{"id":"97e00721-5316-4709-827d-b5a43567d973","arxiv_id":"1908.03892","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For generic determinantal varieties, the log canonical threshold is preserved under generic linkage.","lead":"The paper proves that a generic rank-bounded matrix variety and its generic link have the same log canonical threshold, a singularity invariant. It gives the first known non-trivial class where an earlier inequality is actually an equality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the only soft spot is the imported local linkage claim [Niu14, Claim 3.1.2(3)], which I have no concrete reason to doubt.","rationale":"The reader's weakest assumption correctly identifies [Niu14, Claim 3.1.2(3)] as the point where the proof is most exposed. I agree that this is the only step that could silently invalidate Proposition 3.1 and hence Theorem 1. However, the concern is about reliance on a published result rather than an identified internal contradiction. The rest of the argument is coherent: Proposition 2.9 supplies the degree information used to compute orders, Lemma 2.8 gives the required independence of the generating set, Lemma 4.1 handles the exceptional cases, and the final comparison via the Niu inequality is valid. Since no concrete failure of the cited claim is apparent, and since the proof structure would work if that claim holds, I would not change the reader's ACCEPT verdict. The proposed Macaulay2 test is a worthwhile independent check that the central imported step is correct.","tokens_in":10570,"tokens_out":32675,"duration_ms":362156,"concrete_test":"Use Macaulay2 to test the critical imported claim in a nontrivial case: take m=5, n=5, r=3 and the first chart U11. Build the coordinate ring with the 4 by 4 matrix f of new coordinates, let I=I_2(f), and let J be generated by the nine transformed generic equations f_j/y. Check that codim J=9, that J is a complete intersection, and that J:I equals the generic link of I with respect to the same generating set. If these agree with Proposition 2.9(c), the citation is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the order computation in Proposition 3.1 and the reduction in Theorem 1, I do not find a load-bearing flaw. The only step that could break the chain is the cited local claim [Niu14, Claim 3.1.2(3)]: on each affine chart of A'_{i-1}, the transformed equations f_1/y,...,f_c/y must form a regular sequence and I_{Y_{i-1}} = I_{V_{i-1}}:I_{X_{i-1}} must be a generic link of I_{r_i}(M'). This is exactly what lets Proposition 3.1 compare orders via Lemma 2.8 and Proposition 2.9(c). If that claim failed, the formula ord_{E'_i}(Y)=min{r_i,(n_i-r_i+1)(m_i-r_i)(r_i-1)} could fail. However, the cited claim is published, is consistent with Hochster's grade reduction, and the rest of the paper's logic is internally coherent; I see no countervailing evidence. The Johnson log resolution data and the three exceptional cases in Lemma 4.1 also check out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a generic determinantal variety X defined by the r-minors of an m by n matrix of indeterminates, and for its generic link Y in an affine extension, the log canonical thresholds of X and Y are equal. The proof combines Niu's general inequality lct(A',Y) >= lct(A,X) with Johnson's explicit log resolution of determinantal varieties. The main local statement is Proposition 3.1, which computes the order of the generic link along each divisor E'_i as min{r_i, (n_i-r_i+1)(m_i-r_i)(r_i-1)}. Theorem 1 then follows by comparing these orders with the orders of X and by treating three exceptional parameter ranges in Lemma 4.1 and Proposition 4.2.","tokens_in":10766,"tokens_out":38684,"duration_ms":414536,"significance":"If the proof is correct, this is the first known nontrivial class of varieties for which equality holds in the generic-link log canonical threshold inequality. The paper is concise and well structured: it leverages a clean combination of explicit resolution data, linkage theory, and an order computation, and it avoids any fitted constants or numerical normalization. The explicit formula for the order of a generic determinantal link along the divisors of the Johnson resolution is itself a useful contribution. The main proof is coherent modulo two imported results, Johnson's resolution description and a local linkage claim from Niu's paper, both of which are published.","major_comments":[{"comment":"The step 'By Proposition 2.9(c), the ideal I_W is generated by elements of degrees r_i and (n_i-r_i+1)(m_i-r_i)(r_i-1). This proves the claim of I_W in I_Z^q \\ I_Z^{q+1}' is too terse. To establish the non-containment, one must argue that some nonzero homogeneous generator of degree q (or of degree r_i when q = r_i) is not contained in I_Z^{q+1}. This uses that I_Z is generated by the entries of M' in a polynomial ring over the degree-zero base ring \\tilde S_{i-1}. The argument is correct, but it should be written out, and the grading convention should be stated explicitly, because otherwise the phrase 'elements of degree 0' in the case r_i = 1 is potentially confusing.","section":"Section 3, Proposition 3.1"}],"minor_comments":[{"comment":"There are several typos in the abstract and introduction, including 'determinental', 'projctive', 'coornidates', 'quesitons', and 'forth-coming'; a careful proofreading pass is needed.","section":"Abstract and Introduction"},{"comment":"The notations Y_{i-1} and V_{i-1} are used before they are defined; please add a sentence specifying that these are the strict transforms of Y and V under the composition of the first i-1 blow-ups.","section":"Section 3, Proposition 3.1"},{"comment":"The equality ord_{E'_i}(Y) = ord_{E'_i}(Y_{i-1}) is asserted without comment; a brief justification would help, since E'_i is an exceptional divisor of the i-th blow-up and the order is computed in the local ring of A'_{i-1}.","section":"Section 3, Proposition 3.1"},{"comment":"In the first generator of the ideal defining Y, the term x_3^2 appears to be a typo and should likely be x_3^3 to match the defining ideal of X; the displayed lct values should be checked with the corrected generator.","section":"Example 2.4(a)"},{"comment":"Lemma 4.1 is stated without proof; since the equivalence for the shifted parameters is used directly in the proof of Theorem 1, a short proof or a reference would improve readability.","section":"Lemma 4.1"},{"comment":"The phrase 'I_Y in m \\ m^2' is nonstandard for an ideal; it should read 'I_Y subset m \\ m^2' or, equivalently, that the order of I_Y with respect to m is exactly 1.","section":"Remark 4.4"}],"recommendation":"minor_revision","confidential_remarks":"The main mathematical claim appears correct, and the proof structure is sound. The only load-bearing step that needs care is the exact-order justification in Proposition 3.1, which is currently compressed; it can be fixed with an added paragraph. The paper also depends on a claim from the third author's earlier paper [Niu14, Claim 3.1.2(3)], and while I see no circularity or concrete reason to doubt it, the authors should state that claim precisely so that the reader can verify that it applies to the specific affine charts of the Johnson resolution. This is a short, well-focused paper that fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe headline: this is the first nontrivial class where generic linkage is shown to preserve log canonical thresholds. The proof is clear and, as far as I can see, correct.\n\nWhat's actually new: Niu's inequality lct(A',Y) ≥ lct(A,X) was known, and strictness examples existed, but no nontrivial equality class. Theorem 1 fills that gap for generic determinantal varieties. The machinery is Johnson's explicit log resolution plus linkage theory, but the new piece is Proposition 3.1, a closed formula for ord_{E'_i}(Y). That computation is the heart of the paper and it hangs together.\n\nWhat the paper does well: the preliminaries are efficient; Example 2.4 usefully shows equality and strictness can both occur in codimension two; Proposition 2.9 gives the degree generators needed for the order computation; and the reduction in Theorem 1 is clean once Proposition 3.1 is accepted. The three exceptional cases in Lemma 4.1 are correctly identified.\n\nSoft spots: the proof leans on two imported results: Johnson's log resolution data and, more importantly, [Niu14, Claim 3.1.2(3)], which supplies the local statement that after blowing up, the transformed equations remain a regular sequence and the transformed link is again generic. That's a load-bearing black box. I have no concrete counterexample or reason to doubt it, and it is consistent with Hochster's grade reduction, but the paper does not re-prove it. A referee should check that statement against Niu's thesis carefully. There's also a minor notational typo in Lemma 4.1: the second equivalence writes mi − r where it should be mi − ri (the two differ by i−1). This doesn't affect the argument, but it should be fixed. Finally, the proof that a log resolution of (A',Y) extending the given resolution exists is cited rather than shown; standard, but the reader has to take it on faith.\n\nBottom line: the central argument holds up, and the result is a genuine, modest advance. The paper is short and readable, and it deserves a serious referee. I'd send it out.\n\nBest,","headline":"First nontrivial class where generic linkage preserves lct; proof is sound, with one imported claim worth checking.","tokens_in":11311,"tokens_out":3168,"would_cite":true,"duration_ms":28948,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J17","14M06","13C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Generic links preserve the log canonical threshold of determinantal varieties.","keywords":["log canonical threshold","generic linkage","determinantal variety","singularities of pairs","log resolution","order of ideal","linkage"],"falsifier":"Compute the log canonical threshold of the generic link of the $5\\times 3$ determinant variety with $r=2$ (matrices of rank at most 1). The paper's formula predicts $\\operatorname{lct}(A',Y)=15/2$, so any independent computation from a log resolution of the link yielding a different value would falsify the theorem.","tokens_in":10342,"feed_emoji":"🔗","tokens_out":9130,"duration_ms":84782,"temperature":0.7,"pith_summary":"Log canonical thresholds measure how singular a complex variety is: on a log resolution of the pair $(A,X)$, the threshold is the minimum of $(k_i+1)/a_i$, where $a_i$ is the order of vanishing of $X$ along an exceptional divisor $E_i$ and $k_i$ is the corresponding canonical discrepancy. This paper proves that for a generic determinantal variety $X$ — the variety of $m\\times n$ matrices of rank at most $r-1$ — the generic link $Y$ of $X$ has the same log canonical threshold as $X$ itself. Earlier work had established only the inequality $\\operatorname{lct}(A',Y)\\ge\\operatorname{lct}(A,X)$, which is strict for many other classes, such as hypersurfaces. Here equality is proved by computing the orders of $Y$ along the exceptional divisors of an explicit log resolution of $X$, reducing the comparison to a closed formula and an elementary case distinction.","feed_headline":"Generic links keep the log canonical threshold unchanged","feed_subtitle":"For determinantal varieties, generic linkage leaves the log canonical threshold unchanged, turning a known inequality into equality.","key_machinery":"The central object is the explicit log resolution of a generic determinantal variety: blow up $V(I_1(M))$ in the ambient affine space, then blow up the strict transform of $V(I_2(M))$, and continue up to $V(I_r(M))$; on an affine chart of the $i$-th blow-up, the situation repeats with parameters $(m-i+1,n-i+1,r-i+1)$. The generic link of $X$ is the residual ideal of a generic complete intersection formed from generic linear combinations of the generators of $I_X$, and the order comparison is carried by the formula $\\operatorname{ord}_{E'_i}(Y)=\\min\\{r_i,(n_i-r_i+1)(m_i-r_i)(r_i-1)\\}$, where $r_i=r-i+1$, $n_i=n-i+1$, $m_i=m-i+1$. This formula comes from Proposition 2.9(c), which identifies the minimal generating degree of the generic link, together with a local regular-sequence argument showing that after restriction to each affine chart the transformed complete-intersection equations still define a generic link of the transformed determinantal ideal. The cases where the minimum is smaller than $r_i$ are exactly the parameter ranges where the threshold equals the codimension.","core_discovery":"Let $X\\subset A=\\operatorname{Spec}\\mathbb C[M]$ be $V(I_r(M))$ for an $m\\times n$ matrix $M$ of indeterminates with $m\\ge n\\ge r$, and let $Y$ be the generic link of $X$ in $A'=A\\times\\operatorname{Spec}\\mathbb C[T]$. The paper's theorem states that $\\operatorname{lct}(A',Y)=\\operatorname{lct}(A,X)$, where $\\operatorname{lct}(A,X)=\\min_{0\\le t\\le r-1}\\frac{(m-t)(n-t)}{r-t}$. The proof works by pulling back the log resolution of $(A,X)$ — obtained by blowing up, in order, the transformed varieties $V(I_1(M)),\\dots,V(I_r(M))$ — to $A'$, and comparing $\\operatorname{ord}_{E'_i}(Y)$ with $\\operatorname{ord}_{E_i}(X)$. The comparison shows that the orders agree for $i=1,\\dots,r-1$ whenever $(n-r+1)(m-r)(r-1)\\ge r$, while along the last divisor one has $\\operatorname{ord}_{E'_r}(Y)=0$; in the complementary parameter range, $\\operatorname{lct}(A,X)$ equals the codimension of $X$, so the general inequality forces equality.","pith_inferences":["The same order-comparison method suggests that equality should persist under iterated generic links, since after one link the residual ideal in each affine chart remains a generic determinantal link; the first author's forthcoming work reportedly addresses higher generic links.","One could test the equality for other families with explicit log resolutions and known generic-link generating degrees, such as Pfaffian or symmetric determinantal ideals, using the same blow-up data if a factorizing resolution exists.","For small cases such as the $5\\times 3$ minors with $r=2$, the formula predicts $\\operatorname{lct}(A',Y)=15/2$, which could serve as a concrete independent check by computing a log resolution of the explicitly known generic link."],"forward_implications":["For every generic determinantal variety, the log canonical threshold of the generic link equals the log canonical threshold of the original variety.","In the parameter range $(n-r+1)(m-r)(r-1)