The ideal of an integral curve is invertible

Open

OAI.MaximalSeshadri.Geometry.IntegralCurve.invertible_ideal

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

maximal-seshadrioai-math

For an integral curve $C$ on a surface $S$ , the kernel ideal of $C\to S$ pulled back along the identity is invertible: $C$ is Cartier on $S$ .

Notation. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$ . An IntegralCurve on a surface is an integral closed subscheme of topological Krull dimension one, given by a closed immersion.

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.LocalCurve
noncomputable section

end
end MaximalSeshadri.LocalCurve

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.ProjectiveBertini

Formal statement

theorem IntegralCurve.invertible_ideal (S : Surface) (C : IntegralCurve S) :
    InvertiblePullbackIdeal C.embedding.ker (𝟙 S.scheme) := 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.IntegralCurve.invertible_ideal`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/IntegralCurveCartier.lean#L107

View graph