{"id":"9cb86bf7-c8f1-4b39-afde-3787b0c316d8","arxiv_id":"2507.00681","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper computes the Hilbert series of the second-order jet scheme of the variety of 2×n matrices of rank at most one, obtaining a simple closed form: it is exactly the cube of the Hilbert series of the original variety. The result provides new evidence for a conjectured power-law pattern for jet schemes of determinantal varieties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central Hilbert-series formula is not independently verifiable as written: the Gröbner-basis step rests on an undocumented SINGULAR run, and the shelling proof is an abbreviated boundary-case analysis.","rationale":"After checking the arithmetic of §6, the h-vector totals do reproduce the claimed numerator for n≥3, and the n=2 case is consistent, so I found no algebraic contradiction in the final formula itself. The concern is about warrant: the proof's two load-bearing steps, namely the Gröbner-basis assertion and the shelling case analysis, are not presented at a level that a reader can verify. The SINGULAR run is mentioned but not documented, and the shelling proof skips many boundary subcases and compresses them into tables. Because these are the exact steps from which the Hilbert series is derived, the theorem is conditional on their correctness. This matches the reader's assessment, so the verdict should remain conditional.","tokens_in":15364,"tokens_out":20790,"duration_ms":246596,"concrete_test":"Build one reproducible computational pipeline for n=3,...,7: with the grevlex order of §2, compute a Gröbner basis of I^{2,n}_{2,2} independently (e.g., in Macaulay2 or SINGULAR), extract its Stanley–Reisner facets, and check by exhaustive enumeration that the ∗-ordering of §5 is a shelling and that the resulting h-vector matches (1+(n−1)z)^3. If the leading ideal differs from L(Γ), or any facet pair violates the shelling condition, the central claim is falsified; if the pipeline passes for all tested n, the computational cornerstone and the small-n shelling argument are confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem depends on two linked claims. First, Γ in §3 is a Gröbner basis of I^{2,n}_{2,2}; the decisive verification for n=4,...,7 is asserted from a SINGULAR computation with no script or outputs, so the leading ideal L(I) and the facet classification in §4 cannot be reproduced from the manuscript. Second, Theorem 5.4 asserts that the ∗-ordering is a shelling; the proof is a compressed case analysis whose c1/c2 tables have many boundary cases (e.g., vanishing of A1 when a1=ar or a1=n−1,a2=n) and additive conditions that are stated without derivation. If any of these cases hides a pair P>Q with Q containing all of c(P), the shelling condition fails and the h-vector (1,3(n−1),3(n−1)^2,(n−1)^3) in Theorem 6.1 does not follow. The n=2 check is consistent, but it does not exercise the families that make the general proof delicate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15610,"tokens_out":8714,"duration_ms":99231,"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":[{"comment":"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.","section":"§3, after Lemma 3.2"},{"comment":"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.","section":"§4, Propositions 4.2 and 4.3"},{"comment":"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.","section":"§5, Theorem 5.4 and the vanishing/additive condition tables"},{"comment":"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.","section":"§6, h_j(T) counts"}],"minor_comments":[{"comment":"The running title on page 1 reads 'HILBER T SERIES OF SECOND ORDER JETS'; the spacing in 'HILBER T' should be corrected.","section":"Page 1, running title"},{"comment":"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.","section":"Definition 2.11"},{"comment":"In the proof, the sentence 'δP (p) = 1' uses the undefined symbol p; it should be 'δP (v) = 1'.","section":"§5, Proposition 5.3"},{"comment":"The condition for h3(¯A) is typeset as '1 ≤ a1 < ar ≤ n, ≤ a1 < a2 ≤ n'; the second inequality is missing its leading '1 ≤'.","section":"§6, h3(¯A)"},{"comment":"The family is introduced as 'F∗' but later is sometimes called 'F'; please unify the notation.","section":"§4"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal. The main concern is verifiability, not novelty: the authors should be asked to provide the SINGULAR code and output, justify the reduction to n=4,...,7, and expand the shelling proof and the h-vector counts. With those additions, the paper could be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe paper computes the Hilbert series of the second jet scheme of the determinantal variety of 2x2 minors in a 2xn matrix. The main theorem gives the elegant formula ((1+(n-1)z)/(1-z)^{n+1})^3. This is genuinely new: it is the next case after (2,1) and (3,1) jets, and a concrete exact computation in a family where Hilbert functions are known to be hard. The paper also states the attractive conjecture that the Hilbert series of L^m,n_{r,k} is the (k+1)-th power of the Hilbert series of L^m,n_r, which is consistent with every known case.\n\nWhat the paper does well: it sets up a Gröbner basis candidate Γ, identifies the leading ideal, classifies the facets of the Stanley-Reisner complex into five families, and proves shellability via an explicit ordering. The n=2 check works, the h-vector matches the formula, and the combinatorial classification of facets is substantial. As far as I can tell, the structure is correct.\n\nThe soft spots are real and concentrated in two places. First, the proof that Γ is a Gröbner basis for all n relies on an undocumented SINGULAR computation for n=4,...,7: no script, no output, no file. For a paper whose main theorem is a computation, that is a gap. Second, the shelling proof in Theorem 5.4 is a compressed case analysis. The vanishing and additive condition tables are stated without derivation, and several boundary cases are not shown explicitly. If any of those cases permits Q to cover c(P), the shelling fails and the h-vector does not follow. This is load-bearing, not cosmetic.\n\nI do not think the flaws are fatal. The formula is plausible, the method is sound, and the missing details are likely fillable. But as written the paper is not fully rigorous. A referee should ask for the SINGULAR inputs and outputs and for a fuller shelling verification. The conjecture is worth stating even without proof.\n\nThis is for specialists in jet schemes, determinantal varieties, and Stanley-Reisner theory. The paper deserves a serious referee; I would send it out, expecting revision.\n\nRecommendation: conditional accept.","headline":"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.","tokens_in":16112,"tokens_out":2953,"would_cite":true,"duration_ms":30855,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M12","14E18","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper computes the Hilbert series of the second-order jet scheme of the 2×n determinantal variety, showing it equals $((1+(n-1)z)/(1-z)^{n+1})^3$.","keywords":["determinantal varieties","jet schemes","shellability","Hilbert series","Stanley-Reisner complex","Gröbner basis","Cohen-Macaulay","2x2 minors"],"falsifier":"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.","tokens_in":15151,"feed_emoji":"🧊","tokens_out":8536,"duration_ms":81109,"temperature":0.7,"pith_summary":"The paper computes the Hilbert series of the coordinate ring of the second-order jet scheme of the 2×n determinantal variety (the variety of 2×n matrices of rank at most 1). The central result is a closed form: the series equals $((1+(n-1)z)/(1-z)^{n+1})^3$. To get there, the authors prove that the Stanley-Reisner complex of the leading ideal is shellable, which also implies the quotient ring is Cohen-Macaulay of dimension $3n+3$. This matters because Hilbert functions of jet schemes are generally very hard to determine, and this is one of the few nontrivial cases where a complete answer is found.","feed_headline":"Jet scheme Hilbert series is a cube for 2×n matrices","feed_subtitle":"Shellability of the Stanley-Reisner complex yields a Cohen-Macaulay ring and a closed formula.","key_machinery":"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 $h_0=1$, $h_1=3(n-1)$, $h_2=3(n-1)^2$, $h_3=(n-1)^3$.","core_discovery":"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 $r,k,m,n$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Gröbner basis method for the 2×2 determinantal ideal mod t^2, including Lemma 3.2 used to reduce the Gröbner check to n=4,…,7.","marker":"[8]"},{"why":"Characterized the Stanley-Reisner complex of the leading ideal of I^{m,n}_{2,1} and proved its shellability, the pattern this paper extends to the second jet.","marker":"[6]"},{"why":"Gives the theorem that a shellable Stanley-Reisner complex is Cohen-Macaulay and converts the h-vector into the Hilbert series (Proposition 2.13 in the paper).","marker":"[1]"},{"why":"Computed the Hilbert series of L^{m,n}_{2,1}, whose square structure motivates the cube formula and the conjecture.","marker":"[5]"},{"why":"Provides the Hilbert series of classical determinantal varieties, the base case for the conjecture that jets' Hilbert series are powers.","marker":"[3]"}],"fun_headline_variants":["Hilbert series for 2×n jet scheme: a perfect cube","Jet Hilbert series cubed: (1+(n-1)z)/(1-z)^(n+1)","Second-order jets of 2×n minors: Hilbert series is a cube","Cohen-Macaulay second jets: Hilbert series cubed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hilbert series for 2×n jet scheme: a perfect cube","Jet Hilbert series cubed: (1+(n-1)z)/(1-z)^(n+1)","Second-order jets of 2×n minors: Hilbert series is a cube","Cohen-Macaulay second jets: Hilbert series cubed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2138,"prompt_tokens":830,"completion_tokens":1308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":1223}},"tokens_in":446,"tokens_out":1308,"duration_ms":11998,"temperature":1.0,"reasoning_tokens":1223,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:10:10.553107+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Koˇ sir and B","cited_arxiv_id":null,"evidence_quote":"Supplies the Gröbner basis method for the 2×2 determinantal ideal mod t^2, including Lemma 3.2 used to reduce the Gröbner check to n=4,…,7."},{"cited_title":"Jonov, Initial complex associated to a jet scheme of a determinantal variety, J","cited_arxiv_id":null,"evidence_quote":"Characterized the Stanley-Reisner complex of the leading ideal of I^{m,n}_{2,1} and proved its shellability, the pattern this paper extends to the second jet."},{"cited_title":"Bruns and A","cited_arxiv_id":null,"evidence_quote":"Gives the theorem that a shellable Stanley-Reisner complex is Cohen-Macaulay and converts the h-vector into the Hilbert series (Proposition 2.13 in the paper)."},{"cited_title":"Ghorpade, B","cited_arxiv_id":null,"evidence_quote":"Computed the Hilbert series of L^{m,n}_{2,1}, whose square structure motivates the cube formula and the conjecture."},{"cited_title":"Conca and J","cited_arxiv_id":null,"evidence_quote":"Provides the Hilbert series of classical determinantal varieties, the base case for the conjecture that jets' Hilbert series are powers."}],"review_version":1}