Vanishing of higher cohomology on projective curves

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OAI.MaximalSeshadri.Geometry.projective_curve_ext_zero

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

maximal-seshadrioai-math

Let $K$ be an infinite field, $X$ integral Noetherian, $p:X\to\operatorname{Spec}K$ proper, and $\dim X=1$ . Suppose the sections $s$ of a module define a closed immersion to $\operatorname{Proj}K[\sigma]$ . For every quasi-coherent $M$ and $n$ , $\operatorname{Ext}^{n+2}(\mathcal O_X,M)=0$ .

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

section

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Abelian
open MaximalSeshadri.Projective MaximalSeshadri.ModuleFlasque
attribute [local instance] MvPolynomial.gradedAlgebra

variable {X : Scheme.{0}} [IsNoetherian X]

Formal statement

theorem projective_curve_ext_zero {K σ : Type} [Field K] [Infinite K] [Fintype σ]
    [IsIntegral X] (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p]
    (hd : topologicalKrullDim X = 1) {N : X.Modules}
    (s : σ → (MaximalSeshadri.Frames.O X ⟶ N))
    (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [IsClosedImmersion (sectionsMorphism
      (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)]
    (M : X.Modules) [M.IsQuasicoherent] (n : ℕ) (x : cohomology M (n+2)) : x = 0 := by
  sorry

end
end MaximalSeshadri.Geometry
end
end
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.projective_curve_ext_zero`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/CurveCohomology.lean#L63

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