Ampleness from an affine section cover

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OAI.MaximalSeshadri.Geometry.LineBundle.ample_of_affine_section_cover

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

maximal-seshadrioai-math

Let $X$ be integral and compact, $L$ a line bundle, and $s$ a family of global sections whose nonvanishing opens cover $X$ and are affine. Then $L$ is ample.

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

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective MaximalSeshadri.TensorPure

variable {X : Scheme.{0}}

Formal statement

theorem LineBundle.ample_of_affine_section_cover [IsIntegral X] [CompactSpace X]
    (L : LineBundle X) {ι : Type*} (s : ι → (O X ⟶ L.sheaf))
    (hc : (⨆ i, sectionOpen X (s i)) = ⊤)
    (ha : ∀ i, IsAffineOpen (sectionOpen X (s i))) : L.IsAmple := by
  sorry

end
end MaximalSeshadri.Geometry
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_of_affine_section_cover`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Geometry/AmpleFromCover.lean#L56

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