| import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupGeometry |
| import OAI.NumberTheory.PiExponent.Jets.AffineJetFramedPackets |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PiExponent.AdmissibleJetPackets |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry |
| open PiExponent.AdmissibleBlowupGeometry |
| open PiExponent.BlowupJetSurjectivity PiExponent.ExceptionalAffineChart |
| attribute [local irreducible] AdmissibleBlowupGeometry.affineChart AdmissibleBlowupGeometry.hyperplane |
| AdmissibleBlowupGeometry.centerIdeal |
| AffineJetCoefficientInterface.Frame AffineJetCoefficientInterface.Sections |
| AffineJetCoefficientInterface.coefficient |
| variable {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D) |
|
|
| abbrev affineOpen := chartOpen (affineChart d) |
|
|
| def sectionPolynomial (n : ℕ) |
| (e : AffineJetCoefficientInterface.Frame (affineChart d) (hyperplane d)) : |
| AffineJetCoefficientInterface.Sections (hyperplane d) n → PiExponentApprox.FramePolynomial d.m := |
| AffineJetCoefficientInterface.coefficient (affineChart d) (hyperplane d) n e |
|
|
| theorem formalPackets_surjective_of_jetRestriction (n : ℕ) |
| (e : AffineJetCoefficientInterface.Frame (affineChart d) (hyperplane d)) |
| (hjet : Function.Surjective (jetRestriction (centerIdeal d) (hyperplane d) n)) : |
| Function.Surjective (fun s j => |
| JetGeometry.rationalCoefficientPacket d.curveJetWeights (n * (scale d).radius) |
| (FormalLogJet.formalJet (d.curveCenters j) (sectionPolynomial d n e s))) := by |
| have hc : Function.Injective d.curveCenters := by |
| let i : Fin d.m := ⟨0, by have := d.m_pos; omega⟩ |
| intro j k h |
| exact d.curveCenters_injective i (congrFun h i) |
| have he : ∀ i, (scale d).radius ≤ ((scale d).jetPowers i : ℚ) * d.curveJetWeights i := by |
| intro i |
| rw [mul_comm, (scale d).jetPowers_eq] |
| have hs : (((centerIdeal d)^n).support : Set (compactification d)) ⊆ |
| (affineChart d).opensRange := by |
| cases n with |
| | zero => simp |
| | succ n => simpa using centerIdeal_support_subset_chart d |
| exact AffineJetFramedPackets.surjective_of_jetRestriction d.curveCenters hc |
| (logCutoff d) (scale d).jetPowers (scale d).jetPowers_pos |
| d.curveJetWeights d.curveJetWeights_pos (fun i => (logCutoff_strict d i).le) |
| (scale d).radius he n (affineChart d) (centerIdeal d) (hyperplane d) e |
| (centerIdeal_restrict d) hs hjet |
|
|
| end PiExponent.AdmissibleJetPackets |
| end |
|
|
| end OAI |
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|
