Global sections on a surface are finite-dimensional

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OAI.MaximalSeshadri.Geometry.Surface.sections_finite

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

maximal-seshadrioai-math

For $L$ ample on a surface $S$ and any line bundle $M$ , the space of global sections of $M$ is a finite-dimensional $\mathbb C$ -vector space under complexSectionModule.

Notation. 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. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$ .

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 Abelian TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.ProjectiveBertini
open MaximalSeshadri.AnalyticCoordinates MaximalSeshadri.AlgebraicJets
open MaximalSeshadri.ActualInterpolation MaximalSeshadri.LocalComparison

variable {X : Scheme.{0}}

Formal statement

theorem Surface.sections_finite (S : Surface) (L : LineBundle S.scheme) (hL : L.IsAmple)
    (M : LineBundle S.scheme) :
    letI := complexSectionModule S.structureMap M.sheaf
    Module.Finite ℂ (GlobalSections S.scheme M.sheaf) := 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.Surface.sections_finite`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/SectionGermSystem.lean#L72

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