Finiteness of the relation quotient for the plane cover

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OAI.MaximalSeshadri.Geometry.finite_plane_relation_quotient

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

maximal-seshadrioai-math

In the setting of a finite morphism to the plane given by $s_0,s_1,s_2$ on an integral $X$ with nonempty triple overlap, for a line bundle $L$ the quotient of $\operatorname{Hom}$ from planeRelations s to $L$ by relationBoundaries k s L is finite over $K$ .

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionsMorphism k s hs is the morphism from $X$ to $\operatorname{Proj}$ of the graded polynomial ring in variables indexed by the sections, defined by sections whose iso-opens cover $X$ , with $k$ the ring map from the base into $\Gamma(X,\top)$ .

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 CategoryTheory.Limits TopologicalSpace
open MaximalSeshadri.Projective MaximalSeshadri.Frames BaseSections ModuleFlasque ModuleMayerVietoris
open MaximalSeshadri.PlaneCech

variable {K : Type} [Field K] {X : Scheme.{0}}
attribute [local instance] OAI.MaximalSeshadri.Geometry.instLinearCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopSheafOfModulesRingCatSheaf_definitions_1

attribute [local instance] OAI.MaximalSeshadri.Geometry.instHasExtSheafOfModulesOpensCarrierCarrierCommRingCatRingCatSheaf_definitions_1

attribute [local instance] baseHomModule

Formal statement

theorem finite_plane_relation_quotient [IsIntegral X] (k : K →+* Γ(X,⊤)) {M : X.Modules}
    (s : Fin 3 → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [IsFinite (sectionsMorphism k s hs)] [Nonempty (planeTriple s)] (L : LineBundle X) :
    Module.Finite K (BaseHom k (planeRelations s) L.sheaf ⧸ relationBoundaries k s L) := 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.finite_plane_relation_quotient`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/SurfaceH1Finite.lean#L26

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