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Deligne pairing for equidimensional morphisms

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.17410 v1 pith:2VXN4GCZ submitted 2024-11-26 math.AG

Shiquan Li This is my paper · ORCID
classification math.AG MSC 14C1714F0614G40
keywords DelignepairingequidimensionalmorphismnormfunctornormalschemeintersectiontheoryArakelovhermitianlinebundlePicardcategory
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 6 minor

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).

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 (4)
  1. [§4.2, Proposition 4.12] 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.
  2. [§4.3, Theorem 4.24] 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.
  3. [§7.2, Theorem 7.1] 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.
  4. [§8, Theorem 8.5] 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.
minor comments (6)
  1. [Abstract and Introduction] There are several typographical issues, e.g. 'P ic(S)' and 'Z/greaterorequalslant0' in the conventions; these should be cleaned up.
  2. [§4.2, display (4.2)] 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.
  3. [§4.4, Proposition 4.20 and 4.29] 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.
  4. [§5, Theorem 5.3] 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.
  5. [§6, Lemma 8.7] 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.
  6. [References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the equidimensional Deligne pairing is built by descent from a norm sheaf, and presentation-independence is proved by gluing García's prior pairing on generic and codimension-1 stalks; load-bearing inputs are external prior results.

full rationale

This paper's derivation chain is self-contained against external benchmarks; there is no circular step. The new object, the Deligne pairing <L1,...,L(d+1)>_{X/S} for a surjective projective equidimensional morphism f: X -> S over a noetherian normal S, is constructed in Sections 4.2-4.3 by the Elkik-García method: for f-sufficiently ample line bundles one picks presentations (phi_i, F_i), forms the projective bundle P = prod_i P(F_i^vee), produces canonical regular sections sigma_i, takes the finite morphism Z_V -> V cut out by sigma_1,...,sigma_d, applies Grothendieck's norm functor (Theorem 2.7, from [GD61] section 6.5), and descends the resulting sheaf N to a line bundle J on S (Proposition 4.11). The hard step, independence of presentations (Proposition 4.12), is proved by algebraic Hartogs: the stalks of J at the generic point eta and at each codimension-1 point P are identified with García's Deligne pairing [Gar00] for the base-change morphisms f_eta and f_A, using base-change compatibility of the norm functor (Proposition 2.9) and of descent, and the uniqueness of the gluing follows from the standard fact that a line bundle on a normal scheme is determined by its stalks at eta and codimension-1 points. Nothing here assumes the target result; the identification with García's pairing is a theorem about the stalks of the constructed J, and Proposition 4.16 derives coincidence with [Gar00] in the flat case as a consequence, not as an input. No parameter is fitted, and no known result is renamed: the comparison with Nakayama's intersection sheaf [Nak10] is a discussion of differences, not a reduction. The citations to Yuan-Zhang [YZ24] and Yuan [Yua24] (the advisor's prior work, not the present author's) support only flat-case and field-case machinery (Theorem 4.2.3, Lemma 4.2.2, integration formulae) in Sections 6 and 8; these are external, parameter-free results whose assumptions do not include the equidimensional pairing, and the induction extending them is the paper's own, so the citations are not load-bearing self-citations. One genuine gap must be flagged but does not affect the verdict: the text asserts without proof (Section 1.1, page 5, and Proposition 4.12) that f_eta and f_A have finite Tor-dimension, which is load-bearing for applying [Gar00].

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

The central claim rests on standard algebraic geometry background (norm functor, algebraic Hartogs, descent) and on prior results of García, Eriksson-Freixas, and Gabber-Liu-Lorenzini. There are no fitted parameters and no new postulated entities. The most essential or fragile premise is the finite Tor-dimension assertion over generic and codimension-1 points, which is true but not proved in the text.

assumptions (5)
  • domain assumption S is noetherian and normal (Definition 1.1, Theorem 1.2).
    Normality is used for algebraic Hartogs gluing on codimension-1 points and for the norm functor Nm_{X/S} for finite morphisms (Theorem 2.7). Without it, the construction may fail to yield an invertible sheaf.
  • standard math Algebraic Hartogs theorem: an invertible sheaf on a noetherian normal scheme is determined by its generic and codimension-1 restrictions.
    Used in Proposition 4.12 and Proposition 4.17 to glue isomorphisms from the generic fiber and codimension-1 stalks.
  • domain assumption García's Deligne pairing for surjective equidimensional projective morphisms of finite Tor-dimension, including its functorial properties.
    The construction uses [Gar00] as a black box over the generic point and over codimension-1 points; these results are prior theorems, not proved in this paper.
  • standard math Every morphism from a noetherian scheme to a regular scheme (a field or a DVR) has finite Tor-dimension.
    Used to apply García's theory over eta and Spec O_{S,P}. This holds because fields and DVRs have global dimension at most 1, so the structure sheaf has finite flat dimension.
  • domain assumption Existence of f-regular global sections for sufficiently positive line bundles ([GLL15] Theorem 5.1).
    Used in Section 6 and Theorem 7.1 to reduce to f-regular sequences, and in Lemma 8.7 to prove independence of the metric from the chosen section.

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Pith. "Pith review of Deligne pairing for equidimensional morphisms." pith.science (2026). https://pith.science/paper/2VXN4GCZ

@misc{pith2026241117410,
  author       = {Pith},
  title        = {Pith review of: Deligne pairing for equidimensional morphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VXN4GCZ}},
  note         = {Machine review of arXiv:2411.17410}
}
abstract

Let $S$ be a noetherian normal scheme, and let $X\to S$ be a surjective projective morphism of pure relative dimension $d$. We construct a symmetric multi-additive functor $\mathcal{P}\mathrm{ic}(X)^{d+1} \to \mathcal{P}\mathrm{ic}(S)$, and prove its functorial properties. Our construction uses Elkik's and Garc\'ia's ideas, as well as algebraic Hartogs' theorem. Moreover, our results can be used to define arithmetic intersection theory of hermitian line bundles for equidimensional morphisms.

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Works this paper leans on

4 extracted references · 2 canonical work pages

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