Effective sections with prescribed vanishing

Open

OAI.MaximalSeshadri.Geometry.Surface.eventually_effective_with_arbitrary_jet_frames

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

maximal-seshadrioai-math

Let $L$ be ample on $S$ and $M$ a line bundle with $L\cdot M\ge0$ . Let $m_i$ be natural numbers with $\sum_im_i^2<M^2$ , and for each $i$ let $U_i$ be an affine open with étale coordinates and a $\mathbb C$ -point $\rho_i$ . Then there is $N>0$ such that for $n\ge N$ and any frames $e_i$ of $M^n$ on $U_i$ , there is a nonzero section $s$ of $M^n$ with coefficient in $(\ker\rho_i)^{nm_i}$ for all $i$ .

Notation. $L^2$ is selfIntersection, namely $\chi(L^2)-2\chi(L)+\chi(\mathcal O_S)$ . $L\cdot M$ is mixedEuler, namely $\chi(L\otimes M)-\chi(L)-\chi(M)+\chi(\mathcal O_S)$ . 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 $\mathbb C$ .

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.Abelian TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.ProjectiveBertini

Formal statement

theorem Surface.eventually_effective_with_arbitrary_jet_frames (S : Surface)
    (L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme)
    (hm : 0 ≤ mixedEuler S L M) {r : ℕ} (m : Fin r → ℕ)
    (hq : (∑ i, (m i)^2 : ℕ) < selfIntersection S M)
    (U : Fin r → S.scheme.affineOpens)
    (t : ∀ i, Fin 2 → Γ(S.scheme,(U i).1))
    (het : ∀ i, (MvPolynomial.eval₂Hom (openScalars S.structureMap (U i).1) (t i)).Etale)
    (ρ : ∀ i, letI := (openScalars S.structureMap (U i).1).toAlgebra;
      Γ(S.scheme,(U i).1) →ₐ[ℂ] ℂ) :
    ∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N ≤ n →
      ∀ e : ∀ i, (M.pow n).sheaf.restrict (U i).1.ι ≅ O (U i).1.toScheme,
      ∃ s : O S.scheme ⟶ (M.pow n).sheaf, s ≠ 0 ∧
        ∀ i, affineCoefficient (U i) (e i) s ∈
          (RingHom.ker (ρ i))^(n*m i) := 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.eventually_effective_with_arbitrary_jet_frames`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Jets/EffectiveJetFrames.lean#L19

View graph