| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.SmoothAffineRefinement |
| import OAI.AlgebraicGeometry.Seshadri.Projective.ProjectiveCoordinates |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
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|
| namespace MaximalSeshadri.ProjectiveBertini |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace KaehlerDifferential |
| open MaximalSeshadri.Projective MaximalSeshadri.AlgebraicJets |
| attribute [local instance] MvPolynomial.gradedAlgebra |
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| variable {K : Type} [Field K] {X : Scheme} {σ : Type} |
|
|
| structure EtaleProjectiveChart (g : X ⟶ Spec (CommRingCat.of K)) |
| (h : X ⟶ Proj (PolyGrade K σ)) where |
| U : X.affineOpens |
| coord : σ |
| nonempty : Nonempty U.1 |
| φ : PolyChart (R := K) coord →+* Γ(X, U.1) |
| factor : Spec.map (CommRingCat.ofHom φ) ≫ |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X coord) (poly_X_mem coord) (by decide) = |
| U.2.fromSpec ≫ h |
| κ : Type |
| finite : Finite κ |
| a : κ → ChartVariables coord |
| card : Nat.card κ = 2 |
| etale : (MvPolynomial.eval₂Hom (openScalars g U.1) |
| (fun k => -φ (chartCoordinate coord (a k).1))).Etale |
|
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| lemma restrict_projective_factor (h : X ⟶ Proj (PolyGrade K σ)) |
| (U V : X.affineOpens) (e : U.1 ≤ V.1) (i : σ) |
| (φ : PolyChart (R := K) i →+* Γ(X, V.1)) |
| (hφ : Spec.map (CommRingCat.ofHom φ) ≫ |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = |
| V.2.fromSpec ≫ h) : |
| Spec.map (CommRingCat.ofHom ((X.presheaf.map (homOfLE e).op).hom.comp φ)) ≫ |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = |
| U.2.fromSpec ≫ h := by |
| rw [CommRingCat.ofHom_comp, Spec.map_comp, Category.assoc, hφ, ← Category.assoc, |
| CommRingCat.ofHom_hom, V.2.map_fromSpec U.2] |
|
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| lemma etale_aeval_equiv {A B ι : Type} [CommRing A] [CommRing B] |
| [Algebra K A] [Algebra K B] (e : A ≃ₐ[K] B) (v : ι → A) |
| (hv : (MvPolynomial.aeval (R := K) v).toRingHom.Etale) : |
| (MvPolynomial.aeval (R := K) (fun i => e (v i))).toRingHom.Etale := by |
| have hh := RingHom.Etale.respectsIso.1 _ e.toRingEquiv hv |
| change (e.toAlgHom.comp (MvPolynomial.aeval (R := K) v)).toRingHom.Etale at hh |
| rw [MvPolynomial.comp_aeval] at hh |
| exact hh |
|
|
| theorem exists_etale_projective_chart [IsIntegral X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (h : X ⟶ Proj (PolyGrade K σ)) [IsClosedImmersion h] |
| (hbase : h ≫ projectiveToSpec = g) |
| (i : σ) (W : X.Opens) (x : X) (hxW : x ∈ W) |
| (hxi : h x ∈ Proj.basicOpen (PolyGrade K σ) (MvPolynomial.X i)) : |
| ∃ C : EtaleProjectiveChart g h, x ∈ C.U.1 ∧ C.U.1 ≤ W ∧ C.coord = i := by |
| classical |
| obtain ⟨V, φ, hV, hsurj, hφ⟩ := closed_projective_affine_surjective h i |
| have hxV : x ∈ V.1 := by rwa [hV] |
| obtain ⟨U, hxU, hUW, hstd⟩ := exists_standard_smooth_affine_inside g 2 |
| (W ⊓ V.1) x ⟨hxW, hxV⟩ |
| have e : U.1 ≤ V.1 := hUW.trans inf_le_right |
| let : Nonempty V.1 := ⟨⟨x, hxV⟩⟩ |
| let : Nonempty U.1 := ⟨⟨x, hxU⟩⟩ |
| let : Algebra K Γ(X, V.1) := (openScalars g V.1).toAlgebra |
| let : Algebra K Γ(X, U.1) := (openScalars g U.1).toAlgebra |
| let r : Γ(X, V.1) →+* Γ(X, U.1) := (X.presheaf.map (homOfLE e).op).hom |
| let : Algebra Γ(X, V.1) Γ(X, U.1) := r.toAlgebra |
| let : IsScalarTower K Γ(X, V.1) Γ(X, U.1) := |
| IsScalarTower.of_algebraMap_eq' (openScalars_restrict g e).symm |
| let : Algebra.Etale Γ(X, V.1) Γ(X, U.1) := affine_restriction_etale U V e |
| have hconst := projective_affine_constants h g hbase V (spec_openScalars g V) i φ hφ |
| have hspan := differentials_span_projective_chart i φ hsurj hconst |
| have hs := differentials_span_of_formallyEtale (T := Γ(X, U.1)) |
| (fun j : ChartVariables i => -φ (chartCoordinate i j.1)) hspan |
| let φU := r.comp φ |
| let : Algebra.IsStandardSmoothOfRelativeDimension 2 K Γ(X, U.1) := hstd |
| let p : PrimeSpectrum Γ(X, U.1) := U.2.isoSpec.hom ⟨x, hxU⟩ |
| obtain ⟨f, hfp, κ, hκ, a, hcard, het⟩ := exists_local_etale_from_spanning 2 |
| (fun j : ChartVariables i => -φU (chartCoordinate i j.1)) |
| (by simpa only [φU, RingHom.comp_apply, map_neg, RingHom.algebraMap_toAlgebra] using hs) p.asIdeal |
| let T : X.affineOpens := ⟨X.basicOpen f, U.2.basicOpen f⟩ |
| have hxf : x ∈ T.1 := by |
| have hp : U.2.fromSpec p = x := by |
| change (U.2.isoSpec.hom ≫ U.2.fromSpec) ⟨x, hxU⟩ = x |
| rw [U.2.isoSpec_hom_fromSpec] |
| rfl |
| have hm : p ∈ PrimeSpectrum.basicOpen f := hfp |
| rw [← U.2.fromSpec_preimage_basicOpen f] at hm |
| change U.2.fromSpec p ∈ X.basicOpen f at hm |
| rwa [hp] at hm |
| let r' : Γ(X, U.1) →+* Γ(X, T.1) := |
| (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom |
| let : Algebra K Γ(X, T.1) := (openScalars g T.1).toAlgebra |
| let : IsScalarTower K Γ(X, U.1) Γ(X, T.1) := |
| IsScalarTower.of_algebraMap_eq' (openScalars_restrict g (X.basicOpen_le f)).symm |
| let : IsLocalization.Away f Γ(X, T.1) := U.2.isLocalization_basicOpen f |
| let ρ := (IsLocalization.algEquiv (Submonoid.powers f) |
| (Localization.Away f) Γ(X, T.1)).restrictScalars K |
| have het' := etale_aeval_equiv ρ _ het |
| let φT := r'.comp φU |
| have hφU := restrict_projective_factor h U V e i φ hφ |
| refine ⟨{ U := T |
| coord := i |
| nonempty := ⟨⟨x, hxf⟩⟩ |
| φ := φT |
| factor := restrict_projective_factor h T U (X.basicOpen_le f) i φU hφU |
| κ := κ |
| finite := hκ |
| a := a |
| card := hcard |
| etale := ?_ }, hxf, |
| (X.basicOpen_le f).trans (hUW.trans inf_le_left), rfl⟩ |
| convert het' using 1 |
| apply MvPolynomial.ringHom_ext |
| · intro c |
| change MvPolynomial.eval₂Hom _ _ (MvPolynomial.C c) = |
| MvPolynomial.aeval _ (MvPolynomial.C c) |
| rw [MvPolynomial.eval₂Hom_C, MvPolynomial.aeval_C] |
| rfl |
| · intro k |
| change MvPolynomial.eval₂Hom _ _ (MvPolynomial.X k) = |
| MvPolynomial.aeval _ (MvPolynomial.X k) |
| rw [MvPolynomial.eval₂Hom_X', MvPolynomial.aeval_X] |
| change -r' (φU (chartCoordinate i (a k).1)) = |
| ρ (algebraMap Γ(X, U.1) (Localization.Away f) (-φU (chartCoordinate i (a k).1))) |
| rw [← map_neg] |
| exact ((IsLocalization.algEquiv (Submonoid.powers f) |
| (Localization.Away f) Γ(X, T.1)).commutes |
| (-φU (chartCoordinate i (a k).1))).symm |
|
|
| end |
| end MaximalSeshadri.ProjectiveBertini |
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| end |
| end |
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| end OAI |
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