Ample bundles have an affine cover in a common degree

Proved

OAI.MaximalSeshadri.Geometry.LineBundle.ample_common_degree_cover

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

maximal-seshadrioai-math

For a compact scheme $X$ and $L$ ample, there are $n>0$ and finitely many sections of $L^n$ whose nonvanishing opens cover $X$ and are affine and nonempty.

Notation. sectionOpen X s is the same union for a global section $s$ : its nonvanishing open. IsAmple means that for each point $x$ and open $V\ni x$ , some positive power has a global section with nonvanishing open containing $x$ , inside $V$ , and affine.

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
noncomputable section
open CategoryTheory
open CategoryTheory.Category CategoryTheory.Functor
universe v u v₁ v₂ u₁ u₂
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace TopologicalSpace.Opens
open scoped AlgebraicGeometry
open MaximalSeshadri.Frames
variable {X : Scheme}

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.InvertibleLocal
noncomputable section
open CategoryTheory AlgebraicGeometry
open MaximalSeshadri.Geometry
variable {X Y : Scheme}

end
end MaximalSeshadri.InvertibleLocal

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace TopologicalSpace.Opens
open scoped AlgebraicGeometry
open MaximalSeshadri.Frames
variable {X : Scheme}

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.TensorLocalization
noncomputable section
open CategoryTheory MonoidalCategory
universe v' u'
variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
  {D : Type u'} [Category.{v'} D] (F : C ⥤ D)

end
end MaximalSeshadri.TensorLocalization

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace MonoidalCategory
open scoped AlgebraicGeometry
variable {X : Scheme}

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace MonoidalCategory BraidedCategory
open scoped AlgebraicGeometry
variable {X : Scheme}

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace TopologicalSpace.Opens
open scoped AlgebraicGeometry
variable {X : Scheme}

Formal statement

theorem LineBundle.ample_common_degree_cover [CompactSpace X] (L : LineBundle X)
    (hL : LineBundle.IsAmple X L) :
    ∃ n : ℕ, 0 < n ∧ ∃ l : ℕ,
      ∃ s : Fin l → GlobalSections X (modulePow X L.sheaf n),
        (⨆ i, sectionOpen X (s i)) = ⊤ ∧
        (∀ i, IsAffineOpen (sectionOpen X (s i))) ∧
        (∀ i, (sectionOpen X (s i) : Set X).Nonempty) := 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.LineBundle.ample_common_degree_cover`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Sheaves/PowerExtension.lean#L360

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