Ample line bundles give projective embeddings

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OAI.MaximalSeshadri.Geometry.LineBundle.projective_embedding

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

maximal-seshadrioai-math

Let $X$ be integral and compact with $p:X\to\operatorname{Spec}\mathbb C$ proper, and $L$ ample. Then there exist $N$ and a closed immersion $e:X\to\mathbb P^N_{\mathbb C}$ compatible with the maps to $\operatorname{Spec}\mathbb C$ .

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.

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 CategoryTheory AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective MaximalSeshadri.SectionOpens

Formal statement

theorem LineBundle.projective_embedding {X : Scheme.{0}} [IsIntegral X] [CompactSpace X]
    (p : X ⟶ Spec (.of ℂ)) [IsProper p] (L : LineBundle X) (hL : L.IsAmple) :
    ∃ N : ℕ, ∃ e : X ⟶ complexProjectiveSpace N,
      IsClosedImmersion e ∧ e ≫ projectiveSpaceToSpec N = p := 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.projective_embedding`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Blowup/PointBlowupExistence.lean#L53

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