The section ideal in any local frame

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OAI.MaximalSeshadri.Projective.sectionIdeal_on_any_frame

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

maximal-seshadrioai-math

For a finite family $s$ of sections of $M$ with iso-opens covering $X$ , coefficients $v$ , and an affine open $U$ with a trivialization $e$ of $M$ on $U$ , the ideal of sectionIdeal k s hs v on $U$ is generated by the coefficient, in the frame $e$ , of $\sum_j k(v_j)s_j$ .

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.Frames
noncomputable section
open AlgebraicGeometry CategoryTheory
variable {X Y : Scheme}

end
end MaximalSeshadri.Frames

namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.IdealPullback
attribute [local instance] MvPolynomial.gradedAlgebra
variable {K σ : Type} [Field K] [Fintype σ] {X : Scheme}

Formal statement

theorem sectionIdeal_on_any_frame {M : X.Modules} (k : K →+* Γ(X, ⊤))
    (s : σ → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    (v : σ → K) (U : X.affineOpens) (e : M.restrict U.1.ι ≅ O U.1.toScheme) :
    (sectionIdeal k s hs v).ideal U = Ideal.span
      {U.1.topIso.hom (coefficient e (restrictSection U.1.ι (sectionCombination k s v)))} := 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_on_any_frame`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/PrincipalSectionIdeal.lean#L84

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