REVIEW 4 major objections 6 minor 10 references
Hilbert series of second order jets of determinantal varieties
T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read This paper computes the Hilbert series of the second-order jet scheme of the 2×n determinantal variety, showing it equals $((1+(n-1)z)/(1-z)^{n+1})^3$.
desk verdict A plausible new Hilbert-series computation for second jets of 2x2 minors, with a real proof gap in the shelling argument and an undocumented SINGULAR step; worth refereeing, but the details need to be supplied before the theorem is fully supported. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The proof works through the Stanley-Reisner complex $\Delta_0$ of the leading ideal of $I^{2,n}_{2,2}$. Its facets fall into five families, called $\bar{A},\bar{C},\bar{D},\bar{E},\bar{F}$, each described by three column 'axes'; the authors define a total order on facets, the $*$-ordering, and check that every facet in the order attaches to earlier facets through exactly one new vertex, which is the defining property of a shelling. A standard theorem on shellable complexes then converts the numbers $h_j$ of facets attaching through $j$ new vertices into the Hilbert series. The bulk of the paper is a case analysis, summarized in vanishing and additive tables, that verifies these attachment counts and yields $h_0=1$, $h_1=3(n-1)$, $h_2=3(n-1)^2$, $h_3=(n-1)^3$.
What would settle it
Compute the Hilbert series of $L^{2,4}_{2,2}$ with an independent computer algebra system (for example, one other than SINGULAR) and check whether the leading ideal matches the ideal generated by the seven families of leading monomials listed in Section 3; a mismatch for $n=4$ would break the reduction to all $n$. Alternatively, test the shelling condition directly on the pairs of facets covered by the additive conditions table, looking for a pair where the later facet's attachment set has size other than one.
Extended reading notes
Core claim
The main theorem asserts that, for the ideal $I^{2,n}_{2,2}$ generated by the degree-0, -1, and -2 coefficients of the $2\times2$ minors of a $2\times n$ matrix, the Stanley-Reisner complex of the leading ideal is shellable and the Hilbert series of the quotient ring—the coordinate ring of the second jet scheme $L^{2,n}_{2,2}$—is $((1+(n-1)z)/(1-z)^{n+1})^3$. Equivalently, the ring is Cohen-Macaulay of dimension $3n+3$ with $h$-vector $(1,3(n-1),3(n-1)^2,(n-1)^3)$. The same formula matches the pattern that the Hilbert series of the $k$-th jet is the $(k+1)$-th power of the Hilbert series of the original variety, which the authors state as a conjecture for all $r,k,m,n$.
Load-bearing premise
The proof assumes that the case-by-case checks in Section 5, which verify that each later facet attaches through exactly one new vertex, are complete and correct, and that the unshown SINGULAR computation for $n=4,\dots,7$ really determines the Gröbner basis; if either assumption fails, the Hilbert series formula does not follow.
Editorial extensions
If this is right
- The Hilbert series of $L^{2,n}_{2,2}$ is now known exactly for every $n$: $((1+(n-1)z)/(1-z)^{n+1})^3$.
- The coordinate ring of the second jet scheme is Cohen-Macaulay of dimension $3n+3$, a property that does not follow from the usual geometric description of jet schemes.
- The $h$-vector of the quotient is $(1,3(n-1),3(n-1)^2,(n-1)^3)$, which is the cube of the $h$-vector $(1,n-1)$ appearing in the first-order jet case.
- The formula confirms the authors' conjecture in this case: the Hilbert series of $L^{2,n}_{2,2}$ is the third power of the Hilbert series of $L^{2,n}_2$.
- The shellability proof supplies a combinatorial model (the $*$-ordering on five facet families) that can be used to extract further algebraic information about the jet scheme, such as a Stanley decomposition.
Reading between the lines
- If the conjecture extends beyond the computed cases, it predicts that the Hilbert series of any jet scheme of a determinantal variety is a power of the original's Hilbert series; a natural combinatorial interpretation would be that the Stanley-Reisner complex of the $k$-th jet's leading ideal is a join of $k+1$ copies of the original complex.
- The reduction of the Gröbner basis check to $n=4,\dots,7$ suggests a general finite-verification strategy: for fixed $r$ and $k$, the number of columns $n$ that must be checked by computer is bounded, so the conjecture could be tested automatically for the next small cases such as $(r,k)=(3,2)$ or $(2,3)$.
- The case analysis here is tailored to $2\times n$; adapting the facet-family method to three rows would likely require many more facet types, so a purely combinatorial proof of the general conjecture would need a different organizing principle.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the second-order jet scheme L_{2,2}^{2,n} of the 2×n determinantal variety. It constructs an explicit generating set Γ for the jet ideal, asserts that Γ is a Gröbner basis (with a SINGULAR check for n=4,...,7), classifies the facets of the Stanley-Reisner complex of the leading ideal into five families, defines an ordering of these facets, proves shellability, and derives the Hilbert series ((1+(n-1)z)/(1-z)^{n+1})^3. The proof proceeds through standard Stanley-Reisner theory, and the main result is an elegant closed form consistent with the conjectural power pattern.
Significance. If correct, the main theorem is a significant contribution: it gives one of the few explicit Hilbert series for a higher-order jet scheme of a determinantal variety, establishes Cohen-Macaulayness of the second jet coordinate ring (dimension 3n+3, h-vector (1,3(n-1),3(n-1)^2,(n-1)^3)), and provides concrete support for Conjecture 6.2. The strategy—reducing the Gröbner-basis verification to small n, classifying facets combinatorially, and using shellability—is appropriate. The paper does not fit the Hilbert series from data, and the conjecture is not used in the proof. However, as written the two load-bearing steps are not fully documented or demonstrated.
major comments (4)
- [§3, after Lemma 3.2] The reduction to the cases n=4,...,7 is not justified, and the computer verification is not reproducible. The lemma concerns replacing a common submatrix by another submatrix of the same size; to cover all pairs α,β∈Γ one would need to know why the 2×8 case is not needed, but the text simply says 'In conclusion, we only have to prove that Γ is a Gröbner basis for the case 4 ≤ n ≤ 7.' Moreover, no SINGULAR script, log, or list of S-polynomial checks is provided, so the assertion 'The results confirmed that L(I_{2,2}^{2,n}) is the ideal generated by all leading monomials of polynomials in Γ' cannot be checked. This is load-bearing because the leading ideal and all subsequent facet classifications depend on it.
- [§4, Propositions 4.2 and 4.3] The facet classification is presented as a sequence of 'straightforward to verify' assertions and compressed claims. For example, in Proposition 4.2 the claims in case (i) are proved only in outline, and Proposition 4.3 relies on a separate 'Claim' with a one-paragraph proof. Since the five families ¯A, ¯C, ¯D, ¯E, ¯F are used in the shelling proof and in the Hilbert-series computation, the exhaustion argument needs to be written out in full detail rather than left to the reader.
- [§5, Theorem 5.4 and the vanishing/additive condition tables] The shelling proof is a compressed case analysis. The vanishing conditions and additive conditions tables list boundary cases (for instance, A1 vanishes when a1=ar or a1=n−1,a2=n, but is restored as a c2-point under additive conditions), yet the same-family part of the proof of Theorem 5.4 asserts facts such as 'A1 always exists' without deriving how the additive conditions are used. The 'Claim' in Proposition 4.3 and the abbreviated cases for P∈¯D,Q∈¯E also leave several subcases implicit. Because the conclusion that Q cannot cover c(P) is exactly the shelling condition, these missing derivations are load-bearing.
- [§6, h_j(T) counts] The h_j(T) values are asserted without derivation. For example, h2(¯C)=n(n−1) is given as a sum of three binomial terms with conditions that are not explained, and h2(¯D) and h3(¯F) involve cancellations of binomial terms whose combinatorial origin is not shown. Since these numbers are the only bridge from the shelling to Theorem 6.1, all counts need to be justified. The paper also does not provide an explicit independent check for n=2 (the complete-intersection case) or for n=3, which would help confirm the formula before the delicate shelling arguments are invoked.
minor comments (6)
- [Page 1, running title] The running title on page 1 reads 'HILBER T SERIES OF SECOND ORDER JETS'; the spacing in 'HILBER T' should be corrected.
- [Definition 2.11] The quantifier in the shelling definition is garbled: 'there exist v ∈ Fj − Fi and k < j satisfying Fj − Fk = {v} for all 1 ≤ i ≤ j ≤ e' is not the standard condition. Please restate it precisely.
- [§5, Proposition 5.3] In the proof, the sentence 'δP (p) = 1' uses the undefined symbol p; it should be 'δP (v) = 1'.
- [§6, h3(¯A)] The condition for h3(¯A) is typeset as '1 ≤ a1 < ar ≤ n, ≤ a1 < a2 ≤ n'; the second inequality is missing its leading '1 ≤'.
- [§4] The family is introduced as 'F∗' but later is sometimes called 'F'; please unify the notation.
- [References] Reference [2] is listed as 'to appear' with an arXiv number; please update with the final publication data if it has appeared by the time of publication.
Circularity Check
No circular dependency: the Hilbert series is derived from an explicit Gröbner basis, facet classification, and shelling proof, not from fitting or from the stated conjecture.
full rationale
The derivation chain is self-contained with respect to the claimed Hilbert series. The main theorem's formula is obtained from the h-vector counts in Section 6, which in turn follow from the explicit c1/c2 vanishing and additive tables in Section 5 and the facet classification in Section 4. These tables are computed from the definition of the ∗-ordering and the combinatorial structure of the leading ideal, not by assuming the final Hilbert series. The conjecture in Section 6 is stated after the theorem and is not invoked in any proof. The only self-citation is reference [2] by two of the current authors, and it is used solely to quote a comparison result for a different case (L^{3,n}_{3,1}); it does not support any load-bearing step of the present computation. The SINGULAR verification for n = 4,...,7 is asserted without a reproducible script, which is a verifiability or correctness-risk concern, but it is not circular: the computation checks the Gröbner-basis claim directly rather than encoding the target Hilbert series. No step was found in which a variable is defined in terms of the desired output, in which a fitted parameter is renamed as a prediction, or in which a self-citation supplies the essential premise. The paper's core derivation is therefore not circular; the score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Lemma 3.2 from [8] applies: a Gröbner basis verification can be reduced from arbitrary n to the cases 4 ≤ n ≤ 7.
- ad hoc to paper The SINGULAR computations for n = 4, 5, 6, 7 confirm that L(I^{2,n}_{2,2}) is generated by the leading monomials of Γ.
- ad hoc to paper The h_j(T) values in Section 6, derived from the vanishing and additive condition tables, are correct.
Cite this review
Pith. "Pith review of Hilbert series of second order jets of determinantal varieties." pith.science (2026). https://pith.science/paper/5ODRPBJG
@misc{pith2026250700681,
author = {Pith},
title = {Pith review of: Hilbert series of second order jets of determinantal varieties},
year = {2026},
howpublished = {\url{https://pith.science/paper/5ODRPBJG}},
note = {Machine review of arXiv:2507.00681}
}
abstract
In this paper, we will investigate the jet schemes of determinantal varieties. It is quite often the case that the geometric information concerning the jet schemes of an algebraic variety can be described, but the more refined algebraic information is quite mysterious. For example, it is known that computing the Hilbert function associated to a natural grading on these jet schemes is a very hard problem. The present paper handles a few such computations. It succeeds in computing the Hilbert functions of the second order jet schemes in the case of maximal minors of a $2\times n$ matrix.
Reference graph
Works this paper leans on
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On jet schemes of determinantal varieties
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Reviewed August 6, 2026 · model on record in the stance chip above.
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