A positive line bundle with a section on a projective curve

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OAI.MaximalSeshadri.Geometry.projective_curve_positive_bundle

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

maximal-seshadrioai-math

For $X$ integral Noetherian, $p:X\to\operatorname{Spec}\mathbb C$ proper with $\dim X=1$ , and sections $a$ of a module $A$ giving a closed immersion, there are a line bundle $H$ , a section $s$ , and a point $x$ with $x$ in the iso-open of $s$ and $\chi(H)-\chi(\mathcal O_X)>0$ .

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.

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.Limits AlgebraicGeometry TopologicalSpace Opposite
open MaximalSeshadri.Projective MaximalSeshadri.Frames MaximalSeshadri.SectionOpens
attribute [local instance] MvPolynomial.gradedAlgebra

variable {X : Scheme.{0}} [IsIntegral X] [IsNoetherian X]
variable {σ : Type} [Fintype σ]
variable (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
    (hd : topologicalKrullDim X = 1) {A : X.Modules}
    (a : σ → (O X ⟶ A)) (ha : (⨆ i, isoOpen (a i)) = ⊤)
    [IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)]
include hd a ha

Formal statement

theorem projective_curve_positive_bundle :
    ∃ (H : LineBundle X) (s : O X ⟶ H.sheaf) (x : X),
      x ∈ isoOpen s ∧ 0 < eulerCharacteristic p 1 H.sheaf - eulerCharacteristic p 1 (O X) := 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_curve_positive_bundle`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Intersection/AllCurveDegree.lean#L27

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