Integral quartic sections of an ample power

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OAI.MaximalSeshadri.Geometry.Surface.ample_quartic_integral

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

maximal-seshadrioai-math

For $L$ ample on $S$ there are $d>0$ , $N,n$ , sections $s$ of $L^d$ indexed by $\operatorname{Option}(\operatorname{Fin}N)$ with iso-opens covering $S$ , an equivalence $e$ from $\operatorname{Option}(\operatorname{Fin}n)$ to QuarticIndex (Fin N), and a point $y$ such that $y$ is the only point outside centeredOpen s, the sections define a closed immersion, and for every nonzero polynomial $P$ there is $v$ with $P(v)\ne0$ such that doublePointQuarticIdeal k s hs (v ∘ e.symm) has integral subscheme, smooth over $\mathbb C$ on the centered open.

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 $\Gamma(X,\top)$ . 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. 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 $\mathbb C$ . centeredOpen s is the union of the iso-opens of $s_{\mathrm{some}\,i}$ . QuarticIndex σ is $\sigma\times\sigma\times\operatorname{Option}\sigma\times\operatorname{Option}\sigma$ ; doublePointQuartics are the corresponding degree-4 mixed products of the sections.

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.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Geometry MaximalSeshadri.Hartogs
open MaximalSeshadri.ProjectiveBertini
attribute [local instance] MvPolynomial.gradedAlgebra

end
end MaximalSeshadri.Projective

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

attribute [local instance] MvPolynomial.gradedAlgebra

Formal statement

theorem Surface.ample_quartic_integral (S : Surface)
    (L : LineBundle S.scheme) (hL : LineBundle.IsAmple S.scheme L) :
    ∃ d : ℕ, 0 < d ∧ ∃ N n : ℕ,
      ∃ s : Option (Fin N) → GlobalSections S.scheme (modulePow S.scheme L.sheaf d),
      ∃ hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤,
      ∃ e : Option (Fin n) ≃ QuarticIndex (Fin N), ∃ y : S.scheme,
        (∀ x : S.scheme, x ∉ centeredOpen s ↔ x = y) ∧
        IsClosedImmersion (sectionsMorphism
          (S.structureMap.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom) s hs) ∧
        ∀ P : MvPolynomial (Option (Fin n)) ℂ, P ≠ 0 →
          ∃ v : Option (Fin n) → ℂ, MvPolynomial.aeval v P ≠ 0 ∧
            let k := S.structureMap.appTop.hom.comp
              (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom
            let V := centeredOpen s
            let J := doublePointQuarticIdeal k s hs (v ∘ e.symm)
            IsIntegral J.subscheme ∧
              Smooth ((J.comap V.ι).subschemeι ≫ V.ι ≫ S.structureMap) := 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.Surface.ample_quartic_integral`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Projective/GlobalIntegralQuartic.lean#L68

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