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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 46 47 48 49 50 51 | import OAI.NumberTheory.PiExponent.Approximation.AdmissibleMatrixFrame
import OAI.NumberTheory.PiExponent.Approximation.AdmissibleMatrixInterpolation
import OAI.NumberTheory.PiExponent.Approximation.WeightedGlobalSectionBound
import OAI.NumberTheory.PiExponent.Jets.AdmissibleJetPackets
import OAI.NumberTheory.PiExponent.Jets.AdmissibleJetSurjectivity
namespace OAI
noncomputable section
namespace PiExponent.AdmissibleMatrixInterpolation
open AlgebraicGeometry CategoryTheory Filter
open PiExponentSeshadri.Geometry
open DeterminantContradiction AdmissibleBlowupGeometry AdmissibleJetPackets
attribute [local irreducible] WeightedCompactification.lineBundle
WeightedCompactification.affineChartMap
AffineJetCoefficientInterface.Frame AffineJetCoefficientInterface.Sections
AffineJetCoefficientInterface.coefficient
variable {ν : ℝ} (d : FixedData ν)
theorem eventual_packets_of_eventual_jetRestriction
(hjet : ∀ᶠ n : ℕ in atTop, Function.Surjective
(BlowupJetSurjectivity.jetRestriction (centerIdeal d) (hyperplane d) n)) :
∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n) := by
obtain ⟨N,hN⟩ := WeightedGlobalSectionBound.eventual_admissible_supportBound d
obtain ⟨e⟩ := actualFrame_exists d
filter_upwards [hjet, eventually_ge_atTop N] with n hn hnN
apply packetMap_surjective_of_family d n (sectionPolynomial d n e)
· intro s
exact hN n hnN e s
· exact formalPackets_surjective_of_jetRestriction d n e hn
theorem globalInterpolation_of_eventual_jetRestriction
(hjet : ∀ ν : ℝ, 2 < ν → ∀ d : FixedData ν,
∀ᶠ n : ℕ in atTop, Function.Surjective
(BlowupJetSurjectivity.jetRestriction (centerIdeal d) (hyperplane d) n)) :
GlobalInterpolationStatement := by
apply globalInterpolation_of_eventual_packets
intro ν hν d
exact eventual_packets_of_eventual_jetRestriction d (hjet ν hν d)
theorem globalInterpolation : GlobalInterpolationStatement := by
apply globalInterpolation_of_eventual_jetRestriction
intro ν _ d
exact AdmissibleJetSurjectivity.eventually_jetRestriction_surjective d
end PiExponent.AdmissibleMatrixInterpolation
end
end OAI
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