File size: 3,959 Bytes
b797a5b | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupPolarization
import OAI.NumberTheory.PiExponent.Geometry.AdmissibleCurveCoordinates
namespace OAI
noncomputable section
namespace PiExponent.AdmissibleCurveModel
open AlgebraicGeometry CategoryTheory TopologicalSpace
open AdmissibleBlowupGeometry CurveNormalizationModel
variable {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D)
structure ModelData (C : NumericalAmpleness.IntegralCurve (blowup d)) where
E : Type
[field : Field E]
[algebra : Algebra ℂ E]
[essFiniteType : Algebra.EssFiniteType ℂ E]
coordinates : Fin (d.m+1) → E
coordinates_generate : IntermediateField.adjoin ℂ (Set.range coordinates) = ⊤
trdeg_one : Algebra.trdeg ℂ E = 1
parameter : E
parameter_transcendental : Transcendental ℂ parameter
[parameterFinite : FiniteDimensional (IntermediateField.adjoin ℂ {parameter}) E]
normalization : parameterCurve parameter parameter_transcendental ⟶ C.scheme
[normalizationFinite : IsFinite normalization]
normalization_over : normalization ≫ C.embedding ≫ structureMap d =
parameterCurveStructureMap parameter parameter_transcendental
chart : C.scheme.Opens
chart_nonempty : chart ≠ ⊥
[chartIso : IsIso (normalization ∣_ chart)]
generic_coordinates : parameterCurveGenericPoint parameter parameter_transcendental ≫
(normalization ≫ C.embedding ≫ projection d) =
CurveMonomialMap.genericMonomialMap (exponents d) (constantIndex d)
((scale d).exponents_constant d.curveDegreeWeights_pos) (coordinateIndex d)
((scale d).exponents_coordinate d.curveDegreeWeights_pos) coordinates
theorem existsModelData (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ t : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {t})
(hm : ∃ c : C.scheme, (C.embedding ≫ projection d) c ∈ (affineChart d).opensRange) :
Nonempty (ModelData d C) := by
let := AdmissibleCurveCoordinates.sourceAlgebra d C
have hp := AdmissibleCurveCoordinates.field_properties d C hn hm
let := hp.1
let z := AdmissibleCurveCoordinates.coordinates d C hn hm
have hfinite := CurveParameterFinite.finite_over_every_parameter ℂ C.scheme.functionField hp.2.1
obtain ⟨i,hi⟩ := CurveInequality.nonconstant_coordinates z hp.2.2 hp.2.1
let f := z i
have hf : Transcendental ℂ f := hi
let := hfinite f hf
obtain ⟨g,hg,hgeneric,hgfinite,W,hW,hWiso⟩ :=
CurveNormalizationMorphism.exists_intrinsic_normalization_morphism
(C.embedding ≫ structureMap d) C.dimension f hf
let := hgfinite
let := hWiso
have hWne : W ≠ ⊥ := by
intro he
simp only [he, Opens.mem_bot] at hW
have hc : parameterCurveGenericPoint f hf ≫ (g ≫ C.embedding ≫ projection d) =
CurveMonomialMap.genericMonomialMap (exponents d) (constantIndex d)
((scale d).exponents_constant d.curveDegreeWeights_pos) (coordinateIndex d)
((scale d).exponents_coordinate d.curveDegreeWeights_pos) z := by
change parameterCurveGenericPoint f hf ≫ (g ≫ C.embedding ≫ projection d) =
Spec.map (CommRingCat.ofHom (MvPolynomial.aeval z).toRingHom) ≫ affineChart d
rw [← Category.assoc, ← Category.assoc, hgeneric]
exact (AdmissibleCurveCoordinates.coordinates_generic_map d C hn hm).symm
exact ⟨{ E := C.scheme.functionField
field := inferInstance
algebra := inferInstance
essFiniteType := hp.1
coordinates := z
coordinates_generate := hp.2.2
trdeg_one := hp.2.1
parameter := f
parameter_transcendental := hf
parameterFinite := inferInstance
normalization := g
normalizationFinite := hgfinite
normalization_over := hg
chart := W
chart_nonempty := hWne
chartIso := hWiso
generic_coordinates := hc }⟩
end PiExponent.AdmissibleCurveModel
end
end OAI
|