| import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupGeometry |
| import OAI.NumberTheory.PiExponent.Ampleness.CurveBlowupImage |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory AlgebraicGeometry TopologicalSpace |
|
|
| namespace PiExponent.AdmissibleCurveImage |
| open AdmissibleBlowupGeometry CurveBlowupImage |
| variable {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D) |
|
|
| theorem center_point_isClosed (x : compactification d) |
| (hx : x ∈ (centerIdeal d).support) : IsClosed ({x} : Set (compactification d)) := by |
| change x ∈ ((centerIdeal d).support : Set (compactification d)) at hx |
| rw [centerIdeal_support d] at hx |
| obtain ⟨j, z, rfl⟩ := Set.mem_iUnion.mp hx |
| let q := CompactLogJetIdeal.point (d.curveCenters j) ≫ affineChart d |
| let : IsClosedImmersion q := CompactJetIdeal.section_isClosedImmersion |
| (compactificationStructureMap d) q (by |
| dsimp [q] |
| rw [Category.assoc, affineChart_over, CompactLogJetIdeal.point_section]) |
| have he : Set.range q = {q z} := by |
| ext x |
| constructor |
| · rintro ⟨z',rfl⟩ |
| exact congrArg q (Subsingleton.elim z' z) |
| · rintro rfl |
| exact ⟨z,rfl⟩ |
| rw [← he] |
| exact q.isClosedEmbedding.isClosed_range |
|
|
| theorem meets_center_complement (C : NumericalAmpleness.IntegralCurve (blowup d)) |
| (hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) : |
| ∃ c : C.scheme, (C.embedding ≫ projection d) c ∈ (centerIdeal d).support.compl := |
| exists_image_outside_closed_points (C.embedding ≫ projection d) |
| (centerIdeal d).support (center_point_isClosed d) hn |
|
|
| instance projection_restrict_isIso : |
| IsIso (projection d ∣_ (centerIdeal d).support.compl) := |
| blowup_restrict_isIso (isBlowup d) |
|
|
| abbrev imageCurve (C : NumericalAmpleness.IntegralCurve (blowup d)) |
| (hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) := |
| CurveBlowupImage.imageCurve (projection d) C hn |
|
|
| def functionFieldIso (C : NumericalAmpleness.IntegralCurve (blowup d)) |
| (hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) : |
| (imageCurve d C hn).scheme.functionField ≅ C.scheme.functionField := |
| imageFunctionFieldIso (projection d) C hn (centerIdeal d).support.compl |
| (meets_center_complement d C hn) |
|
|
| theorem functionFieldIso_generic (C : NumericalAmpleness.IntegralCurve (blowup d)) |
| (hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) : |
| Spec.map (functionFieldIso d C hn).hom ≫ |
| (imageCurve d C hn).scheme.fromSpecStalk (genericPoint (imageCurve d C hn).scheme) ≫ |
| (imageCurve d C hn).embedding = |
| C.scheme.fromSpecStalk (genericPoint C.scheme) ≫ C.embedding ≫ projection d := |
| imageFunctionFieldIso_generic (projection d) C hn (centerIdeal d).support.compl |
| (meets_center_complement d C hn) |
|
|
| theorem image_meets_chart (C : NumericalAmpleness.IntegralCurve (blowup d)) |
| (hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) |
| (hm : ∃ c : C.scheme, (C.embedding ≫ projection d) c ∈ (affineChart d).opensRange) : |
| ∃ c : (imageCurve d C hn).scheme, |
| (imageCurve d C hn).embedding c ∈ (affineChart d).opensRange := by |
| obtain ⟨c,hc⟩ := hm |
| refine ⟨CurveBlowupImage.imageMap (projection d) C hn c, ?_⟩ |
| simpa only [← Scheme.Hom.comp_apply, CurveBlowupImage.imageMap_comp] using hc |
|
|
| end PiExponent.AdmissibleCurveImage |
|
|
| end |
|
|
| end OAI |
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|
