Clearing denominators from an edge to a vertex

Proved

OAI.MaximalSeshadri.Projective.planePair_clearing

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

maximal-seshadrioai-math

For sections $s_0,s_1,s_2$ with affine iso-opens and a line bundle $L$ , in the Laurent module on the triple overlap, every $x\in$ planePair k s L i j admits $d\in\mathbb N$ such that the Laurent monomial $T^{d(w_j-w_i)}$ moves $x$ into planeVertex k s L i, where $w_0=(0,0),w_1=(1,0),w_2=(0,1)$ .

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. planeVertex and planePair are the images in the sections over the triple overlap (planeTriple) of sections over the chart of $s_i$ , and over the overlap of the charts of $s_i,s_j$ , respectively.

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.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Geometry MaximalSeshadri.Geometry.BaseSections MaximalSeshadri.Frames
open MaximalSeshadri.LaurentPlane MaximalSeshadri.PlaneCech

variable {K : Type} [Field K] {X : Scheme.{0}} {M : X.Modules}

Formal statement

theorem planePair_clearing (k : K →+* Γ(X,⊤))
    (s : Fin 3 → (O X ⟶ M)) (haff : ∀ i, IsAffineOpen (SectionOpens.isoOpen (s i)))
    (L : LineBundle X) :
    letI := laurentModule ((planeTriple s).ι.appTop.hom.comp k) (planeX s) (planeY s)
      (L.sheaf.restrict (planeTriple s).ι)
    ∀ i j x, x ∈ planePair k s L i j →
      ∃ d : ℕ, T (K := K) (d • (weight j-weight i)) • x ∈ planeVertex k s L i := by
  sorry

end
end MaximalSeshadri.Projective
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.planePair_clearing`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/PlanePairToric.lean#L31

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