| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.SmoothAffineRefinement |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.SurfaceCurveDegree |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.ProjectiveBertini MaximalSeshadri.Projective MaximalSeshadri.Frames |
| open MaximalSeshadri.AnalyticCoordinates MaximalSeshadri.AlgebraicJets |
| open MaximalSeshadri.LocalComparison MaximalSeshadri.NodalLocal |
| open scoped Topology |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| lemma topIso_baseScalars (p : X ⟶ Spec (CommRingCat.of ℂ)) (U : X.Opens) : |
| U.topIso.hom.hom.comp (baseScalars (U.ι ≫ p)) = openScalars p U := by |
| ext c |
| rw [baseScalars_comp_actual] |
| change (U.ι.appTop ≫ U.topIso.hom) (baseScalars p c) = _ |
| simp only [Scheme.Opens.ι_appTop, Scheme.Opens.topIso_hom] |
| have H := openScalars_restrict p (show U ≤ ⊤ from le_top) |
| have hb : openScalars p ⊤ = baseScalars p := by |
| unfold openScalars baseScalars |
| change ((p.appTop ≫ X.presheaf.map (𝟙 (Opposite.op (⊤ : X.Opens))))).hom.comp _ = _ |
| rw [CategoryTheory.Functor.map_id, Category.comp_id] |
| rw [hb] at H |
| exact (congrArg (fun restriction : Γ(X,⊤) ⟶ Γ(X,U) => restriction (baseScalars p c)) |
| (X.presheaf.map_comp (homOfLE (show U.ι ''ᵁ ⊤ ≤ ⊤ from le_top)).op |
| (eqToHom U.ι_image_top.symm).op)).symm.trans |
| (congrArg (fun scalars : ℂ →+* Γ(X,U) => scalars c) H) |
|
|
| def openTopAlgEquiv (p : X ⟶ Spec (CommRingCat.of ℂ)) (U : X.Opens) : |
| letI : Algebra ℂ Γ(X,U) := (openScalars p U).toAlgebra |
| letI : Algebra ℂ Γ(U.toScheme,⊤) := (baseScalars (U.ι ≫ p)).toAlgebra |
| Γ(X,U) ≃ₐ[ℂ] Γ(U.toScheme,⊤) := by |
| letI : Algebra ℂ Γ(X,U) := (openScalars p U).toAlgebra |
| letI : Algebra ℂ Γ(U.toScheme,⊤) := (baseScalars (U.ι ≫ p)).toAlgebra |
| refine { U.topIso.symm.commRingCatIsoToRingEquiv with commutes' := ?_ } |
| intro c |
| apply (ConcreteCategory.bijective_of_isIso U.topIso.hom).injective |
| change U.topIso.hom (U.topIso.inv (openScalars p U c)) = |
| U.topIso.hom (baseScalars (U.ι ≫ p) c) |
| rw [← CommRingCat.comp_apply, U.topIso.inv_hom_id] |
| exact (congrArg (fun z : ℂ →+* Γ(X,U) => z c) (topIso_baseScalars p U)).symm |
|
|
| lemma chart_topIso_naturality (j : Y ⟶ X) (U : X.Opens) : |
| U.topIso.hom ≫ j.app U ≫ (j ⁻¹ᵁ U).topIso.inv = (j ∣_ U).appTop := by |
| have H := (arrowResLEAppIso j U (j ⁻¹ᵁ U) le_rfl).hom.w |
| change U.topIso.hom ≫ j.appLE U (j ⁻¹ᵁ U) le_rfl = |
| (j.resLE U (j ⁻¹ᵁ U) le_rfl).appTop ≫ (j ⁻¹ᵁ U).topIso.hom at H |
| rw [Scheme.Hom.resLE_eq_morphismRestrict, Scheme.Hom.appLE_eq_app] at H |
| apply (cancel_mono (j ⁻¹ᵁ U).topIso.hom).mp |
| simpa only [Category.assoc, Iso.inv_hom_id, Category.comp_id] using H |
|
|
| def structuralOpenChartMap (p : X ⟶ Spec (CommRingCat.of ℂ)) (j : Y ⟶ X) (U : X.Opens) : |
| letI : Algebra ℂ Γ(X,U) := (openScalars p U).toAlgebra |
| letI : Algebra ℂ Γ((j ⁻¹ᵁ U).toScheme,⊤) := |
| (baseScalars ((j ⁻¹ᵁ U).ι ≫ (j ≫ p))).toAlgebra |
| Γ(X,U) →ₐ[ℂ] Γ((j ⁻¹ᵁ U).toScheme,⊤) := by |
| letI : Algebra ℂ Γ(X,U) := (openScalars p U).toAlgebra |
| letI : Algebra ℂ Γ((j ⁻¹ᵁ U).toScheme,⊤) := |
| (baseScalars ((j ⁻¹ᵁ U).ι ≫ (j ≫ p))).toAlgebra |
| letI : Algebra ℂ Γ(U.toScheme,⊤) := (baseScalars (U.ι ≫ p)).toAlgebra |
| exact (structuralChartMap p j U).comp (openTopAlgEquiv p U).toAlgHom |
|
|
| lemma structuralOpenChartMap_apply (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (j : Y ⟶ X) (U : X.Opens) (f : Γ(X,U)) : |
| letI : Algebra ℂ Γ(X,U) := (openScalars p U).toAlgebra |
| letI : Algebra ℂ Γ((j ⁻¹ᵁ U).toScheme,⊤) := |
| (baseScalars ((j ⁻¹ᵁ U).ι ≫ (j ≫ p))).toAlgebra |
| structuralOpenChartMap p j U f = (j ⁻¹ᵁ U).topIso.inv (j.app U f) := by |
| change (j ∣_ U).appTop (U.topIso.inv f) = _ |
| rw [← chart_topIso_naturality] |
| simp only [CommRingCat.comp_apply] |
| rw [U.topIso.inv_hom_id_apply] |
|
|
| lemma structuralOpenChartMap_ker (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (I : X.IdealSheafData) (U : X.affineOpens) : |
| letI : Algebra ℂ Γ(X,U.1) := (openScalars p U.1).toAlgebra |
| letI : Algebra ℂ Γ((I.subschemeι ⁻¹ᵁ U.1).toScheme,⊤) := |
| (baseScalars ((I.subschemeι ⁻¹ᵁ U.1).ι ≫ (I.subschemeι ≫ p))).toAlgebra |
| RingHom.ker (structuralOpenChartMap p I.subschemeι U.1).toRingHom = I.ideal U := by |
| ext t |
| change structuralOpenChartMap p I.subschemeι U.1 t = 0 ↔ t ∈ I.ideal U |
| rw [structuralOpenChartMap_apply, ← I.ker_subschemeι_app U] |
| change (I.subschemeι ⁻¹ᵁ U.1).topIso.inv (I.subschemeι.app U.1 t) = 0 ↔ |
| I.subschemeι.app U.1 t = 0 |
| exact map_eq_zero_iff _ (ConcreteCategory.bijective_of_isIso |
| (I.subschemeι ⁻¹ᵁ U.1).topIso.inv).injective |
|
|
| lemma structuralOpenChartMap_surjective (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (j : Y ⟶ X) [IsClosedImmersion j] (U : X.affineOpens) : |
| letI : Algebra ℂ Γ(X,U.1) := (openScalars p U.1).toAlgebra |
| letI : Algebra ℂ Γ((j ⁻¹ᵁ U.1).toScheme,⊤) := |
| (baseScalars ((j ⁻¹ᵁ U.1).ι ≫ (j ≫ p))).toAlgebra |
| Function.Surjective (structuralOpenChartMap p j U.1) := by |
| let : Algebra ℂ Γ(X,U.1) := (openScalars p U.1).toAlgebra |
| let : Algebra ℂ Γ((j ⁻¹ᵁ U.1).toScheme,⊤) := |
| (baseScalars ((j ⁻¹ᵁ U.1).ι ≫ (j ≫ p))).toAlgebra |
| let : Algebra ℂ Γ(U.1.toScheme,⊤) := (baseScalars (U.1.ι ≫ p)).toAlgebra |
| exact (structuralChartMap_surjective p j U).comp (openTopAlgEquiv p U.1).surjective |
|
|
| theorem IntegralCurve.nodal_branch_sum_le_degree_on_open (S : Surface) |
| (C : IntegralCurve S) (A : LineBundle S.scheme) |
| (hA : LineBundle.IsAmple S.scheme A) |
| (L : LineBundle S.scheme) (s : O S.scheme ⟶ L.sheaf) |
| (U : S.scheme.affineOpens) [Nonempty (C.embedding ⁻¹ᵁ U.1).toScheme] |
| [IsDomain Γ(S.scheme,U.1)] |
| (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme) |
| (hR : ringKrullDim Γ(S.scheme,U.1) ≤ 2) |
| (q : letI : Algebra ℂ Γ(S.scheme,U.1) := (openScalars S.structureMap U.1).toAlgebra |
| (ℂ × ℂ) → (Γ(S.scheme,U.1) →ₐ[ℂ] ℂ)) |
| (hq : ∀ t, AnalyticAt ℂ (fun z => q z t) 0) |
| (hjets : letI : Algebra ℂ Γ(S.scheme,U.1) := (openScalars S.structureMap U.1).toAlgebra |
| ∀ n : ℕ, |
| RingHom.ker ((Ideal.Quotient.mk (IsLocalRing.maximalIdeal |
| (MvPowerSeries (Fin 2) ℂ)^n)).comp (analyticTaylor q hq).toRingHom) = |
| (RingHom.ker (q 0))^n ∧ |
| Function.Surjective ((Ideal.Quotient.mk (IsLocalRing.maximalIdeal |
| (MvPowerSeries (Fin 2) ℂ)^n)).comp (analyticTaylor q hq).toRingHom)) |
| (g : Γ(S.scheme,U.1)) (hg0 : g ≠ 0) [hgp : (Ideal.span {g}).IsPrime] |
| (hker : RingHom.ker (C.embedding.app U.1).hom = Ideal.span {g}) |
| (u : (ℂ × ℂ) → ℂ) (hu : AnalyticAt ℂ u 0) (hu0 : u 0 ≠ 0) |
| (hg : ∀ᶠ z in 𝓝 0, q z g = u z*z.1*z.2) |
| (hf : U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) ∉ Ideal.span {g}) : |
| letI : Algebra ℂ Γ(S.scheme,U.1) := (openScalars S.structureMap U.1).toAlgebra |
| let f := U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) |
| (((restrictX ℂ (bivariateTaylor q hq f)).order.toNat + |
| (restrictZ ℂ (bivariateTaylor q hq f)).order.toNat : ℕ) : ℤ) ≤ |
| curveDegree S L C := by |
| let : Algebra ℂ Γ(S.scheme,U.1) := (openScalars S.structureMap U.1).toAlgebra |
| let : Algebra ℂ Γ((C.embedding ⁻¹ᵁ U.1).toScheme,⊤) := |
| (baseScalars ((C.embedding ⁻¹ᵁ U.1).ι ≫ (C.embedding ≫ S.structureMap))).toAlgebra |
| let : IsAffine (C.embedding ⁻¹ᵁ U.1).toScheme := U.2.preimage C.embedding |
| let : IsNoetherian (C.embedding ⁻¹ᵁ U.1).toScheme := {} |
| let f := U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) |
| let π := structuralOpenChartMap S.structureMap C.embedding U.1 |
| have hπker : RingHom.ker π.toRingHom = Ideal.span {g} := by |
| ext t |
| change π t = 0 ↔ t ∈ Ideal.span {g} |
| rw [← hker] |
| change structuralOpenChartMap S.structureMap C.embedding U.1 t = 0 ↔ |
| C.embedding.app U.1 t = 0 |
| rw [structuralOpenChartMap_apply] |
| exact map_eq_zero_iff _ (ConcreteCategory.bijective_of_isIso |
| (C.embedding ⁻¹ᵁ U.1).topIso.inv).injective |
| obtain ⟨eF,he⟩ := exists_restricted_pullback_frame C.embedding U.1 e s |
| have hcoeff : π f = coefficient eF |
| (restrictSection (C.embedding ⁻¹ᵁ U.1).ι (pullbackSection C.embedding s)) := by |
| rw [he] |
| change (C.embedding ∣_ U.1).appTop |
| (U.1.topIso.inv (U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)))) = _ |
| rw [U.1.topIso.hom_inv_id_apply] |
| have hs : pullbackSection C.embedding s ≠ 0 := by |
| intro hz |
| apply hf |
| rw [← hπker] |
| change π f = 0 |
| rw [hcoeff, hz, restrictSection_zero, coefficient_zero] |
| obtain ⟨σ,hσ,M,a,ha,hclosed⟩ := C.exists_projective_sections S A hA |
| let := hσ |
| let := hclosed |
| exact projective_nodal_degree_bound (C.embedding ≫ S.structureMap) C.dimension |
| a ha (L.pullback C.embedding) (pullbackSection C.embedding s) hs |
| (C.embedding ⁻¹ᵁ U.1).ι eF hR q hq hjets f g hg0 u hu hu0 hg hf |
| π (structuralOpenChartMap_surjective S.structureMap C.embedding U) hπker hcoeff |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
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| end |
| end |
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| end OAI |
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