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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.NumberTheory.PiExponent.Ampleness.WeightedAffineFrame
import OAI.NumberTheory.PiExponent.Jets.AffineJetCoefficientInterface
namespace OAI
noncomputable section
namespace PiExponent.AdmissibleMatrixInterpolation
open AdmissibleBlowupGeometry AlgebraicGeometry CategoryTheory PiExponentSeshadri.Geometry
attribute [local irreducible] WeightedCompactification.lineBundle
WeightedCompactification.affineChartMap
AffineJetCoefficientInterface.Frame
theorem frame_exists_of_actual {X : Scheme} {R : Type} [CommRing R]
(j : Spec (CommRingCat.of R) ⟶ X) [IsOpenImmersion j] (A : LineBundle X)
(h : Nonempty (A.sheaf.restrict j.opensRange.ι ≅
structureSheaf j.opensRange.toScheme)) :
Nonempty (AffineJetCoefficientInterface.Frame j A) := by
unfold AffineJetCoefficientInterface.Frame
exact h
theorem monomialFrame_exists {R ι σ : Type} [CommRing R]
(a : σ → ι →₀ ℕ) (z : σ) (hz : a z = 0)
(coordinate : ι → σ) (hcoordinate : ∀ i, a (coordinate i) = Finsupp.single i 1) :
Nonempty (AffineJetCoefficientInterface.Frame
(WeightedCompactification.affineChartMap (R := R) a z hz coordinate hcoordinate)
(WeightedCompactification.lineBundle (R := R) a)) := by
exact frame_exists_of_actual
(X := Proj (WeightedCompactification.imageGrade (R := R) a))
(R := MvPolynomial ι R)
(WeightedCompactification.affineChartMap (R := R) a z hz coordinate hcoordinate)
(WeightedCompactification.lineBundle (R := R) a)
(WeightedAffineFrame.affineFrame_nonempty (R := R) a z hz coordinate hcoordinate)
theorem actualFrame_exists {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D) :
Nonempty (AffineJetCoefficientInterface.Frame (affineChart d) (hyperplane d)) := by
exact monomialFrame_exists (R := ℂ)
(exponents d) (constantIndex d)
((scale d).exponents_constant d.curveDegreeWeights_pos)
(coordinateIndex d) ((scale d).exponents_coordinate d.curveDegreeWeights_pos)
end PiExponent.AdmissibleMatrixInterpolation
end
end OAI
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