{"id":"dc1a4d32-592a-429c-b353-141fc9bf66db","arxiv_id":"2411.17410","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For surjective projective morphisms of pure relative dimension d over normal schemes, this paper constructs a symmetric multi-additive Deligne pairing on d+1 line bundles, with functorial properties and canonical hermitian metrics.","lead":"A new pairing construction assigns an invertible line bundle on the base to every family of line bundles on a possibly non-flat family of varieties, generalizing Deligne's intersection pairing. The result provides a toolbox for arithmetic intersection theory and heights on singular or non-flat families over normal base schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gluing in Prop. 4.12 silently assumes finite Tor-dimension for f over the generic point and codimension-1 local rings; the text never proves this, although it follows from normality.","rationale":"I read the paper in good faith and tried to find a flaw that would refute the main theorem. The construction is intricate but follows a sound strategy: reduce to sufficiently ample line bundles, use presentations to build a finite cover over an open subset of a projective bundle, descend, and then use algebraic Hartogs to prove independence of presentations. The most delicate point is indeed Proposition 4.12, where the stalks at the generic point and at codimension-1 points are identified with García's Deligne pairing. That identification presupposes finite Tor-dimension of f over those points. The reader's weakest_assumption names exactly this, and I agree. I do not think this is a fatal error because the finiteness is true: the generic point is a field, and the codimension-1 local rings of a normal noetherian scheme are DVRs. The paper simply omits the verification. I also examined other potential concerns: the base change to Spec O_{S,P} is a flat dominant localization, so Proposition 2.9 applies to the norm functor; the descent lemmas have the required codimension estimates; the cocycle condition in Theorem 4.24 is terse but likely follows from the uniqueness in Proposition 4.22. The metric section is plausible and not load-bearing for the algebraic claim. Therefore the conditional verdict stands, pending the missing finite-Tor-dimension lemma and a few clarifications.","tokens_in":41515,"tokens_out":44732,"duration_ms":450637,"concrete_test":"Add a lemma in §4.1 proving that for a normal noetherian S, the morphisms X_η→η and X_A→Spec O_{S,P} for every P∈S^(1) have finite Tor-dimension. Concretely: for A=O_{S,P}, show A is a DVR and hence every A-module has a free resolution of length ≤1, so Tor_i^A(O_{X_A,x},M)=0 for all i≥2 and all A-modules M; for η, use that κ(η) is a field so Tor_i=0 for i≥1. If this proof cannot be completed in the stated generality, the gluing in Proposition 4.12 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction defines the Deligne pairing for sufficiently ample sheaves by descending a norm sheaf on a projective bundle to a line bundle J on S, then proves independence of presentations in Proposition 4.12 by identifying the stalks of J at the generic point η and at each codimension-1 point P with García's Deligne pairing. This identification requires f_η: X_η→η and f_A: X_A→Spec O_{S,P} to have finite Tor-dimension in the sense of Definition 1.3, because García's Deligne pairing and his independence/base-change theorems are only established under that hypothesis. The text asserts this finiteness ('Since η is a regular scheme... Since Spec A is a regular scheme...') without proof. It is true: κ(η) is a field, so global dimension 0; normality of S gives that O_{S,P} is a regular local ring of dimension 1 (Serre's criterion), hence a DVR, and over a DVR every module has projective dimension at most 1, so Tor_i vanishes for i≥2. But this is a load-bearing step: if the finiteness failed at some codimension-1 point, the isomorphisms ψ_η and ψ_P would not be defined, and the algebraic Hartogs argument that glues different presentations would collapse. The exposition therefore has a genuine missing verification at a point on which the whole construction depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a Deligne pairing functor ⟨L1,...,L_{d+1}⟩_{X/S}: Pic(X)^{d+1} → Pic(S) for a surjective projective morphism f: X → S of pure relative dimension d, where S is noetherian and normal. The construction first treats f-sufficiently ample line bundles by using presentations to build a projective bundle P → S, forming regular sections, taking norms over the finite part Z_V → V, descending to S up to ξ-torsion, and then gluing the resulting local pairings. Independence of presentations is proved in Proposition 4.12 by comparing stalks at the generic point and at codimension-1 points with García's Deligne pairing and applying algebraic Hartogs. The extension to general line bundles is made by differences and by gluing over an affine open cover (Theorem 4.24). The paper also proves functorial properties (base change, pullback of f^*M, projection formula, divisor sequence, birational invariance, and the d=0 norm case), constructs canonical sections of the pairing from f-regular sequences, and defines canonical smooth metrics for hermitian line bundles over C, with compatibility and isometry statements (Theorems 8.5, 8.9, 8.10).","tokens_in":41752,"tokens_out":3378,"duration_ms":33902,"significance":"If the construction is correct, this is a substantial generalization of Deligne pairing: it removes flatness and finite Tor-dimension assumptions, replacing them by normality of the base scheme. The reduction to García's pairing on the generic fiber and at codimension-1 points is a conceptually clear and potentially powerful method, and it provides a more geometric alternative to Nakayama's intersection sheaves. The paper also extends the hermitian (Arakelov) theory of Deligne pairings to equidimensional morphisms. The claims include machine-checkable functoriality diagrams and explicit reduction steps to prior work, which is a strength. However, the current exposition leaves several load-bearing verifications terse or delegated, so the result is plausible but not yet fully established in the written form.","major_comments":[{"comment":"The proof of independence of presentations relies on the assertion that the morphisms f_η: X_η → η and f_A: X_A → Spec O_{S,P} (for P ∈ S^(1)) have finite Tor-dimension, so that García's Deligne pairing and his results apply. The text states this only implicitly ('Since η is a regular scheme... Since Spec A is a regular scheme...') and does not prove it. This verification is load-bearing: without finite Tor-dimension, the isomorphisms ψ_η and ψ_P are not defined, and the algebraic Hartogs argument that glues different presentations collapses. The missing proof is short: κ(η) is a field, and normality of S implies O_{S,P} is a DVR (Serre's criterion), so every module has projective dimension at most 1. Please add this argument or an explicit reference.","section":"§4.2, Proposition 4.12"},{"comment":"The gluing over an affine open cover asserts the cocycle condition θ_{i'i''} ∘ θ_{ii'} = θ_{ii''} on triple intersections without proof. The canonical isomorphisms of Proposition 4.12 are defined via the isomorphisms on the generic and codimension-1 stalks, but the cocycle property is not automatic from that definition; it needs a verification (or a reference to a uniqueness statement that forces it). Since the existence of the global pairing depends on this glueing, the argument should be completed.","section":"§4.3, Theorem 4.24"},{"comment":"The proof that an isomorphism u_i: L_i → L'_i induces a canonical isomorphism ⟨L_1,...,L_{d+1}⟩_{X/S} → ⟨L'_1,...,L'_{d+1}⟩_{X/S} is terse. In particular, the reduction to f-regular sections via Lemma 7.3 assumes the existence of global sections s_i whose zero loci have the correct relative dimensions; Lemma 7.3 is only cited to [GLL15] and the argument 'apply Lemma 7.3 for L_1,...,L_d in turn' is not spelled out. Since Theorem 7.4 states a full functor on Picard categories, this step should be expanded so that the compatibility with the previously constructed isomorphisms (Σ, γ, base change, etc.) is verifiable.","section":"§7.2, Theorem 7.1"},{"comment":"The uniqueness and independence of the canonical metric are not fully formalized. The proof constructs the metric when L_1 admits an f-regular section s_1, then shows independence of the choice of s_1 using Lemmas 8.6 and 8.7, and finally extends to general L_1 by differences. However, the statement asserts a unique smooth metric satisfying (1)-(5), and the proof does not isolate the uniqueness argument from the existence argument in a way that makes clear that any metric satisfying the listed axioms must coincide with the constructed one. In particular, the 'unique' part should be justified explicitly, perhaps by showing that the Chern form condition (1) together with the section formula (2) determines the metric on a dense open set, and then invoking continuity.","section":"§8, Theorem 8.5"}],"minor_comments":[{"comment":"There are several typographical issues, e.g. 'P ic(S)' and 'Z/greaterorequalslant0' in the conventions; these should be cleaned up.","section":"Abstract and Introduction"},{"comment":"The definition of V_i uses a strict inequality 'dim f_P^{-1}(x) ∩ Z_i ≤ d−i' after the preceding equality; this is fine but the line break and notation could be clarified.","section":"§4.2, display (4.2)"},{"comment":"The commutative diagrams are hard to read because the horizontal arrows are not labeled consistently with the surrounding text (e.g. the left vertical arrow in (4.12) is described as 'the unique canonical isomorphism in the symmetric monoidal category Pic(S)', but the diagram labels only some arrows). Please label all arrows or explain the convention.","section":"§4.4, Proposition 4.20 and 4.29"},{"comment":"In the proof, the notation 'g(η') ∈ U' and 'P = g(P') ∈ V' mixes set-theoretic and scheme-theoretic language; it would be clearer to write the base change of points explicitly.","section":"§5, Theorem 5.3"},{"comment":"In Lemma 8.7, the notation 'Y1 = Z(s1)' and 'Y2 = Z(s2)' uses subscripts that are then used for two different purposes (the zero loci of s1 and s2); later in Lemma 8.6, 'Y = Y1 ∩ Y2' is used. This is potentially confusing; consider renaming the zero loci.","section":"§6, Lemma 8.7"},{"comment":"Some references are incomplete or informal: [Yua24] is cited but not explicitly used in the main text, and several arXiv/URL references would benefit from journal or volume information where available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and useful generalization, and the main strategy is sound. The missing verification in Proposition 4.12 is genuinely a load-bearing point, but it is also easily fixable; the other terse arguments (Theorem 4.24 cocycle, Theorem 7.1 reduction, Theorem 8.5 uniqueness) are local and can be addressed in revision. If the author supplies the missing details, I would be inclined to accept. The scope fits math.AG well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. Shiquan Li has the first Deligne pairing for surjective projective equidimensional morphisms over normal noetherian bases that does not require flatness or finite Tor-dimension. That's a real advance: it upgrades Nakayama's intersection sheaf to a symmetric multi-additive functor on Picard categories and adds the standard functorial properties plus hermitian metrics. The strategy—construct on presentations, descend the norm sheaf, then glue using algebraic Hartogs—is sensible, and the paper is honest that it reduces to García's pairing on the generic fiber and at codimension-one points.\n\nThe stress-test note is on target but the flaw is smaller than it sounds. Prop. 4.12 invokes García's results for f_η and f_A without proving they have finite Tor-dimension. That is a genuine missing verification. It is also true: κ(η) is a field, and normality of S makes O_{S,P} a DVR, so Tor_i vanishes for i≥2. The author should add a sentence (they almost do in the introduction) but the theorem stands.\n\nBigger soft spots: (i) Theorem 4.24 asserts the cocycle condition θ_{i'i''}∘θ_{ii'}=θ_{ii''} without proof. Uniqueness from Prop. 4.17 should force it, but the argument is not written and the reader cannot check it without doing work. (ii) Theorem 7.1's reduction to f-regular sections via Lemma 7.3 and [GLL15, Thm 5.1] is terse; the moving-lemma application should be explicit. (iii) In Theorem 8.5 the existence is shown by induction and independence from the chosen regular section is proved, but the uniqueness clause is not demonstrated as cleanly as the rest. These are exposition gaps, not detected errors. The 'good base change' restriction (generic point flat, codim-1 points finite Tor-dimension) is a real limitation but it is clearly stated.\n\nThe citation pattern is fine: García, Elkik, Ducrot, EF24 are the right prior work, and the comparison with García's pairing is one of the best parts. I found no circularity and no invented entities. The paper deserves a serious referee: the result is important, the architecture of the proof is plausible, and the missing details are fixable. My advice: send to a strong algebraic-geometry journal, and ask the author to expand 4.12, 4.24, 7.1, and the uniqueness part of 8.5.","headline":"A genuine extension of Deligne pairing to non-flat equidimensional morphisms over normal schemes; the construction is credible and the gaps are real but repairable.","tokens_in":42301,"tokens_out":2615,"would_cite":true,"duration_ms":24677,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C17","14F06","14G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any projective equidimensional family over a normal scheme, this paper constructs a Deligne pairing with all the expected functorial properties, and a canonical metric over C.","keywords":["Deligne pairing","equidimensional morphism","norm functor","normal scheme","intersection theory","Arakelov theory","hermitian line bundle","Picard category"],"falsifier":"Take a non-flat surjective projective morphism of pure relative dimension d over a noetherian normal scheme S, choose two different presentations of an f-sufficiently ample line bundle, and compute the descended line bundle J given by Proposition 4.11 for each presentation; the theorem asserts these are canonically isomorphic after restriction to the generic point and every codimension-one point of S. If for one such example the two line bundles do not glue through the unique Hartogs isomorphism (for instance, if the stalks over a codimension-one point are not canonically identified), Proposition 4.12 and Theorem 1.2 would fail.","tokens_in":41275,"feed_emoji":"🧮","tokens_out":9862,"duration_ms":99799,"temperature":0.7,"pith_summary":"The paper constructs, for every surjective projective morphism f:X→S of noetherian schemes of pure relative dimension d with normal base S, a Deligne pairing: a symmetric multi-additive functor sending d+1 line bundles on X to a line bundle on S. This extends the classical norm functor for finite morphisms and the Deligne pairing for flat families to equidimensional families that need not be flat. The author proves the pairing satisfies base change, pullback, projection formula, divisor sequence, birational invariance, and norm reduction; when f is flat it agrees with earlier definitions. It also constructs canonical sections from suitable section sequences and, over C, a canonical smooth metric whose first Chern form is the fiber integral of the product of the input Chern forms. These tools are intended for arithmetic intersection theory of hermitian line bundles for equidimensional morphisms.","feed_headline":"Deligne pairing constructed for non-flat equidimensional families","feed_subtitle":"Extends the norm functor and flat Deligne pairing, with canonical metrics for hermitian line bundles.","key_machinery":"The machinery is the norm functor Nm_{X'/X} for finite morphisms over normal schemes (from EGA II §6.5), García's Deligne pairing for morphisms of finite Tor-dimension applied after base change to the generic point and to codimension-one local rings, and algebraic Hartogs' theorem used to glue the resulting line bundles across codimension-one points. The descent step uses the fact that an invertible sheaf on an open subset of a projective space bundle whose complement in each fiber has codimension at least two (or three in one variant) descends to the base up to twists by O(1)'s. A relative-dimension-one gluing lemma (Proposition 3.3, identifying two norm functors through the determinants of two endomorphisms whose cokernels are isomorphic) is the key local step that makes the construction work.","core_discovery":"The central claim is that for a surjective projective morphism f:X→S of noetherian schemes of pure relative dimension d, with S normal, there exists a symmetric multi-additive functor ⟨L1,...,L_{d+1}⟩_{X/S}: Pic(X)^{d+1}→Pic(S) satisfying six functorial properties: good base change, pullback of f^*M, projection formula, divisor sequence, birational invariance, and reduction to the norm functor when d=0. The construction handles f-sufficiently ample line bundles by forming a projective space bundle P over S from presentations of the bundles, building a regular sequence of sections on X×_S P, taking the norm functor of the restriction of the remaining bundle to the finite part of the intersection, descending to S up to ξ-torsion, and proving independence of presentations via algebraic Hartogs' theorem using agreement over the generic point and codimension-one local rings with García's Deligne pairing. General bundles are then obtained by formal differences and gluing. When f is flat, the constructed pairing is identical to the original Deligne pairing of Deligne, Elkik, García, Ducrot, and Eriksson–Freixas. Over C, the pairing of hermitian line bundles carries a unique smooth metric satisfying the curvature identity c1(⟨L1,...,L_{d+1}⟩,‖·‖)=∫_{X/S}∏_{i=1}^{d+1} c1(L_i,‖·‖_i) and compatible with isometries and good base change.","pith_inferences":["Over a noetherian normal base, flatness can be replaced by the much weaker 'pure relative dimension' condition; the same Hartogs-glued construction may extend to non-noetherian normal bases if an analogue of the Hartogs theorem holds, since the norm-functor part is already set up without noetherian assumptions.","The canonical metric built here for non-flat families suggests one can define heights and intersection numbers for cycles in families with non-reduced or singular fibers, not just smooth ones.","Because the proof reduces all presentations to checks at codimension-one points, future generalizations could aim to define the pairing whenever the base is normal and the morphism has pure relative dimension, with any finite-Tor-dimension hypothesis verified only at those points."],"forward_implications":["The pairing reduces to the Grothendieck norm functor when d=0, so for finite morphisms over normal schemes it recovers the norm for possibly non-flat extensions.","For flat f, the construction is identical to the Deligne pairings of Deligne, Elkik, García, Ducrot, and Eriksson–Freixas, so all six functorial properties hold for those classical pairings as well.","The canonical sections of Theorem 1.5 give a way to evaluate the pairing on f-regular section sequences, with regularity of the output section guaranteed when the zero loci of all d+1 sections are disjoint.","Over C, the canonical metric satisfies the exact curvature identity c1(⟨L1,...,L_{d+1}⟩,‖·‖)=∫_{X/S} ∏_{i=1}^{d+1} c1(L_i,‖·‖_i), and the metric respects isometries and good base change, enabling analytic computations.","The construction provides a framework for defining arithmetic intersection theory of hermitian line bundles for equidimensional morphisms that need not be flat."],"supporting_citations":[{"why":"Provides the Deligne pairing for finite Tor-dimension morphisms used at the generic point and codimension-one local rings, and the regularity/descent propositions the construction relies on.","marker":"[Gar00]"},{"why":"Supplies the method of constructing intersection bundles via projections and the descent lemma for invertible sheaves on open subsets of projective bundles.","marker":"[Elk89]"},{"why":"Defines the norm functor for finite morphisms over normal schemes, which is the d=0 case and the building block for the relative-dimension-one gluing.","marker":"[GD61]"},{"why":"The original Deligne pairing for smooth projective families of curves, the model being generalized here.","marker":"[Del73]"},{"why":"Recent construction of Deligne pairing for flat projective morphisms without noetherian assumptions; used for compatibility and for section-sequence results in the flat case.","marker":"[EF24]"},{"why":"Provides the determinant-formula presentation of Deligne pairing in the flat case, used to identify the new construction with the classical one.","marker":"[Duc05]"},{"why":"Gives the moving lemma and adelic line-bundle techniques used to produce f-regular sequences and to construct the canonical metric.","marker":"[YZ24]"},{"why":"The hypersurface moving lemma used to find regular global sections whose zero loci have the correct relative dimension.","marker":"[GLL15]"},{"why":"Contains the method for constructing canonical metrics on intersection bundles by induction using f-regular sections and the Poincaré–Lelong formula.","marker":"[Elk90]"},{"why":"Earlier intersection sheaves over normal schemes, a related construction with stronger assumptions that the paper generalizes.","marker":"[Nak10]"}],"fun_headline_variants":["Deligne pairing for non-flat equidimensional morphisms","Symmetric multi-additive Deligne pairing, equidimensional case","Beyond flatness: Deligne pairing for pure relative dimension","New Deligne pairing via algebraic Hartogs for equidimensional maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on S being noetherian and normal so that algebraic Hartogs' theorem can glue the pairings defined over the generic point and over each codimension-one local ring, and it silently uses that each of those base-changed morphisms has finite Tor-dimension so that García's Deligne pairing applies there.","fun_headline_variants_meta":{"raw":{"variants":["Deligne pairing for non-flat equidimensional morphisms","Symmetric multi-additive Deligne pairing, equidimensional case","Beyond flatness: Deligne pairing for pure relative dimension","New Deligne pairing via algebraic Hartogs for equidimensional maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1442,"prompt_tokens":955,"completion_tokens":487,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":418}},"tokens_in":571,"tokens_out":487,"duration_ms":5106,"temperature":1.0,"reasoning_tokens":418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:09:54.360487+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-flat surjective projective morphism of pure relative dimension d over a noetherian normal scheme S, choose two different presentations of an f-sufficiently ample line bundle, and compute the descended line bundle J given by Proposition 4.11 for each presentation; the theorem asserts these are canonically isomorphic after restriction to the generic point and every codimension-one point of S. If for one such example the two line bundles do not glue through the unique Hartogs isomorphism (for instance, if the stalks over a codimension-one point are not canonically identified), Proposition 4.12 and Theorem 1.2 would fail.","supporting_citations":[],"review_version":1}