Section opens as basic opens of the Rees Proj

Open

OAI.MaximalSeshadri.ReesGrading.relative_generator_section_open

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

maximal-seshadrioai-math

Let $I$ be an ideal of a ring $R$ , $f:B\to\operatorname{Spec}R$ , $J,\iota$ presenting the pullback of $I$ along $f$ , $\beta:B\to\operatorname{Proj}\bigoplus I^n$ a lift of $f$ , and $a\in I$ . If a section $s$ of $J$ satisfies $s\circ\iota=$ multiplication by the image of $a$ , then the iso-open of $s$ is the preimage under $\beta$ of the basic open of the degree-one generator attached to $a$ .

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism.

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.ReesGrading
noncomputable section

open CategoryTheory AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Geometry MaximalSeshadri.Frames MaximalSeshadri.SpecMaps

variable {R : Type} [CommRing R] (I : Ideal R)
variable {B : Scheme.{0}} (f : B ⟶ Spec (.of R))
  (J : LineBundle B) (ι : J.sheaf ⟶ O B)
  (hJ : PresentsPullbackIdeal (IdealPullback.specIdeal I) f J ι)
include hJ

Formal statement

theorem relative_generator_section_open (β : B ⟶ affineBlowup I) (hβ : β ≫ projection I = f) (a : I)
    (s : O B ⟶ J.sheaf)
    (hs : s ≫ ι = scalarEnd ((coordinate f) a.val)) :
    SectionOpens.isoOpen s = β ⁻¹ᵁ Proj.basicOpen (piece I) (generator I a) := by
  sorry

end
end MaximalSeshadri.ReesGrading
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.ReesGrading.relative_generator_section_open`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Blowup/RelativeReesSections.lean#L56

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