Tensor Euler difference on a curve as a length

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OAI.MaximalSeshadri.Geometry.projective_tensor_euler_difference

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

maximal-seshadrioai-math

On a projective curve $X$ over $\mathbb C$ with line bundles $L,M$ and a nonzero section $s$ of $L$ : $\chi(L\otimes M)-\chi(M)$ equals the $\mathbb C$ -dimension of the global sections of the cokernel of $M\to L\otimes M$ (sectionMultiply L M s).

Notation. $\chi$ is eulerCharacteristic: the alternating sum of $\mathbb C$ -dimensions of $\operatorname{Ext}^n(\mathcal O,-)$ , for $n\le2$ on a surface and $n\le1$ on a curve. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionsMorphism k s hs is the morphism from $X$ to $\operatorname{Proj}$ of the graded polynomial ring in variables indexed by the sections, defined by sections whose iso-opens cover $X$ , with $k$ the ring map from the base into $\Gamma(X,\top)$ .

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.

Preamble

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section
noncomputable section

section

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.SectionOpens MaximalSeshadri.Projective

variable {X : Scheme.{0}}

variable [IsIntegral X] [IsNoetherian X]

Formal statement

theorem projective_tensor_euler_difference {σ : Type} [Fintype σ]
    (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
    (hd : topologicalKrullDim X = 1) {A : X.Modules}
    (a : σ → (O X ⟶ A)) (ha : (⨆ i, SectionOpens.isoOpen (a i)) = ⊤)
    [IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)]
    (L M : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) :
    eulerCharacteristic p 1 (L.tensor M).sheaf - eulerCharacteristic p 1 M.sheaf =
      letI := Module.compHom Γ(cokernel (sectionMultiply L M s), ⊤) (baseScalars p)
      (Module.finrank ℂ Γ(cokernel (sectionMultiply L M s),⊤) : ℤ) := by
  sorry

end
end MaximalSeshadri.Geometry
end
end
end
end OAI

Source

OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces, September 23, 2026, https://github.com/openai/math (paper: preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf; Lean: lean/OAI/AlgebraicGeometry/Seshadri, Apache License 2.0); Lean declaration `OAI.MaximalSeshadri.Geometry.projective_tensor_euler_difference`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Sheaves/TensorCurveEuler.lean#L135

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