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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
end
end
end OAI
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