Section ideals are effective Cartier

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OAI.MaximalSeshadri.Projective.sectionIdeal_effective_cartier

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

maximal-seshadrioai-math

Let $X$ be integral, $L$ a line bundle, and $s$ a finite family of sections with iso-opens covering $X$ and nonzero combination $\sum_j k(v_j)s_j$ . Then sectionIdeal k s hs v pulls back along the identity to an invertible ideal.

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionIdeal k s hs v is the comap along sectionsMorphism of the ideal sheaf of the hyperplane with coefficients $v$ ; sectionCombination k s v is $\sum_j k(v_j)s_j$ .

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 AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Frames
variable {X Y : Scheme}

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Geometry
variable {K σ : Type} [Field K] [Fintype σ] {X : Scheme}

Formal statement

theorem sectionIdeal_effective_cartier [IsIntegral X] (L : LineBundle X)
    (k : K →+* Γ(X, ⊤)) (s : σ → (O X ⟶ L.sheaf))
    (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) (v : σ → K)
    (hv : sectionCombination k s v ≠ 0) :
    InvertiblePullbackIdeal (sectionIdeal k s hs v) (𝟙 X) := by
  sorry

end
end MaximalSeshadri.Projective
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.Projective.sectionIdeal_effective_cartier`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/CartierSection.lean#L68

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