import Mathlib import OAI.AlgebraicGeometry.NumericalDimension.FiniteCurveMaps import OAI.AlgebraicGeometry.NumericalDimension.SurfaceIntersection namespace OAI /-! Curve Projection. -/ open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory variable {X C Y : Scheme} [IsIntegral X] [IsIntegral C] [IsIntegral Y] [IsLocallyNoetherian X] [IsLocallyNoetherian C] [IsLocallyNoetherian Y] [StalkwiseNormal Y] [CompactSpace X] [CompactSpace C] (sX : X ⟶ Spec (.of ℂ)) (sC : C ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 1 sX] [SmoothOfRelativeDimension 1 sC] [IsProper sX] [IsProper sC] include sX sC in theorem properCurveDegree_comp_finite (f : X ⟶ C) [IsFinite f] [IsDominant f] (i : C ⟶ Y) (D : cartierDivisors (X := Y)) : properCurveDegree (f ≫ i) D = (curveFieldDegree f : ℤ) * properCurveDegree i D := by let : Smooth sX := SmoothOfRelativeDimension.smooth 1 sX let : Smooth sC := SmoothOfRelativeDimension.smooth 1 sC let : StalkwiseNormal X := ⟨integrallyClosed_stalk_of_smooth_field sX⟩ let : StalkwiseNormal C := ⟨integrallyClosed_stalk_of_smooth_field sC⟩ obtain ⟨E,hE⟩ := exists_pulledCartierRepresentative i D.1 D.2 obtain ⟨F,hF⟩ := exists_cartierPullback f E hE.isCartier rw [properCurveDegree_eq sX (f ≫ i) D F (hE.comp_dominant f i hF), properCurveDegree_eq sC i D E hE] exact smoothCurve_degree_cartierPullback sX sC f hF end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory theorem curveFieldDegree_pos {X Y : Scheme} [IsIntegral X] [IsIntegral Y] (f : X ⟶ Y) [IsFinite f] [IsDominant f] : 0 < curveFieldDegree f := by obtain ⟨V,hV,hη,_⟩ := exists_isAffineOpen_mem_and_subset (show f (genericPoint X) ∈ (⊤ : Y.Opens) from trivial) let : Nonempty V := ⟨⟨f (genericPoint X),hη⟩⟩ let : Nonempty (f ⁻¹ᵁ V) := ⟨⟨genericPoint X,hη⟩⟩ let hU := hV.preimage f let := (f.app V).hom.toAlgebra let : Module.Finite Γ(Y,V) Γ(X,f ⁻¹ᵁ V) := f.finite_app V hV let : IsSchemeTheoreticallyDominant f := .of_isDominant f let : Module.IsTorsionFree Γ(Y,V) Γ(X,f ⁻¹ᵁ V) := Module.isTorsionFree_iff_algebraMap_injective.mpr (f.app_injective V) rw [curveFieldDegree_eq_affine f V hV hU] exact Module.finrank_pos end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory lemma dominantFunctionFieldMap_spec {X Y : Scheme} [IsIntegral X] [IsIntegral Y] (f : X ⟶ Y) [IsDominant f] : Spec.map (CommRingCat.ofHom (dominantFunctionFieldMap f)) ≫ Y.fromSpecStalk (genericPoint Y) = X.fromSpecStalk (genericPoint X) ≫ f := by change Spec.map ((Y.presheaf.stalkCongr (Inseparable.of_eq (dominant_genericPoint f).symm)).hom ≫ f.stalkMap (genericPoint X)) ≫ _ = _ rw [Spec.map_comp, Category.assoc] change Spec.map (f.stalkMap (genericPoint X)) ≫ Spec.map (Y.presheaf.stalkSpecializes (Inseparable.of_eq (dominant_genericPoint f)).specializes) ≫ Y.fromSpecStalk (genericPoint Y) = _ rw [Scheme.SpecMap_stalkSpecializes_fromSpecStalk, Scheme.SpecMap_stalkMap_fromSpecStalk] end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory universe u theorem valuationRing_stalk_of_smooth_curve {k : Type u} [Field k] {C : Scheme.{u}} [IsIntegral C] (sC : C ⟶ Spec (.of k)) [SmoothOfRelativeDimension 1 sC] (x : C) : ValuationRing (C.presheaf.stalk x) := by let : Smooth sC := SmoothOfRelativeDimension.smooth 1 sC let : IsLocallyNoetherian C := LocallyOfFiniteType.isLocallyNoetherian sC let : IsIntegrallyClosed (C.presheaf.stalk x) := integrallyClosed_stalk_of_smooth_field sC x let : Ring.KrullDimLE 1 (C.presheaf.stalk x) := krullDimLE_of_coheight_le (coheight_le_of_smooth_dimension sC 1 x) by_cases hfield : IsField (C.presheaf.stalk x) · let := hfield.toField infer_instance · let : Ring.DimensionLEOne (C.presheaf.stalk x) := ⟨fun {I} hI hprime => Ring.krullDimLE_one_iff_of_noZeroDivisors.mp inferInstance I hI hprime⟩ let : IsDedekindDomain (C.presheaf.stalk x) := { } let : IsDiscreteValuationRing (C.presheaf.stalk x) := ((IsDiscreteValuationRing.TFAE (C.presheaf.stalk x) hfield).out 3 1).mp (show IsDedekindDomain (C.presheaf.stalk x) from inferInstance) infer_instance end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory AlgebraicGeometry.Scheme universe u lemma partial_fromFunctionField_eq {C Y : Scheme.{u}} [IsIntegral C] (f : C.PartialMap Y) {x : C} (hx : x ∈ f.domain) : f.fromFunctionField = Spec.map (CommRingCat.ofHom (algebraMap (C.presheaf.stalk x) C.functionField)) ≫ f.fromSpecStalkOfMem hx := by have hη : genericPoint C ∈ f.domain := (genericPoint_specializes x).mem_open f.domain.2 hx have h : f.domain.fromSpecStalkOfMem (genericPoint C) hη = Spec.map (CommRingCat.ofHom (algebraMap (C.presheaf.stalk x) C.functionField)) ≫ f.domain.fromSpecStalkOfMem x hx := by apply (cancel_mono f.domain.ι).mp rw [Category.assoc, Scheme.Opens.fromSpecStalkOfMem_ι, Scheme.Opens.fromSpecStalkOfMem_ι] exact (C.SpecMap_stalkSpecializes_fromSpecStalk (genericPoint_specializes x)).symm change f.domain.fromSpecStalkOfMem (genericPoint C) hη ≫ f.hom = _ rw [h, Category.assoc] rfl end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory AlgebraicGeometry.Scheme universe u theorem exists_extension_from_functionField_of_smooth_curve {k : Type u} [Field k] {C Y : Scheme.{u}} [IsIntegral C] (sC : C ⟶ Spec (.of k)) [SmoothOfRelativeDimension 1 sC] (sY : Y ⟶ Spec (.of k)) [IsProper sY] (g : Spec C.functionField ⟶ Y) (hg : g ≫ sY = C.fromSpecStalk (genericPoint C) ≫ sC) : ∃ f : C ⟶ Y, C.fromSpecStalk (genericPoint C) ≫ f = g ∧ f ≫ sY = sC := by let : Y.IsSeparated := by constructor rw [← Limits.terminal.comp_from sY] infer_instance let r := RationalMap.ofFunctionField sC sY g hg have hr : r.fromFunctionField = g := RationalMap.fromFunctionField_ofFunctionField _ _ _ _ have hdomain : r.domain = ⊤ := by apply top_unique intro x _ let : ValuationRing (C.presheaf.stalk x) := valuationRing_stalk_of_smooth_curve sC x have hv : ValuativeCriterion.Existence sY := by have h : UniversallyClosed sY := inferInstance rw [UniversallyClosed.eq_valuativeCriterion] at h exact h.1 have hw : g ≫ sY = Spec.map (CommRingCat.ofHom (algebraMap (C.presheaf.stalk x) C.functionField)) ≫ (C.fromSpecStalk x ≫ sC) := by rw [hg, ← Category.assoc] exact congrArg (fun t => t ≫ sC) (C.SpecMap_stalkSpecializes_fromSpecStalk (genericPoint_specializes x)).symm obtain ⟨l, hlg, hl⟩ := (hv { R := C.presheaf.stalk x commRing := inferInstanceAs (CommRing (C.presheaf.stalk x)) domain := inferInstanceAs (IsDomain (C.presheaf.stalk x)) K := C.functionField i₁ := g i₂ := C.fromSpecStalk x ≫ sC commSq := ⟨hw⟩ }).exists_lift let p := PartialMap.ofFromSpecStalk sC sY l hl have hx : x ∈ p.domain := PartialMap.mem_domain_ofFromSpecStalk _ _ _ _ apply RationalMap.mem_domain.mpr refine ⟨p, hx, RationalMap.eq_of_fromFunctionField_eq _ _ ?_⟩ rw [RationalMap.fromFunctionField_toRationalMap, partial_fromFunctionField_eq p hx, PartialMap.fromSpecStalkOfMem_ofFromSpecStalk, hr] exact hlg let p := r.toPartialMap have hpdom : p.domain = ⊤ := hdomain let : IsIso p.domain.ι := by rw [hpdom] change IsIso C.topIso.hom infer_instance let f := inv p.domain.ι ≫ p.hom have hf : C.fromSpecStalk (genericPoint C) ≫ f = g := by have hη : genericPoint C ∈ p.domain := by rw [hpdom]; trivial have hi : C.fromSpecStalk (genericPoint C) ≫ inv p.domain.ι = p.domain.fromSpecStalkOfMem (genericPoint C) hη := by apply (cancel_mono p.domain.ι).mp simp only [Category.assoc, IsIso.inv_hom_id, Category.comp_id, Scheme.Opens.fromSpecStalkOfMem_ι] change C.fromSpecStalk (genericPoint C) ≫ inv p.domain.ι ≫ p.hom = g rw [← Category.assoc, hi] change p.fromFunctionField = g rw [← RationalMap.fromFunctionField_toRationalMap, RationalMap.toRationalMap_toPartialMap, hr] refine ⟨f, hf, ?_⟩ let : IsDominant (C.fromSpecStalk (genericPoint C)) := ⟨by have hden : Dense ({genericPoint C} : Set C) := dense_iff_closure_eq.mpr (genericPoint_spec C) exact hden.mono (Set.singleton_subset_iff.mpr ⟨_, C.fromSpecStalk_closedPoint⟩)⟩ apply ext_of_isDominant (C.fromSpecStalk (genericPoint C)) rw [← Category.assoc, hf, hg] end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory theorem curve_lift_through_proper_functionField_iso {C T N : Scheme} [IsIntegral C] [IsIntegral T] [IsIntegral N] [NoetherianSpace C] [Nontrivial T] (sC : C ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 1 sC] [IsProper sC] (sT : T ⟶ Spec (.of ℂ)) [IsProper sT] (ν : N ⟶ T) [IsProper ν] [IsDominant ν] (hν : Function.Bijective (dominantFunctionFieldMap ν)) (g : C ⟶ T) [IsDominant g] (hg : g ≫ sT = sC) : ∃ τ : C ⟶ N, τ ≫ ν = g ∧ IsFinite τ ∧ IsDominant τ := by let e := RingEquiv.ofBijective (dominantFunctionFieldMap ν) hν let a : N.functionField →+* C.functionField := (dominantFunctionFieldMap g).comp e.symm.toRingHom have ha : a.comp (dominantFunctionFieldMap ν) = dominantFunctionFieldMap g := by ext r exact congrArg (dominantFunctionFieldMap g) (e.symm_apply_apply r) let G : Spec C.functionField ⟶ N := Spec.map (CommRingCat.ofHom a) ≫ N.fromSpecStalk (genericPoint N) have hG : G ≫ ν = C.fromSpecStalk (genericPoint C) ≫ g := by dsimp only [G] rw [Category.assoc, ← dominantFunctionFieldMap_spec ν, ← Category.assoc] erw [← Spec.map_comp, ← dominantFunctionFieldMap_spec g] congr 2 exact CommRingCat.hom_ext ha have hGovern : G ≫ (ν ≫ sT) = C.fromSpecStalk (genericPoint C) ≫ sC := by rw [← Category.assoc, hG, Category.assoc, hg] obtain ⟨τ,hτ,hτover⟩ := exists_extension_from_functionField_of_smooth_curve sC (ν ≫ sT) G hGovern have heq : τ ≫ ν = g := by apply ext_of_isDominant_of_isSeparated sT (by simpa only [Category.assoc, hg] using hτover) (C.fromSpecStalk (genericPoint C)) rw [← Category.assoc, hτ, hG] have : IsDominant (Spec.map (CommRingCat.ofHom a)) := by constructor apply (PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical _).mpr exact ((RingHom.injective_iff_ker_eq_bot a).mp a.injective).le.trans bot_le have : IsDominant G := inferInstance have : IsDominant (C.fromSpecStalk (genericPoint C) ≫ τ) := by rw [hτ] infer_instance have : IsDominant τ := IsDominant.of_comp (C.fromSpecStalk (genericPoint C)) τ have : IsProper (τ ≫ (ν ≫ sT)) := by rw [hτover] infer_instance have : IsProper τ := IsProper.of_comp τ (ν ≫ sT) have : Surjective ν := inferInstance let : Nontrivial N := ν.surjective.nontrivial exact ⟨τ,heq,finite_of_proper_dominant_from_smooth_curve sC τ,inferInstance⟩ end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory TopologicalSpace lemma reduced_ker_radical {X Y : Scheme} [IsReduced X] (f : X ⟶ Y) [QuasiCompact f] : f.ker.radical = f.ker := by apply le_antisymm · intro U a ha rw [Scheme.IdealSheafData.radical_ideal, Scheme.Hom.ker_apply] at ha rw [Scheme.Hom.ker_apply, RingHom.mem_ker] obtain ⟨n,hn⟩ := ha exact IsNilpotent.eq_zero ⟨n, by simpa only [RingHom.mem_ker, map_pow] using hn⟩ · exact Scheme.IdealSheafData.le_radical _ theorem exists_lift_closed_of_reduced {X Y Z : Scheme} [IsReduced X] (f : X ⟶ Y) [QuasiCompact f] (i : Z ⟶ Y) [IsClosedImmersion i] (h : Set.range f ⊆ Set.range i) : ∃ g : X ⟶ Z, g ≫ i = f := by have hs : f.ker.support ≤ i.ker.support := by change (f.ker.support : Set Y) ⊆ i.ker.support rw [Scheme.Hom.support_ker, Scheme.Hom.support_ker] exact closure_mono h have hk : i.ker ≤ f.ker := by rw [← reduced_ker_radical f, ← Scheme.IdealSheafData.vanishingIdeal_support] exact Scheme.IdealSheafData.le_support_iff_le_vanishingIdeal.mp hs exact ⟨IsClosedImmersion.lift i f hk, IsClosedImmersion.lift_fac i f hk⟩ end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory TopologicalSpace theorem factor_dominant_pointClosure {C X : Scheme} [IsIntegral C] (g : C ⟶ X) [QuasiCompact g] : ∃ h : C ⟶ pointClosure (g (genericPoint C)), h ≫ pointClosureι (g (genericPoint C)) = g ∧ IsDominant h := by let p := g (genericPoint C) have hr : Set.range g ⊆ Set.range (pointClosureι p) := by rw [pointClosure_range] rintro x ⟨y,rfl⟩ exact specializes_iff_mem_closure.mp ((genericPoint_specializes y).map g.continuous) obtain ⟨h,hh⟩ := exists_lift_closed_of_reduced g (pointClosureι p) hr refine ⟨h,hh,⟨?_⟩⟩ rw [denseRange_iff_closure_range, (pointClosureι p).isEmbedding.closure_eq_preimage_closure_image] rw [← Set.range_comp, ← TopCat.coe_comp, ← Scheme.Hom.comp_base, hh] have hc : closure (Set.range g) = closure ({p} : Set X) := by simpa only [Set.image_univ] using ((genericPoint_spec C).image g.continuous).symm rw [hc, ← pointClosure_range] exact Set.eq_univ_of_forall (fun x => ⟨x,rfl⟩) end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory TopologicalSpace theorem curve_factors_through_normalized_image {C X : Scheme} [IsIntegral C] (sC : C ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 1 sC] [IsProper sC] (sX : X ⟶ Spec (.of ℂ)) [IsProper sX] (g : C ⟶ X) (hg : g ≫ sX = sC) (hnc : ¬ IsClosed ({g (genericPoint C)} : Set X)) : let T := pointClosure (g (genericPoint C)) let ν := (T.fromSpecStalk (genericPoint T)).fromNormalization ∃ τ : C ⟶ (T.fromSpecStalk (genericPoint T)).normalization, τ ≫ ν ≫ pointClosureι (g (genericPoint C)) = g ∧ IsFinite τ ∧ IsDominant τ := by let T := pointClosure (g (genericPoint C)) let i := pointClosureι (g (genericPoint C)) let n := T.fromSpecStalk (genericPoint T) have : IsLocallyNoetherian C := LocallyOfFiniteType.isLocallyNoetherian sC have : CompactSpace C := QuasiCompact.compactSpace_of_compactSpace sC have : AlgebraicGeometry.IsNoetherian C := ⟨⟩ have : IsProper (g ≫ sX) := by rw [hg]; infer_instance have : IsProper g := IsProper.of_comp g sX have : Nontrivial T := pointClosure_nontrivial _ hnc have : IsProper (i ≫ sX) := inferInstance have : IsLocallyNoetherian T := LocallyOfFiniteType.isLocallyNoetherian (i ≫ sX) have : CompactSpace T := QuasiCompact.compactSpace_of_compactSpace (i ≫ sX) have : AlgebraicGeometry.IsNoetherian T := ⟨⟩ have : IsFinite n.fromNormalization := finite_genericNormalization (i ≫ sX) obtain ⟨h,hh,hdom⟩ := factor_dominant_pointClosure g have hhov : h ≫ (i ≫ sX) = sC := by rw [← Category.assoc, hh, hg] obtain ⟨τ,hτ,hfin,hτdom⟩ := curve_lift_through_proper_functionField_iso sC (i ≫ sX) n.fromNormalization genericNormalization_bijective_functionField h hhov exact ⟨τ, by rw [← Category.assoc, hτ, hh],hfin,hτdom⟩ end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory TopologicalSpace theorem surface_curve_degree_projection {C X : Scheme} [IsIntegral C] [IsIntegral X] [IsLocallyNoetherian C] [IsLocallyNoetherian X] [StalkwiseNormal X] [CompactSpace C] (sC : C ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 1 sC] [IsProper sC] (sX : X ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 2 sX] [IsProper sX] (g : C ⟶ X) (hg : g ≫ sX = sC) (hp : Order.coheight (g (genericPoint C)) = 1) : ∃ d : ℕ, 0 < d ∧ ∀ D : cartierDivisors (X := X), properCurveDegree g D = (d : ℤ) * surfaceCartierPrimeDegree sX D ⟨_,hp⟩ := by let p : PrimeDivisor X := ⟨_,hp⟩ let T := pointClosure p.1 let n := T.fromSpecStalk (genericPoint T) let i := n.fromNormalization ≫ pointClosureι p.1 have hnc : ¬ IsClosed ({p.1} : Set X) := by intro h have he := coheight_eq_of_smooth_closed sX 2 p.1 h rw [p.2] at he norm_num at he obtain ⟨τ,hτ,hfin,hdom⟩ := curve_factors_through_normalized_image sC sX g hg hnc have : SmoothOfRelativeDimension 1 (i ≫ sX) := by simpa only [i,Category.assoc] using surface_prime_normalization_smooth sX p.1 p.2 have : IsProper (i ≫ sX) := by simpa only [i,Category.assoc] using surface_prime_normalization_proper sX p.1 have : IsLocallyNoetherian n.normalization := LocallyOfFiniteType.isLocallyNoetherian (i ≫ sX) have : CompactSpace n.normalization := QuasiCompact.compactSpace_of_compactSpace (i ≫ sX) refine ⟨curveFieldDegree τ,curveFieldDegree_pos τ,?_⟩ intro D change properCurveDegree g D = (curveFieldDegree τ : ℤ) * properCurveDegree i D rw [← properCurveDegree_comp_finite sC (i ≫ sX) τ i D] exact congrArg (fun t => properCurveDegree t D) hτ.symm end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory TopologicalSpace theorem pointClosure_map_functionField_bijective {X Y : Scheme} (f : X ⟶ Y) [IsProper f] (q : X) [IsIso (f.stalkMap q)] : ∃ r : pointClosure q ⟶ pointClosure (f q), r ≫ pointClosureι (f q) = pointClosureι q ≫ f ∧ IsProper r ∧ ∃ (_ : IsDominant r), Function.Bijective (dominantFunctionFieldMap r) := by have : IsIso (f.residueFieldMap q) := (IsLocalRing.ResidueField.mapEquiv (asIso (f.stalkMap q)).commRingCatIsoToRingEquiv).toCommRingCatIso.isIso_hom have he : (pointClosureι q ≫ f) (genericPoint (pointClosure q)) = f q := by rw [Scheme.Hom.comp_apply, pointClosure_generic] have hv := factor_dominant_pointClosure (pointClosureι q ≫ f) rw [he] at hv obtain ⟨r,hr,hdom⟩ := hv have : IsProper (r ≫ pointClosureι (f q)) := by rw [hr]; infer_instance have : IsProper r := IsProper.of_comp r (pointClosureι (f q)) refine ⟨r,hr,inferInstance,hdom,?_⟩ let a : Spec (X.residueField q) ⟶ pointClosure q := (X.fromSpecResidueField q).toImage let b : Spec (Y.residueField (f q)) ⟶ pointClosure (f q) := (Y.fromSpecResidueField (f q)).toImage have ha : a ≫ pointClosureι q = X.fromSpecResidueField q := Scheme.Hom.toImage_imageι _ have hb : b ≫ pointClosureι (f q) = Y.fromSpecResidueField (f q) := Scheme.Hom.toImage_imageι _ have : IsDominant a := inferInstanceAs (IsDominant (X.fromSpecResidueField q).toImage) have : IsPreimmersion a := by have : IsPreimmersion (a ≫ pointClosureι q) := by rw [ha]; infer_instance exact IsPreimmersion.of_comp a (pointClosureι q) have : IsPreimmersion b := by have : IsPreimmersion (b ≫ pointClosureι (f q)) := by rw [hb]; infer_instance exact IsPreimmersion.of_comp b (pointClosureι (f q)) have hd : a ≫ r = Spec.map (f.residueFieldMap q) ≫ b := by apply (cancel_mono (pointClosureι (f q))).mp rw [Category.assoc,hr,← Category.assoc,ha,Category.assoc,hb, Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField] have : IsPreimmersion (a ≫ r) := by rw [hd]; infer_instance have hsur := (dominantFunctionFieldMap_bijective_of_preimmersion (a ≫ r)).2 refine ⟨(dominantFunctionFieldMap r).injective,?_⟩ intro z obtain ⟨w,hw⟩ := hsur (dominantFunctionFieldMap a z) refine ⟨w,(dominantFunctionFieldMap a).injective ?_⟩ simpa only [dominantFunctionFieldMap_comp_apply] using hw end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory lemma curveFieldDegree_eq_one {X Y : Scheme} [IsIntegral X] [IsIntegral Y] (f : X ⟶ Y) [IsDominant f] (hf : Function.Bijective (dominantFunctionFieldMap f)) : curveFieldDegree f = 1 := by let := (dominantFunctionFieldMap f).toAlgebra exact Module.finrank_of_bijective_algebraMap hf lemma dominantFunctionFieldMap_bijective_comp {X Y Z : Scheme} [IsIntegral X] [IsIntegral Y] [IsIntegral Z] (f : X ⟶ Y) (g : Y ⟶ Z) [IsDominant f] [IsDominant g] (hf : Function.Bijective (dominantFunctionFieldMap f)) (hg : Function.Bijective (dominantFunctionFieldMap g)) : Function.Bijective (dominantFunctionFieldMap (f ≫ g)) := by have he : (dominantFunctionFieldMap (f ≫ g) : Z.functionField → X.functionField) = (dominantFunctionFieldMap f) ∘ (dominantFunctionFieldMap g) := funext (dominantFunctionFieldMap_comp_apply f g) rw [he] exact hf.comp hg lemma dominantFunctionFieldMap_bijective_of_comp {X Y Z : Scheme} [IsIntegral X] [IsIntegral Y] [IsIntegral Z] (f : X ⟶ Y) (g : Y ⟶ Z) [IsDominant f] [IsDominant g] (hfg : Function.Bijective (dominantFunctionFieldMap (f ≫ g))) : Function.Bijective (dominantFunctionFieldMap f) := by refine ⟨(dominantFunctionFieldMap f).injective,?_⟩ intro x obtain ⟨z,hz⟩ := hfg.2 x exact ⟨dominantFunctionFieldMap g z, by simpa only [dominantFunctionFieldMap_comp_apply] using hz⟩ end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne lemma IsCartierPullback.order_eq_of_equation {X Y : Scheme} [IsIntegral X] [IsIntegral Y] [IsLocallyNoetherian X] [IsLocallyNoetherian Y] [StalkwiseNormal Y] {f : X ⟶ Y} [IsDominant f] {D : WeilDivisor Y} {P : WeilDivisor X} (hP : IsCartierPullback f D P) {U : Y.Opens} {a : Y.functionField} (ha : a ≠ 0) (hDa : ∀ q : PrimeDivisor Y, q.1 ∈ U → D q = Y.ord a q.1) (p : PrimeDivisor X) (hpU : f p.1 ∈ U) : P p = X.ord (dominantFunctionFieldMap f a) p.1 := by obtain ⟨V, _, hpV, b, hb, hDb, hPb⟩ := hP p.1 rw [hPb p hpV] exact pulled_equation_order_eq f hb ha hDb hDa p hpV hpU end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory variable {C X Y : Scheme} [IsIntegral C] [IsIntegral X] [IsIntegral Y] [IsLocallyNoetherian C] [IsLocallyNoetherian X] [IsLocallyNoetherian Y] [StalkwiseNormal Y] theorem IsPulledCartierRepresentative.of_cartierPullback (g : C ⟶ X) (f : X ⟶ Y) [IsDominant f] {D : WeilDivisor Y} {P : WeilDivisor X} {E : WeilDivisor C} (hP : IsCartierPullback f D P) (hE : IsPulledCartierRepresentative (g ≫ f) D E) : IsPulledCartierRepresentative g P E := by obtain ⟨U,_hU,hηU,a,ha,hDa,hE⟩ := hE obtain ⟨V,hV,hηV,hVU⟩ := exists_isAffineOpen_mem_and_subset (show g (genericPoint C) ∈ f ⁻¹ᵁ U from hηU) refine ⟨V,hV,hηV,dominantFunctionFieldMap f a,(map_ne_zero _).mpr ha,?_,?_⟩ · exact fun p hp => hP.order_eq_of_equation ha hDa p (hVU hp) intro x obtain ⟨U',_hU',hxU,b,hb,hDb,u,hu,heu,hEu⟩ := hE x obtain ⟨V',hV',hxV,hV'U⟩ := exists_isAffineOpen_mem_and_subset (show g x ∈ f ⁻¹ᵁ U' from hxU) refine ⟨V',hV',hxV,dominantFunctionFieldMap f b,(map_ne_zero _).mpr hb, (fun p hp => hP.order_eq_of_equation hb hDb p (hV'U hp)), f.stalkMap (g (genericPoint C)) u,hu.map _,?_,?_⟩ · calc _ = dominantFunctionFieldMap f (algebraMap (Y.presheaf.stalk ((g ≫ f) (genericPoint C))) Y.functionField u) := (dominantFunctionFieldMap_algebraMap f (g (genericPoint C)) u).symm _ = _ := by rw [heu,map_div₀] · intro p hp rw [hEu p (hV'U hp),Scheme.Hom.stalkMap_comp] rfl end NumericalDimensionOne open AlgebraicGeometry CategoryTheory open scoped TensorProduct nonZeroDivisors open scoped TensorProduct open AlgebraicGeometry CategoryTheory TopologicalSpace open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace namespace NumericalDimensionOne open AlgebraicGeometry CategoryTheory variable {C X Y : Scheme} [IsIntegral C] [IsIntegral X] [IsIntegral Y] [IsLocallyNoetherian C] [IsLocallyNoetherian X] [IsLocallyNoetherian Y] [StalkwiseNormal X] [StalkwiseNormal Y] [CompactSpace C] (sC : C ⟶ Spec (.of ℂ)) [SmoothOfRelativeDimension 1 sC] [IsProper sC] include sC in theorem properCurveDegree_ambientPullback (g : C ⟶ X) (f : X ⟶ Y) [IsDominant f] (D : cartierDivisors (X := Y)) (P : WeilDivisor X) (hP : IsCartierPullback f D.1 P) : properCurveDegree g ⟨P,hP.isCartier⟩ = properCurveDegree (g ≫ f) D := by obtain ⟨E,hE⟩ := exists_pulledCartierRepresentative (g ≫ f) D.1 D.2 rw [properCurveDegree_eq sC (g ≫ f) D E hE, properCurveDegree_eq sC g ⟨P,hP.isCartier⟩ E (hE.of_cartierPullback g f hP)] end NumericalDimensionOne end OAI