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Log canonical thresholds of generic links of determinantal varieties

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Pith's one-line read Generic links preserve the log canonical threshold of determinantal varieties.

desk verdict First nontrivial class where generic linkage preserves lct; proof is sound, with one imported claim worth checking. read the letter →

arxiv 1908.03892 v2 pith:YQBOEJWC submitted 2019-08-11 math.AG math.AC

Youngsu Kim , Lance Edward Miller , Wenbo Niu This is my paper · ORCID
classification math.AGmath.AC MSC 14J1714M0613C40
keywords logcanonicalthresholdgenericlinkagedeterminantalvarietysingularitiesofpairsresolutionorderideal
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

The whole comparison rests on the local fact, imported from earlier work, that after restricting to a coordinate chart of each blow-up, the transformed complete-intersection equations divided by the appropriate powers of exceptional coordinates still form a regular sequence and define a generic link of the transformed determinantal ideal; if that failed, the order formula for the generic link would break down.

Editorial extensions

If this is right

  • 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 $(n-r+1)(m-r)(r-1)<r$, which includes $r=1$, square matrices with $m=r$, and maximal minors of $m\times(m-1)$ matrices, the threshold equals the codimension, so equality with the generic link is forced.
  • In the nondegenerate range, the order of the link along the last exceptional divisor is zero, while along all earlier divisors the orders of $X$ and $Y$ agree, by Corollary 4.3.
  • The previously known inequality $\operatorname{lct}(A',Y)\ge\operatorname{lct}(A,X)$ is therefore an equality for this whole family, giving the first nontrivial class of varieties for which equality is known.

Reading between the lines

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

  • 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.
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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

1 major / 6 minor

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.

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 (1)
  1. [Section 3, Proposition 3.1] 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.
minor comments (6)
  1. [Abstract and Introduction] There are several typos in the abstract and introduction, including 'determinental', 'projctive', 'coornidates', 'quesitons', and 'forth-coming'; a careful proofreading pass is needed.
  2. [Section 3, Proposition 3.1] 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.
  3. [Section 3, Proposition 3.1] 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}.
  4. [Example 2.4(a)] 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.
  5. [Lemma 4.1] 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.
  6. [Remark 4.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and imports only independent external results, none of which assume the target equality.

full rationale

The paper's central comparison is Theorem 1, which proves lct(A',Y)=lct(A,X) for a generic determinantal variety X. The proof uses the inequality lct(A',Y) >= lct(A,X) from [Niu14, Prop. 3.7] as an input, not as the conclusion. The reverse inequality is obtained by explicit order computations: Proposition 3.1 computes ord_{E'_i}(Y)=min{r_i,(n_i-r_i+1)(m_i-r_i)(r_i-1)} using Proposition 2.9(c), a standard graded canonical-module and linkage degree computation, together with Johnson's explicit log resolution; Theorem 1 then compares these orders with ord_{E_i}(X)=r_i. The only results imported from the third author's prior work are [Niu14, Prop. 3.7] and [Niu14, Claim 3.1.2(3)]: the former is a general inequality for generic links and does not assume equality for determinantal varieties, and the latter is a local regular-sequence and linkage-preservation statement on blow-up charts, which likewise does not contain the equality being proved. These are parameter-free results with stated assumptions disjoint from the target theorem, so they constitute independent support rather than a self-citation chain. No fitted constants, no renamed predictions, and no self-definitional reductions occur. The three exceptional cases in Proposition 4.2 are checked by direct substitution into Equation (3). Hence the derivation is not circular.

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

The central claim rests on no fitted constants and no invented objects. The deductions use standard resolution theory, classical structure of determinantal ideals, and two published results from Niu's earlier paper; those are external inputs rather than assumptions containing the target conclusion.

assumptions (5)
  • domain assumption Iterated blowups along strict transforms of V(I_i(M)) resolve (A,X) with divisor data a_{i,X}=r-i+1 and k_i=(m-i+1)(n-i+1)-1 (Theorem 2.5).
    Quoted from [Joh03] and [Doc13]; the proof does not rederive this log resolution and uses its orders for the lct formula.
  • domain assumption lct(A',Y) >= lct(A',V) = lct(A,X) for a generic link Y of X (Equation (1)).
    Quoted from [Niu14, Prop. 3.7]; this prior result by the third author supplies one side of the equality.
  • domain assumption For a determinantal ideal, the canonical module of S/I_X is generated in degree (r-1)m and I_Y/I_V is its shift, so I_Y/I_V is generated in degree (n-r+1)(m-r)(r-1).
    Imported from [BH92] and [Mig98] as Proposition 2.9; used to compute the vanishing order of the generic link at the blow-up center.
  • domain assumption In each blow-up chart, the transformed regular sequence f_j/y remains regular and I_{Y_{i-1}} equals the colon ideal I_{V'_{i-1}} : I_{X_{i-1}}, so the transformed link is again a generic link.
    Cited from [Niu14, Claim 3.1.2(3)] in Proposition 3.1; this is the step connecting local orders of Y to Proposition 2.9(c).
  • standard math Log resolutions exist for pairs (A,X) on smooth quasi-projective varieties over C.
    Background assumption in Definition 2.3 and in the cited resolution theory; not special to this paper.

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Pith. "Pith review of Log canonical thresholds of generic links of determinantal varieties." pith.science (2026). https://pith.science/paper/YQBOEJWC

@misc{pith2026190803892,
  author       = {Pith},
  title        = {Pith review of: Log canonical thresholds of generic links of determinantal varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQBOEJWC}},
  note         = {Machine review of arXiv:1908.03892}
}
read the original abstract

We show that the log canonical threshold of a generic determinantal variety and its generic link are the same.

Discussion (0). Continue with ORCID to comment.

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