner basis of I^{m,m}_{m,k} and Proposition 3.7 of [4] for the Hilbert series of a shellable Stanley-Reisner ring; these are independent external results, and the shelling is proved directly in Section 3. Theorem B is established by defining five explicit polynomial families (a,b,c,d,e), proving membership in I^{3,n}_{3,1} (Proposition 4.1), invoking the external Lemma 4.2 of [2] to reduce the S-pair verification to matrices of size 3x4 through 3x9, and reporting a Singular verification for those finite cases in Section 4. The facet description and shelling in Sections 5-6 are then developed from the leading ideal, and Section 7 counts h-vector contributions and derives the Hilbert series. The square relation is an output of this computation, not an input: the numerator coefficients are computed from the shelling data and only afterwards identified with the square of the base Hilbert series. No parameter is fitted to the target data and no result is defined in terms of its own conclusion. The main weakness is a reproducibility gap (the Singular computation for n=4,...,9 is not accompanied by scripts, logs, or certificates, and the quoted Lemma 4.2 does not explicitly state that the term order is preserved under relabeling), but this is a verification or correctness concern, not circular reasoning. There are no self-citations by the present authors, and no load-bearing claim is reduced to a citation of the authors' own prior work.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters and no invented entities. The burden is carried by external theorems and by two unverified internal assertions: the Singular computation in Section 4 and the pairwise disjointness of D_i in Section 3.

assumptions (8)

standard math Standard definitions and properties of jet schemes, Gröbner bases, Stanley-Reisner rings, and shellability
Used throughout Sections 2 through 7 as background.
domain assumption Docampo's orbit classification and irreducible component theorem for jet schemes of determinantal varieties
Theorem 1.1 and Propositions 8.1 through 8.5 from [12] are the foundation for Section 8.
domain assumption Košir-Sethuraman's Gröbner basis theorem for L^{m,m}_{m,k}
Proposition 3.1 from [14] supplies the initial ideal used in Theorem A.
domain assumption Lemma 4.2 of Košir and Sethuraman [2] on replacing submatrices
Used in Section 4 to reduce the Gröbner basis check for arbitrary n to matrices of size at most 3 by 10.
ad hoc to paper Singular computation verifies the Gröbner basis for n = 4,...,9
The paper states this computational verification but provides no code, logs, or certificates, so the reader cannot independently check it.
ad hoc to paper The sets D_i of variables in the leading monomials of f^(i) are pairwise disjoint
Asserted without proof in Section 3 during the proof of Proposition 3.6; the shelling and Hilbert series of Theorem A depend on it.
domain assumption Proposition 5.1 from [4] and Lemma 7.1 from [5] on paths and lattice path counts
Used to characterize facets and count c(Q) in Sections 5 and 7.
domain assumption Proposition 8.3 from [12]: λ ◁ μ iff closure(C_μ) is contained in closure(C_λ)
Used in Theorem C to identify zero sets of polynomial families with orbit closures.

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Pith. "Pith review of On jet schemes of determinantal varieties." pith.science (2026). https://pith.science/paper/7CPYABBW

@misc{pith2026250622898,
  author       = {Pith},
  title        = {Pith review of: On jet schemes of determinantal varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CPYABBW}},
  note         = {Machine review of arXiv:2506.22898}
}

abstract

Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the $k$-th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix as $\mathscr{L}^{m,n}_{r,k}$. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are 3 while $k$ is 1, we establish a correspondence between the defining ideals of $\mathscr{L}^{m,n}_{r,k}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of $\mathscr{L}^{m,n}_{r,k}$ accordingly. Moreover, for general $\mathscr{L}^{m,n}_{r,k}$, \cite{12} provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components. Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Hilbert series of second order jets of determinantal varieties
math.AG 2025-07 conditional novelty 6.0 of 10

For the 2×n rank-1 determinantal variety, the Hilbert series of its second jet scheme equals the cube of the original Hilbert series, specifically ((1+(n-1)z)/(1-z)^(n+1))^3.

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Works this paper leans on

17 extracted references · 14 canonical work pages · cited by 1 Pith paper

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Pith tools

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