A section vanishing to positive order vanishes on a curve

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OAI.MaximalSeshadri.Geometry.Surface.section_contains_curve_of_order

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

maximal-seshadrioai-math

Let $M$ be a line bundle on a surface $S$ , $s\ne0$ a section, $U$ an affine open with trivialization $e$ , $\rho$ a $\mathbb C$ -point of $\Gamma(U)$ , and $n>0$ with the coefficient of $s$ in $(\ker\rho)^n$ . Then there is an integral curve $C$ on $S$ on which $s$ pulls back to zero.

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.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.ProjectiveBertini

Formal statement

theorem Surface.section_contains_curve_of_order (S : Surface) (M : LineBundle S.scheme)
    (s : O S.scheme ⟶ M.sheaf) (hs : s ≠ 0)
    (U : S.scheme.affineOpens) (e : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
    (ρ : letI := (openScalars S.structureMap U.1).toAlgebra;
      Γ(S.scheme,U.1) →ₐ[ℂ] ℂ) (n : ℕ) (hn : 0 < n)
    (ho : affineCoefficient U e s ∈ (RingHom.ker ρ)^n) :
    ∃ C : IntegralCurve S, pullbackSection C.embedding s = 0 := 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.Surface.section_contains_curve_of_order`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/SectionCurveBound.lean#L55

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