Embedding by mixed sections

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OAI.MaximalSeshadri.Geometry.LineBundle.mixed_section_embedding

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

maximal-seshadrioai-math

Let $X$ be integral and compact, proper over a commutative ring $K$ , $L,M$ line bundles, $s_i$ a finite family of sections of $L$ with nonvanishing opens affine, nonempty and covering $X$ , and trivializations of $M$ on those opens. Then there are $n>0$ and finitely many sections of $L^n\otimes M$ with iso-opens covering $X$ defining a closed immersion to $\operatorname{Proj}$ 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. sectionOpen X s is the same union for a global section $s$ : its nonvanishing open. 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 Opposite MvPolynomial
open MaximalSeshadri.Frames MaximalSeshadri.TensorPure MaximalSeshadri.Projective

variable {X : Scheme.{0}} {K ι : Type} [CommRing K] [Fintype ι]

Formal statement

theorem LineBundle.mixed_section_embedding [IsIntegral X] [CompactSpace X]
    (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] (L M : LineBundle X)
    (s : ι → GlobalSections X L.sheaf)
    (hc : (⨆ i, sectionOpen X (s i)) = ⊤)
    (haff : ∀ i, IsAffineOpen (sectionOpen X (s i)))
    (hne : ∀ i, (sectionOpen X (s i) : Set X).Nonempty)
    (d : ∀ i, M.sheaf.restrict (sectionOpen X (s i)).ι ≅ O (sectionOpen X (s i)).toScheme) :
    ∃ n : ℕ, 0 < n ∧ ∃ σ : Type, ∃ _ : Fintype σ,
      ∃ t : σ → GlobalSections X ((L.pow n).tensor M).sheaf,
      ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤,
        IsClosedImmersion (sectionsMorphism
          (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) t ht) := 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.mixed_section_embedding`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Projective/MixedEmbedding.lean#L19

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