REVIEW 3 major objections 4 minor 16 references
Q-divisor and Ampleness
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read A divisor on a smooth projective variety is ample exactly when every subvariety is the cosupport of a multiplier ideal sheaf attached to one of its rational multiples.
desk verdict Plausible but incomplete: the proof of Theorem 1.4 has a load-bearing gap in the (SoO) inheritance step, and the model-categorical theorem rests on an unverified preprint. 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 load-bearing object is the multiplier ideal sheaf $\mathcal{I}(D)$ for an effective Q-divisor $D$, defined via a log resolution as $f_*\mathcal{O}_Y(K_{Y/X}-\lfloor D\rfloor)$; its cosupport is the locus where the sheaf is not the full structure sheaf. The (SoO) condition demands that every subvariety be such a cosupport for some $D$ in $\operatorname{Div}_{\mathbb{Q}\ge 0}(X,L)$, the set of Q-divisors Q-equivalent to a nonnegative rational multiple of $L$. The second half of the paper uses the category $\operatorname{Div}_{\mathbb{Q}\ge 0}(X,L)$, its finite-limit completion $\mathbb{M}(X,L)$ by finite intersections, and the functor $H=\operatorname{supp} \mathcal{I}(-)$; the criterion from the author's companion work says a functor with finite limits that is essentially surjective and has a lifting property generates a homotopical presentation, and Lemma 4.9 verifies the lifting property in a simple way because limits in $\operatorname{Op}(X)$ are intersections.
What would settle it
A decisive test would be to exhibit a smooth projective variety $X$ and a non-ample divisor $L$ such that every subvariety of $X$ is the cosupport of $\mathcal{I}(D)$ for some $D\in \operatorname{Div}_{\mathbb{Q}\ge 0}(X,L)$; by Theorem 1.4 such an example cannot exist, so finding one would refute the equivalence. A more targeted check is to look for a rational nef threshold $r$ for a divisor $L$ satisfying (SoO): if the perturbed divisor $k(rH+L)$ fails (SoO) for all large divisible $k$, the proof's key step is invalid.
Extended reading notes
Core claim
The central discovery is Theorem 1.4: for a complex smooth projective variety $X$ and a divisor $L$, $L$ is ample if and only if $L$ satisfies (SoO), meaning every subvariety $Z$ of $X$ equals $\operatorname{cosupp} \mathcal{I}(D)$ for some effective Q-divisor $D$ that is Q-linearly equivalent to a nonnegative rational multiple of $L$. The forward direction is proved by constructing, from global generation of $\mathcal{O}_X(nL)\otimes \mathcal{I}_Z$, a divisor $H$ whose $(1-\varepsilon)H$ has cosupport exactly $Z$. The reverse direction proceeds in three steps: (SoO) forces $L$ to be big; then, assuming $L$ nef, (SoO) forces $L\cdot C>0$ for every curve $C$ and eventually $L^{\dim V}\cdot V>0$ for every subvariety $V$, making $L$ ample; finally, assuming $L$ not nef, the paper uses asymptotic multiplier ideals to show that the number of negative curve classes would be both infinite (from the irrationality of the nef threshold) and finite (from Nadel vanishing and boundedness), a contradiction. Theorem 1.7 then rephrases (SoO) as the statement that the functor $H(D)=\operatorname{supp} \mathcal{I}(D)$ is essentially surjective onto Zariski open sets, and uses a criterion for homotopical presentations to conclude that $H$ generates a Quillen equivalence between a localized universal model category and the model category of simplicial presheaves on $\operatorname{Op}(X)$.
Load-bearing premise
The proof that (SoO) forces $L$ to be nef relies on the unproven assertion that if the smallest real number $r$ making $rH+L$ nef is rational, then a sufficiently large divisible multiple $k(rH+L)$ also satisfies (SoO), so that the already-proved nef case applies and forces $r$ to be irrational; if this propagation step fails, the contradiction argument collapses.
Editorial extensions
If this is right
- If Theorem 1.4 holds, ampleness of $L$ can be checked by a purely local singularity-theoretic condition: every subvariety must be cut out by some multiplier ideal attached to a multiple of $L$.
- Since (SoO) is insensitive to replacing $L$ by a positive rational multiple, the criterion shows ampleness is detected by the entire semigroup of Q-divisors generated by $L$.
- Theorem 1.7 says the Zariski topology of $X$ is determined, up to homotopical presentation, by the multiplier ideals of multiples of $L$; this gives a model-categorical route to recovering the variety from its divisor data.
- The proof that (SoO) implies nefness implies that any $L$ with (SoO) has no curve with negative intersection with $L$, and more generally no subvariety with negative top intersection, so (SoO) rules out all negative classes.
Reading between the lines
- The paper leaves open whether (SoO) can be weakened to checking only curves or only subvarieties of a fixed dimension; a likely testable statement is that checking all curves suffices in dimension two.
- The use of multiplier ideals suggests a connection to log canonical thresholds and minimal centers; one could try to characterize ampleness via the existence of suitable minimal centers for every subvariety.
- The homotopical presentation criterion might extend to singular varieties if one replaces multiplier ideals with non-lc ideals, as the paper itself remarks; testing this on mild singularities (e.g., canonical or log canonical) would be a concrete next step.
- The proof's 'propagation' step could be examined computationally for explicit threefolds to see whether rational nef thresholds really do preserve (SoO); if a counterexample exists, the theorem might still hold via a different argument.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes two characterizations of ampleness for divisors on smooth projective complex varieties: Theorem 1.4 characterizes ampleness by the property (SoO) that every subvariety is the cosupport of a multiplier ideal sheaf of some effective Q-divisor Q-linearly equivalent to a nonnegative rational multiple of L, and Theorem 1.7 characterizes ampleness by the condition that a certain multiplier-ideal functor generates a homotopical presentation of the Zariski open-site category Op(X). The proof of Theorem 1.4 proceeds in three steps: (SoO) implies bigness, (SoO)+nef implies ampleness, and finally (SoO) implies nefness. The proof of Theorem 1.7 uses a finite-limit completion of the divisor category and a general homotopical-presentation criterion imported from the author's preprint [Lee24].
Significance. If the results hold, Theorem 1.4 gives an elegant singularity-theoretic characterization of ampleness via multiplier ideal cosupports, and Theorem 1.7 provides a bridge to model-categorical presentations, reinforcing earlier work by the author on Grothendieck topoi. The paper uses standard multiplier-ideal techniques (Lazarsfeld's books, Nadel vanishing) and Dugger's universal model categories. However, the main algebraic proof is not self-contained: a key inheritance step in Section 2.3 is stated without proof and is load-bearing, and the statement of Theorem 1.7 contains a support-versus-open-set inconsistency. The model-categorical half is also heavily dependent on the companion preprint [Lee24], whose results are not proved in this manuscript.
major comments (3)
- [Section 2.3 (proof of Theorem 1.4)] The proof of (2)=>(1) of Theorem 1.4 relies on the assertion that if r is rational, then k(rH+L) also satisfies (SoO) for sufficiently large and divisible k. This assertion is not proved. Given (SoO) for L, for a subvariety Z one only knows the existence of D_Z ~_Q dL with cosupp I(D_Z)=Z, with no bound on d. The evident construction E_Z = D_Z + bH_0 with H_0 in |H| forces ck=d and b=dp/q if E_Z ~_Q c k(rH+L); the cosupport remains Z only when b<1, since otherwise the round-down of E_Z acquires a component along H_0. Nothing in Section 2.3 or in Definition 1.3 provides the needed bound d<q/p. The irrationality of r is then used to produce infinitely many integral curve classes in L^{<0}, which is essential for ruling out cases (2.63) and (2.64) of Lemma 2.6. Without the unproved inheritance step, the proof does not rule out rational r and the contradiction in Section 2.3 collapses. This is a load-bearing gap in the proof of the nontrivial direction of Theorem 1.4.
- [Theorem 1.7 and Section 4] The functor H in Theorem 1.7 is declared to map D to supp I(D), the support of the multiplier ideal sheaf, which is a closed subset of X, while the target category Op(X) consists of Zariski open sets. The proof then defines H(D) by 'x in H(D) iff I(D)_x = O_{X,x}', which is the complement of the support and is open. As printed, the theorem statement and proof are inconsistent: H does not land in Op(X) if H(D)=supp I(D), and the equivalence (2)<->(3) in the proof only makes sense for the open-locus reading. The intended definition is presumably H(D) = X \ supp I(D). This must be corrected before Theorem 1.7 can be evaluated.
- [Section 3 and proof of Theorem 1.7] The proof of Theorem 1.7 invokes Theorem 3.3 as the criterion for generating a homotopical presentation, and Theorem 3.5 is used in Section 5. Both are imported verbatim from the author's preprint [Lee24] without proof. Since Theorem 1.7 is a central claim of the paper, the model-categorical part is only as solid as the companion preprint. The authors should either include proofs of Theorems 3.3 and 3.5, or clearly delineate which results are established in this paper and verify that [Lee24] is an acceptable reference for the editors; as it stands, the main model-categorical characterization of ampleness is not self-contained.
minor comments (4)
- [Section 2.2.1] The notation O_X(k) is used without defining the underlying divisor; it should be O_X(kA) for a fixed very ample divisor A, and the following line 'H_1,...,H_{dim X-1} in |O_X(k)⊗I_x|' needs the same clarification.
- [Throughout, especially Sections 2.3 and 4] The symbol H is used both for the divisor K_X+A in Section 2.3 and for the functor Div_{Q>=0}(X,L)->Op(X) in Sections 4 and 5; this overloaded notation is confusing and should be changed, e.g., by renaming the functor.
- [Section 2.3] In the sentence 'If r is rational then, since kprH `Lq also satisfies (SoO) for a sufficiently large and divisible k', the phrase 'large and divisible' should explicitly say 'divisible by the denominator of r' for clarity.
- [General presentation] The manuscript contains many typographical errors and OCR artifacts (e.g., 'effective Q-divisorsD', 'Zp}L}q', inconsistent use of m0 vs m0, and misplaced parentheses). A careful proofreading pass is needed before resubmission.
Circularity Check
No significant circularity: Theorem 1.4 is derived from the definition of (SoO) rather than assumed, Theorem 1.7's categorical input is supported by external references, and the §2.3 (SoO)-transfer assertion is an unproved gap rather than a circular reduction.
full rationale
The paper's main geometric criterion, Theorem 1.4, is not circular: (SoO) is defined independently of ampleness, and the converse is proved through three steps (bigness, nef implies ample, and nefness). The proof of nefness in §2.3 contains a load-bearing but unproved assertion: 'If r is rational then, since kprH `Lq also satisfies (SoO) for a sufficiently large and divisible k, rH `L is ample by (ii), a contradiction to the minimality of r.' This is a genuine omitted proof — no argument is given that (SoO) transfers from L to k(rH+L) — but it is a correctness gap, not a circularity, because the asserted transfer is not equivalent to the input by construction. For Theorem 1.7, the proof reduces to Theorem 1.4 and a categorical criterion (Theorem 3.3). Although Theorem 3.3 is stated from the author's preprint [Lee24] without proof, the paper explicitly notes it is 'well-known from the infinity categorical perspective ([Cis19], [Lur09])', so the self-citation is backed by independent external references and is not the sole basis of the argument. The construction of M(X,L) from [Lee22] is redefined in the present text (Definitions 4.1–4.7), and the equivalence (2)⇔(3) in Theorem 1.7 is a direct consequence of the definitions of H and H^int. No parameter is fitted and no known result is renamed. Accordingly, the circularity score is 0; the §2.3 gap should be weighed as a correctness risk, not as circularity.
Assumptions & free parameters
assumptions (5)
- standard math Standard multiplier ideal theory from [Laz04b] (log resolutions, Nadel vanishing, asymptotic multiplier ideals).
- ad hoc to paper Theorem 3.3 from [Lee24]: a finite-limit-preserving functor satisfying essential surjectivity and lifting properties generates a homotopical presentation.
- ad hoc to paper Theorem 3.5 from [Lee24]: admissible subcategory version of the homotopical presentation criterion.
- ad hoc to paper The assertion that if r is rational then k(rH+L) satisfies (SoO) for sufficiently large and divisible k.
- standard math Kleiman's criterion and the finiteness of numerical classes of curves of bounded degree.
Cite this review
Pith. "Pith review of Q-divisor and Ampleness." pith.science (2026). https://pith.science/paper/HTV4THBN
@misc{pith2026241118081,
author = {Pith},
title = {Pith review of: Q-divisor and Ampleness},
year = {2026},
howpublished = {\url{https://pith.science/paper/HTV4THBN}},
note = {Machine review of arXiv:2411.18081}
}
read the original abstract
We give two criteria for a divisor on complex smooth projective variety to be ample using the multiplier ideal sheaf and the model category.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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