| import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupPolarization |
| import OAI.NumberTheory.PiExponent.Geometry.AdmissibleCurveCoordinates |
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| namespace OAI |
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| noncomputable section |
| namespace PiExponent.AdmissibleCurveModel |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open AdmissibleBlowupGeometry CurveNormalizationModel |
| variable {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D) |
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| 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 |
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| end |
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| end OAI |
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