openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /AdmissibleMatrixFrame.lean
| 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 | |