{"id":"87c2f75d-2d7d-4f8e-a3ae-34e88d030ba4","arxiv_id":"2505.18729","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the product operator on (1,1)-classes is exactly the span of the prime divisors annihilated by the collection.","lead":"On a smooth projective variety, multiplying (n-2) free divisor classes gives a Lefschetz-type map; this paper proves that, in the supercritical range, its kernel is spanned exactly by the prime divisors killed by the product. This settles the algebraic analogue of a convex-geometry open problem in the free case and yields a new proof of extremal cases of the Alexandrov-Fenchel inequality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The induction in Theorem 3.6 assumes without proof that restricting a supercritical free collection to H ∈ |L_{n-2}| preserves supercriticality; this is the load-bearing step and needs a Hall-Rado verification.","rationale":"The reader's weakest assumption correctly identifies the restriction-supercriticality assertion in Theorem 3.6 as the least secure step. My analysis confirms it is genuinely unproven in the text and is load-bearing: the induction hypothesis can only be invoked on H if the restricted collection is supercritical. However, the concern is not that the statement is false; a fairly direct Hall-Rado argument using Lemma 2.7 appears to prove it, given the mixed supercritical condition nd(L_I + L_{n-2}) ≥ |I|+3 and the monotonicity of numerical dimension under adding nef classes. Thus the gap is a missing justification rather than a fatal flaw. I also checked the subsequent conversion of E_i to codimension-one W_i: since Null_2(L) is defined with Zariski closure, it may have codimension-one components, and a general H cannot contain a codimension-two component, so the asserted containment is coherent, though it should be spelled out. The positivity of constants in Lemma 2.13 is not a serious issue because the constants are positive rationals. Overall, the central theorem is likely correct, but the paper as written is conditional on filling the omitted restriction-supercriticality proof. The reader's conditional verdict is appropriate, so no change is needed.","tokens_in":18148,"tokens_out":47671,"duration_ms":357962,"concrete_test":"Verify the omitted restriction claim: fix a nonempty I ⊂ [n-3] and form the multiset of nef classes consisting of |I|+2 copies of L_I and one copy of L_{n-2} on X. Using Lemma 2.7, prove that the product L_I^{|I|+2} · L_{n-2} is positive by checking that for every submultiset T, nd(sum of classes in T) ≥ |T|; the cases with r copies of L_I and the L_{n-2} use nd(L_I + L_{n-2}) ≥ |I|+3. If this check succeeds, the restriction supercriticality holds and Theorem 3.6's induction is valid; if a counterexample to this positivity is found, the induction fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.6, after restricting α to a general H ∈ |L_{n-2}|, the authors state: \"As L is supercritical, the restriction collection (L_1|_H, ..., L_{n-3}|_H) is supercritical on H,\" with no proof. This is the pivot of the induction: the induction hypothesis applies to H only if, for every nonempty I ⊂ [n-3], nd_H(L_I|_H) ≥ |I|+2, i.e., (L_I)^{|I|+2} · L_{n-2} > 0. The original supercriticality gives nd_X(L_I) ≥ |I|+2, which alone yields only nd_H(L_I|_H) ≥ |I|+1 in general; the missing increment of 1 must come from the mixed condition nd_X(L_I + L_{n-2}) ≥ |I|+3. A short argument via Lemma 2.7 and the monotonicity nd(rL_I + L_{n-2}) ≥ nd(L_I + L_{n-2}) appears to close the gap, but the argument is absent from the paper. Without it, the induction step is unsupported. A secondary unproven point is the use of codimension-one components of Null_2(L) to write E_i = H ∩ W_i; this is consistent with the closure in Definition 2.8, but the dimensional transition from codimension-two E_i to codimension-one W_i should be justified explicitly.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the kernel of the Lefschetz-type operator given by cup product with L = L_1 · ... · L_{n-2}, where each L_i is a free divisor class on a smooth projective variety X of dimension n and the collection is supercritical in the sense that nd(L_I) ≥ |I| + 2 for every nonempty I. Theorem A (restated as Theorem 3.6) asserts that ker L is spanned by the classes of prime divisors D with L · [D] = 0, resolving the Shenfeld–van Handel question in this setting. The proof has a 3-fold baby case (Proposition 3.1), an intermediate statement for one free class of numerical dimension at least 3 (Proposition 3.4), and an induction on dimension in which a general member H ∈ |L_{n-2}| is used to reduce to a supercritical collection on H. The paper then derives applications to extremals of the Alexandrov–Fenchel inequality for rational polytopes (Theorem 4.5) and to extremals of the Khovanskii–Teissier inequality (Corollary 4.8).","tokens_in":18462,"tokens_out":20683,"duration_ms":177771,"significance":"If Theorem A is correct, it is a substantial advance: it extends the previous ordered-positivity result [HX23, Theorem A] to arbitrary supercritical collections and gives a complete algebraic characterization of the kernel in the free case, with a clean statement in terms of prime divisors. The toric application reproduces a deep convex-geometric theorem of Shenfeld–van Handel by algebro-geometric means, and the Khovanskii–Teissier application is a useful new equality case. The proof relies on standard tools (Hodge index, Hall–Rado for nef classes, Bertini, and a recent homology lemma of Huang–Huh–Michałek–Wang–Wang) and is mostly self-contained. However, the induction in Theorem 3.6 contains a load-bearing assertion about preservation of supercriticality under restriction that is not proved; this must be supplied before the main theorem can be considered established.","major_comments":[{"comment":"The proof states: \"As L is supercritical, the restriction collection (L_1|_H, ..., L_{n-3}|_H) is supercritical on the lower dimensional variety H.\" This is the pivot of the induction and is not immediate. Supercriticality on X gives nd_X(L_I) ≥ |I| + 2 for every I ⊂ [n-3], but the induction hypothesis on H needs nd_H(L_I|_H) ≥ |I| + 2, i.e. (L_I)^{|I|+2} · L_{n-2} > 0. The former alone gives only nd_H(L_I|_H) ≥ |I| + 1 in general; the missing unit must come from the mixed condition nd_X(L_I + L_{n-2}) ≥ |I| + 3 together with freeness. No argument connecting these facts is given. Please add a lemma (for instance, using the Hall–Rado criterion of Lemma 2.7 and the monotonicity nd(rL_I + L_{n-2}) ≥ nd(L_I + L_{n-2})) and verify the strict positivity, or restructure the induction so that this restriction property is built in.","section":"§3.3, proof of Theorem 3.6, after Eq. (19)"},{"comment":"The passage from E_i, a prime divisor of H with L_I|_H · [E_i]_H = 0, to a codimension-one component W_i of Null_2(L) with E_i = H ∩ W_i needs a fuller justification. From L · [E_i]_X = 0 one knows E_i ⊂ Null_2(L). Since H is chosen to meet every component of Null_2(L) transversally at a general point, E_i cannot itself be a component of Null_2(L), but the text does not explain why Null_2(L) must contain a component of dimension n-1 containing E_i, nor why the constant c_i in Eq. (22) is strictly positive for every irreducible component Z of H ∩ W. These facts are used to write [E_i]_X = c_i L_{n-2} · [W_i]_X and hence to transfer the equality from H to X; please make the dimensional and positivity transitions explicit, including the role of the closed union in Definition 2.8.","section":"§3.3, proof of Theorem 3.6, Eqs. (21)–(23)"},{"comment":"In the case k ≥ 2, the proof asserts without comment that the collection {A^{(k-1)}, B^{(n-k-1)}} is supercritical once nd(A) ≥ k+1, nd(B) ≥ n-k+1 and nd(A+B) = n. This is true, but it uses the observation that for any subcollection with a' ≥ 1 copies of A and b' ≥ 1 copies of B, the class a'A + b'B = (A+B) + (a'-1)A + (b'-1)B is big, while boundary subcollections are handled by nd(A) ≥ k+1 and nd(B) ≥ n-k+1. Please include this one-line argument; as written, the supercriticality is asserted rather than demonstrated.","section":"§4.2, proof of Corollary 4.8"}],"minor_comments":[{"comment":"The title contains a typo, \"CHARACTERIZA TION\" should read \"CHARACTERIZATION\".","section":"Title and abstract"},{"comment":"The phrase \"there exits I\" should be \"there exists I\".","section":"§2.4, proof of Lemma 2.9"},{"comment":"In Theorem 3.6, condition (c) invokes Lemma 2.13 with W a component of Null_2(L); since the lemma is stated for an arbitrary prime divisor D in an ambient projective variety, the application is legitimate only because each W is a prime divisor of X and L_{n-2} is free. This should be said explicitly, and the constant c should be stated as positive rather than merely real, since the positivity is used later.","section":"§2.5, Lemma 2.13 and §3.3, condition (c)"},{"comment":"The proof of the claim uses the decomposition of Lemma 2.3 and then adds finite movable classes to form a spanning set D_2; the sentence \"we can add finite prime divisor classes and finite movable classes\" is terse and could be clarified by explaining that every class in H^{1,1}(X) is a numerical combination of effective divisors and movable classes.","section":"§3.1, proof of Proposition 3.1"},{"comment":"When Remark 2.11 is invoked to conclude that each [D] in Eq. (30) spans an extremal ray and is torus-invariant, the text could note explicitly that an extremal ray of the pseudo-effective cone of a toric variety is generated by a torus-invariant divisor class; this is standard but is used for a nontrivial conclusion.","section":"§4.1, proof of Theorem 4.5"}],"recommendation":"major_revision","confidential_remarks":"The main result is attractive and the overall strategy is plausible, but the induction in Theorem 3.6 relies on an unproved assertion about restriction of supercritical collections that is genuinely load-bearing. In my view this is fixable within the scope of the paper, so I recommend major revision rather than rejection. The applications to the Alexandrov–Fenchel and Khovanskii–Teissier equalities appear sound once the main theorem is repaired, and the paper is appropriately positioned relative to the prior work [HX22, HX23, SvH23]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a real advance, and I think the main theorem is true, but the proof as written skips the one step that makes the induction work, and the authors need to supply it.\n\nWhat's new: HX23 needed nd(L_i) ≥ i+2; Theorem A removes that in the free case and gets the full supercritical statement. The proof architecture is genuinely different: prove the 3-fold case, then restrict to a general H in |L_{n-2}| and use homology comparison to lift the decomposition from H to X. The applications are also new: a toric proof of the Shenfeld–van Handel extremal characterization for rational polytopes, and a KT equality case. I checked the toric dictionary and it looks sound.\n\nWhere the paper is soft: The pivotal assertion in Theorem 3.6 is that the supercriticality of L on X implies supercriticality of (L_1|_H,...,L_{n-3}|_H) on H. That is used to apply the induction hypothesis, but it is stated without proof. From nd_X(L_I) ≥ |I|+2 you only get nd_H(L_I|_H) ≥ |I|+1 for a general H; you need one extra increment, which has to come from nd_X(L_I + L_{n-2}) ≥ |I|+3. The stress-test note says a short argument via Lemma 2.7 and monotonicity nd(rL_I + L_{n-2}) ≥ nd(L_I + L_{n-2}) closes it. I agree that looks plausible, but the paper doesn't include it; the induction is unsupported as written. Likewise, Lemma 2.13 only supplies a real constant c_W, but the proof of Theorem 3.6 needs c_i > 0 to divide. Since H∩W is an effective cycle, positivity should be true, but again the text doesn't say why.\n\nNeither gap looks fatal. They are the kind of thing a careful referee would ask to be written out. The central claim is independent of the target result and the overall strategy is coherent.\n\nWho should read this: anyone working on the (Ker2) problem, hard Lefschetz for nef classes, or on algebraic proofs of AF-type extremal characterizations. It deserves a serious referee. I'd send it out and ask for the restriction lemma to be proven explicitly.","headline":"The main theorem is likely true and a genuine advance, but the induction hinges on an unproved restriction lemma that needs to be supplied.","tokens_in":18975,"tokens_out":2767,"would_cite":true,"duration_ms":23737,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C20","14C30","14M25","52A39"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the induced Lefschetz operator on (1,1)-classes is spanned exactly by the prime divisors that the collection…","keywords":["hard Lefschetz class","free divisor classes","supercritical collection","kernel of Lefschetz operator","Alexandrov-Fenchel inequality","Khovanskii-Teissier inequality","null locus","toric varieties"],"falsifier":"Construct a smooth projective fourfold (for instance toric) with a supercritical pair of free divisor classes $(L_1,L_2)$ and an $(1,1)$-class $\\alpha$ killed by $L_1\\cdot L_2$ but not representable as a real combination of prime divisors killed by $L_1\\cdot L_2$; a toric computation with explicit intersection products would decide this directly, since nef bundles on smooth toric varieties are semiample.","tokens_in":17951,"feed_emoji":"📐","tokens_out":10048,"duration_ms":90793,"temperature":0.7,"pith_summary":"This paper proves that a natural numerical condition controls the kernel of the Lefschetz-type operators coming from products of divisor classes. On a smooth projective variety of dimension $n$, any collection of $n-2$ free divisor classes that is 'supercritical' defines a multiplication operator on $(1,1)$-classes whose kernel is spanned precisely by the prime divisors annihilated by the product. Equivalently, the collection is a hard Lefschetz class exactly when it kills no prime divisor. This settles the algebraic form of the open problem posed in [SvH23] in the free case, and it gives an algebro-geometric proof of the extremal characterization for the Alexandrov-Fenchel inequality for rational polytopes, along with a new extremal characterization for the Khovanskii-Teissier inequality.","feed_headline":"Kernel of Lefschetz operator equals span of killed prime divisors","feed_subtitle":"For supercritical free divisor classes, only prime divisors set the kernel—settling the AF extremal conjecture.","key_machinery":"The load-bearing notion is the supercritical condition $\\operatorname{nd}(\\sum_{i\\in I} L_i)\\ge |I|+2$ for every nonempty $I\\subseteq[n-2]$, which guarantees that the null locus $\\operatorname{Null}_2(\\mathbf{L})$ of codimension-two subvarieties killed by $\\mathbf{L}$ is a proper Zariski closed set. The induction proceeds by cutting with a general $H\\in |L_{n-2}|$; the asserted preservation of supercriticality under restriction lets the induction hypothesis express $\\alpha|_H$ as a combination of prime divisors $E_i$ of $H$ killed by the restricted product. A homology-class lemma (Lemma 2.13, from [HHM+25]) identifies each $E_i$, which lies in a codimension-one component $W_i$ of the null locus, as a positive multiple of $L_{n-2}\\cdot[W_i]$ in homology, so the combination can be pushed forward to $X$. The remaining class is disposed of by the threefold-based Proposition 3.4 for a single free class of numerical dimension at least three. Throughout, the Hall-Rado criterion for nef classes (Lemma 2.7, from [HX22]) and the Hodge-index/Lorentzian proportionality principle (Lemma 2.5) supply the numerical vanishing steps.","core_discovery":"Theorem A (Theorem 3.6) states: for a smooth projective variety $X$ of dimension $n$ and a supercritical collection $\\mathbf{L} = (L_1,\\ldots,L_{n-2})$ of free divisor classes, $\\ker \\mathbf{L} = \\operatorname{span}_{\\mathbb{R}}\\{[D] : D \\in \\operatorname{Prime}(X),\\ \\mathbf{L}\\cdot[D]=0\\}$. In particular, $\\mathbf{L}$ is a hard Lefschetz class if and only if $\\mathbf{L}\\cdot[D]\\neq 0$ for every prime divisor $D$. The same conclusion holds when the classes are only semiample (Remark 1.2). Corollary A derives the equality case of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes: equality holds exactly when the two polytopes share their supporting hyperplanes in all active normal directions. Corollary B characterizes equality in the Khovanskii-Teissier inequality for two free divisor classes by writing their difference as a combination of prime divisors annihilated by the remaining intersection numbers. The proof is inductive, with a threefold base case and a restriction-to-a-general-hypersurface step.","pith_inferences":["The induction step relies on the unproved assertion that supercriticality is preserved under restriction to a general member of the last linear system; verifying this directly for nef (not necessarily free) classes would extend the theorem to the full supercritical nef case and complete the original conjecture.","If the same kernel description holds for nef supercritical collections, the toric proof of the AF-extremal theorem would pass from rational polytopes to arbitrary convex bodies by approximation, matching the known polytopal result of [SvH23].","The restriction-and-lifting mechanism, especially the use of the general-fiber homology lemma, may be reusable for characterizing kernels of Lefschetz operators on higher-degree cohomology $H^{d,d}(X)$ for collections of $n-2d$ classes, where the kernel description is expected to involve 'degenerate' contributions beyond prime divisors.","A natural test is to compute, on a smooth toric fourfold with a supercritical pair of free divisors, whether every class killed by the product is numerically equivalent to a combination of killed torus-invariant divisors; the theorem says yes, and the calculation is purely combinatorial."],"forward_implications":["For any supercritical collection of free (or semiample) divisor classes, the kernel of the Lefschetz operator is a subspace of dimension at most the Picard number, and the annihilated prime divisors lie in the augmented base locus of a big class and span extremal rays of the pseudo-effective cone.","A supercritical free collection is a hard Lefschetz class precisely when no prime divisor is annihilated, giving a purely divisorial criterion for injectivity of the operator.","The Alexandrov-Fenchel equality for a supercritical collection of rational polytopes holds if and only if the two polytopes have identical supporting hyperplanes in every active normal direction, now proved by toric algebraic geometry.","The Khovanskii-Teissier equality for intersection numbers $(A^k\\cdot B^{n-k})$ of two free divisor classes is characterized by $A-cB$ being a real combination of prime divisors $D_i$ with $A^{k-1}\\cdot B^{n-k-1}\\cdot[D_i]=0$.","When the relevant mixed volume or intersection number vanishes, equality holds automatically and the vanishing itself is characterized by a failure of numerical dimension conditions (Remark 4.9)."],"supporting_citations":[{"why":"Supplies the Hall-Rado criterion for nef classes (Lemma 2.7) connecting numerical dimensions of subcollections to nonvanishing of complete intersections.","marker":"[HX22]"},{"why":"Formulates Conjecture 1.1 and the null-locus properness lemma, and proves the characterization under the stronger positivity condition that the new theorem weakens to freeness.","marker":"[HX23]"},{"why":"Poses the algebraic extremal problem and solves it for convex polytopes; the paper's Corollary A reproduces that polytopal result by toric methods.","marker":"[SvH23]"},{"why":"Provides the dimensionality mechanism for mixed discriminants, the linear-algebra precursor of the kernel characterization.","marker":"[Pan85]"},{"why":"Contributes Lemma 2.12/2.13 on the homology class of a general fiber, which converts components of $H\\cap W$ into multiples of $L_{n-2}|_W$ in the lifting step.","marker":"[HHM+25]"},{"why":"Gives the divisorial Zariski decomposition (Lemma 2.3) used to control dualities with the pseudo-effective cone in the threefold baby case.","marker":"[Bou04]"},{"why":"Identifies rational and cohomological equivalence for torus-invariant divisors, needed to translate the cohomological kernel statement into support-function equalities for polytopes.","marker":"[FS97]"}],"fun_headline_variants":["Prime divisors alone decide Lefschetz kernel","Supercritical free divisors: kernel from prime divisors","Kernel spanned by prime divisors killed by L","Hard Lefschetz holds iff no prime divisor killed","Extremals of AF inequality: shared hyperplanes only"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The induction step assumes, without proof in the text, that restricting a supercritical collection of free divisor classes to a general member of the last linear system produces another supercritical collection; if this restriction property fails, the proof of the general case collapses.","fun_headline_variants_meta":{"raw":{"variants":["Prime divisors alone decide Lefschetz kernel","Supercritical free divisors: kernel from prime divisors","Kernel spanned by prime divisors killed by L","Hard Lefschetz holds iff no prime divisor killed","Extremals of AF inequality: shared hyperplanes only"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1276,"prompt_tokens":941,"completion_tokens":335,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":262}},"tokens_in":557,"tokens_out":335,"duration_ms":4139,"temperature":1.0,"reasoning_tokens":262,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:29:36.971828+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a smooth projective fourfold (for instance toric) with a supercritical pair of free divisor classes $(L_1,L_2)$ and an $(1,1)$-class $\\alpha$ killed by $L_1\\cdot L_2$ but not representable as a real combination of prime divisors killed by $L_1\\cdot L_2$; a toric computation with explicit intersection products would decide this directly, since nef bundles on smooth toric varieties are semiample.","supporting_citations":[],"review_version":1}