{"id":"6fb29e90-b73e-4f3d-83fa-4538593788ed","arxiv_id":"2411.18081","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that ampleness of a divisor is equivalent to every subvariety being a cosupport of a multiplier ideal sheaf in the divisor's rational multiples, and equivalently to a model-categorical presentation of the Zariski site.","lead":"This paper gives two new ways to recognize 'ample' divisors on complex smooth projective varieties, one using singularities of divisor multiples and one using model categories. The main result says a divisor is ample exactly when every subvariety appears as a singular locus, called the cosupport, of a multiplier ideal sheaf attached to a rational multiple of the divisor.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.4 hinges on an unproved inheritance step: from (SoO) for L it asserts (SoO) for k(rH+L), which forces r irrational; the natural construction fails without a bound on the coefficient d.","rationale":"The reader's weakest_assumption identifies precisely the same unproved propagation step in §2.3. I agree that this is the most load-bearing gap: the entire argument that r is irrational, and hence the construction of infinitely many negative curve classes, depends on it. The concern is not a disagreement with the consensus or an external objection; it is an internally missing justification for a step that the proof uses as if it were immediate. I also note two secondary issues that do not change the verdict: Theorem 1.7 depends on Theorem 3.3 from the author's own [Lee24], which is not proved here, and the definition of H in Theorem 1.7 says 'support' but the proof uses the open set where I(D)_x = O_{X,x}, i.e. the complement of the support. These are real but less central than the §2.3 gap. The natural candidate for proving the inheritance step reveals a concrete coefficient obstruction, so the step is not merely terse; it is unsupported as written. The theorem may still be true, but the submitted proof of Theorem 1.4 is incomplete. Since the reader's verdict is already REJECT and my concern supports that verdict without changing it, I recommend UNCHANGED.","tokens_in":12841,"tokens_out":8265,"duration_ms":81373,"concrete_test":"Try to prove the missing inheritance lemma: for every subvariety Z with a witness D_Z ~_Q dL and cosupp I(D_Z)=Z, construct a witness E_Z for L' = k(rH+L). The natural candidate E_Z = D_Z + bH_0 forces b = dp/q, so the required condition is d < q/p. Check whether the proof of Theorem 1.4 or the cited [Lee22] supplies any bound on the minimal d for a witness of Z, or an alternative construction avoiding the coefficient obstruction. Concretely, work through the argument on a smooth projective surface with L big and non-nef and r in Q ∩ (0,1), and verify whether every such d can be chosen below q/p; if no such choice is proved, the inheritance step is unsupported and Theorem 1.4 lacks a complete proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §2.3, after defining r as the nef threshold of rH+L, the proof states: '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 inheritance claim is load-bearing: it is the only step that forces r to be irrational, and that irrationality is then used to produce infinitely many curve classes in L^{<0}. The claim is not proved anywhere. To derive it, fix a subvariety Z. From (SoO) for L, there is D_Z ~_Q dL with cosupp I(D_Z)=Z. Write r=p/q. For L' = k(rH+L) = (kp/q)H + kL, one needs E_Z ~_Q cL' with the same cosupport. The only evident construction is E_Z = D_Z + bH_0 with H_0 in |H|. Comparing coefficients forces ck=d and ckp/q=b, hence b = dp/q. The multiplier ideal I(D_Z + bH_0) has the same cosupport as I(D_Z) only when b<1 (so the round-down is unchanged). Thus the construction works only if d < q/p. Nothing in §2.3 or in the definition of (SoO) gives such a bound, nor any bound on d independent of Z and k. Since (SoO) only asserts existence of some d, the implication (SoO)(L) implies (SoO)(k(rH+L)) does not follow from the given argument. This is not a peripheral detail: the rationality of r is the pivot of the contradiction in §2.3, which is the proof of the nontrivial direction of Theorem 1.4.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":13236,"tokens_out":25069,"duration_ms":234399,"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":[{"comment":"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(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":"Theorem 1.7 and Section 4"},{"comment":"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.","section":"Section 3 and proof of Theorem 1.7"}],"minor_comments":[{"comment":"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.","section":"Section 2.2.1"},{"comment":"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":"Throughout, especially Sections 2.3 and 4"},{"comment":"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.","section":"Section 2.3"},{"comment":"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.","section":"General presentation"}],"recommendation":"major_revision","confidential_remarks":"The two main theorems are not established in the current form. The gap in Section 2.3 is the most serious: the inheritance of (SoO) by k(rH+L) is a load-bearing assertion with no proof, and the natural construction fails without a bound on the coefficient d. The support/open-set inconsistency in Theorem 1.7 is fixable in principle, and the dependence on [Lee24] could be clarified, but the Section 2.3 gap requires a substantive new argument. The editors may also wish to verify the status of the author's preprints [Lee22] and [Lee24], since they carry substantial weight in this submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: Lee's paper claims a new characterization of ampleness—via cosupport of multiplier ideals and, separately, a model-categorical presentation. The extension from the nef case in [Lee22] to arbitrary divisors is a genuine new result, and Theorem 1.7 is new. The motivation is good, and the parts showing bigness and the nef case are mostly standard and well-sourced.\n\nThe soft spot is real. In §2.3, the proof asserts that if the nef threshold r is rational, then k(rH+L) also satisfies (SoO) for large divisible k, and uses that to force r irrational. The stress-test note is right: this inheritance step is not proved. Given a subvariety Z with a witness D_Z ~_Q dL, the natural construction for k(rH+L) needs d < q/p, which the definition of (SoO) does not guarantee. Without that bound the step fails, and the contradiction collapses. This is the pivot of the nontrivial direction, not a side remark.\n\nTwo more concerns, both secondary. Theorem 1.7 depends on Theorem 3.3/3.5 imported from the author's preprint [Lee24], which is not verified here. And the definition of H in Theorem 1.7 is confusing: supp I(D) is a closed set, while H is supposed to take values in open sets; the proof later works with the complement, so the notation needs fixing.\n\nThe paper is not incoherent—the author engages seriously with the literature and the claim may well be true. But as it stands, the proof is incomplete. I'd send it to a referee who can stress-test the inheritance step and the [Lee24] imports. A desk rejection would be premature; acceptance without major revision would be wrong.\n\nRecommendation: send to peer review, expect major revision.","headline":"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.","tokens_in":13752,"tokens_out":4351,"would_cite":false,"duration_ms":36712,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C20","14F18","18N40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["ample divisor","multiplier ideal sheaf","cosupport","model category","homotopical presentation","Bousfield localization","Quillen equivalence","asymptotic multiplier ideal"],"falsifier":"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.","tokens_in":12567,"feed_emoji":"🔍","tokens_out":5904,"duration_ms":45039,"temperature":0.7,"pith_summary":"This paper proves that a divisor on a complex smooth projective variety is ample exactly when it satisfies a strong geometric condition: every subvariety of the variety appears as the cosupport of a multiplier ideal sheaf attached to an effective Q-divisor that is a nonnegative rational multiple of the divisor. This gives a purely singularity-theoretic characterization of ampleness, complementing earlier numerical and cohomological ones. The paper also converts this into a categorical criterion: the functor that sends such a Q-divisor to the support of its multiplier ideal sheaf generates a homotopical presentation of the category of Zariski open sets. If correct, the two criteria tie ampleness to the behavior of multiplier ideals and to the homotopy theory of small categories.","feed_headline":"Multiplier ideals give an exact test for ampleness","feed_subtitle":"A divisor on a smooth projective variety is ample precisely when every subvariety appears as the vanishing locus of one of its multiplier…","key_machinery":"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.","core_discovery":"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 $\\operatorname{Op}(X)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the definition and basic properties of multiplier ideal sheaves and Nadel vanishing (Theorem 9.4.8), used throughout Sections 2 and 5.","marker":"[Laz04b]"},{"why":"Provides asymptotic multiplier ideal theory and boundedness results (Corollary 1.4.41, Proposition 2.1.21) that the proof of Theorem 1.4 leans on.","marker":"[Laz04a]"},{"why":"Proved the nef case of the (SoO) criterion and introduced the finite-limit completion $\\mathbb{M}(X,L)$; the present paper extends that result to all divisors.","marker":"[Lee22]"},{"why":"Gives the criteria (Theorem 3.3 and 3.5) for a functor to generate a homotopical presentation; these are the engine behind Theorem 1.7.","marker":"[Lee24]"},{"why":"Introduced the universal model category $\\mathcal{U}(\\mathcal{C})$ of simplicial presheaves with the Bousfield-Kan structure, the setting for homotopical presentations.","marker":"[Dug01]"},{"why":"Founded the theory of model categories, the framework in which the second criterion is formulated.","marker":"[Qui67]"},{"why":"Provides the theory of left Bousfield localization used to define the Quillen adjunction in the definition of homotopical presentation.","marker":"[Hir03]"}],"fun_headline_variants":["Ample iff every subvariety is a multiplier ideal cosupport","Divisor ample iff every subvariety is a multiplier locus","All subvarieties as multiplier supports: new ampleness test","New criterion: L ample iff every subvariety is a cosupport","Multiplier ideal loci detect exactly when a divisor is ample"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ample iff every subvariety is a multiplier ideal cosupport","Divisor ample iff every subvariety is a multiplier locus","All subvarieties as multiplier supports: new ampleness test","New criterion: L ample iff every subvariety is a cosupport","Multiplier ideal loci detect exactly when a divisor is ample"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000529,"raw_usage":{"total_tokens":2524,"prompt_tokens":891,"completion_tokens":1633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":1555}},"tokens_in":507,"tokens_out":1633,"duration_ms":10493,"temperature":1.0,"reasoning_tokens":1555,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:33:01.990788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}