| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Projective.FiniteAtlas |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| 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 |
|
|
| variable {K : Type} [Field K] {X : Scheme} {σ : Type} |
|
|
| theorem etale_chart_of_affine_coordinates [IsIntegral X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (h : X ⟶ Proj (PolyGrade K σ)) |
| (hbase : h ≫ projectiveToSpec = g) |
| (i : σ) (V : X.affineOpens) |
| (φ : PolyChart (R := K) i →+* Γ(X, V.1)) (hsurj : Function.Surjective φ) |
| (hφ : Spec.map (CommRingCat.ofHom φ) ≫ |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = |
| V.2.fromSpec ≫ h) |
| (W : X.Opens) (x : X) (hxW : x ∈ W) (hxV : x ∈ V.1) : |
| ∃ C : EtaleProjectiveChart g h, x ∈ C.U.1 ∧ C.U.1 ≤ W ∧ C.coord = i := by |
| classical |
| 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 |
|
|
| theorem finite_atlas_of_local_coordinates [IsIntegral X] [CompactSpace X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (h : X ⟶ Proj (PolyGrade K σ)) |
| (hbase : h ≫ projectiveToSpec = g) |
| (hlocal : ∀ x : X, ∃ i : σ, ∃ V : X.affineOpens, x ∈ V.1 ∧ |
| ∃ φ : PolyChart (R := K) i →+* Γ(X, V.1), Function.Surjective φ ∧ |
| Spec.map (CommRingCat.ofHom φ) ≫ |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = |
| V.2.fromSpec ≫ h) : |
| Nonempty (OverlappingEtaleProjectiveAtlas g h) := by |
| classical |
| have hchart (W : X.Opens) (x : X) (hxW : x ∈ W) : |
| ∃ C : EtaleProjectiveChart g h, x ∈ C.U.1 ∧ C.U.1 ≤ W := by |
| obtain ⟨i, V, hxV, φ, hsurj, hφ⟩ := hlocal x |
| obtain ⟨C, hxC, hCW, _⟩ := |
| etale_chart_of_affine_coordinates g h hbase i V φ hsurj hφ W x hxW hxV |
| exact ⟨C, hxC, hCW⟩ |
| choose C hC using (fun x : X => hchart ⊤ x |
| (show x ∈ (⊤ : X.Opens) from trivial)) |
| obtain ⟨s, hs⟩ := isCompact_univ.elim_finite_subcover (fun x : X => ((C x).U.1 : Set X)) |
| (fun x => (C x).U.1.isOpen) (by |
| intro x _ |
| exact Set.mem_iUnion.mpr ⟨x, (hC x).1⟩) |
| have hcover : ∀ x : X, ∃ j : s, x ∈ (C j.1).U.1 := by |
| intro x |
| obtain ⟨j, hj, hx⟩ := Set.mem_iUnion₂.mp (hs (Set.mem_univ x)) |
| exact ⟨⟨j, hj⟩, hx⟩ |
| let x₀ : X := Classical.arbitrary X |
| obtain ⟨j₀, _⟩ := hcover x₀ |
| have hov (j : s) : ∃ D : EtaleProjectiveChart g h, |
| D.U.1 ≤ (C j₀.1).U.1 ∧ D.U.1 ≤ (C j.1).U.1 := by |
| have hne (k : s) : ((C k.1).U.1 : Set X).Nonempty := |
| Set.nonempty_coe_sort.mp (C k.1).nonempty |
| obtain ⟨x, hx₀, hx⟩ := nonempty_preirreducible_inter |
| (C j₀.1).U.1.isOpen (C j.1).U.1.isOpen (hne j₀) (hne j) |
| obtain ⟨D, _, hD⟩ := hchart |
| ((C j₀.1).U.1 ⊓ (C j.1).U.1) x ⟨hx₀, hx⟩ |
| exact ⟨D, hD.trans inf_le_left, hD.trans inf_le_right⟩ |
| choose D hD₀ hD using hov |
| refine ⟨{ ι := s ⊕ s |
| finite := inferInstance |
| chart := Sum.elim (fun j => C j.1) D |
| cover := ?_ |
| base := Sum.inl j₀ |
| overlap := fun j => Sum.inr (Sum.elim id id j) |
| left := ?_ |
| right := ?_ }⟩ |
| · intro x |
| obtain ⟨j, hj⟩ := hcover x |
| exact ⟨Sum.inl j, hj⟩ |
| · intro j |
| cases j with |
| | inl j => exact hD₀ j |
| | inr j => exact hD₀ j |
| · intro j |
| cases j with |
| | inl j => exact hD j |
| | inr j => exact le_rfl |
|
|
| end |
| end MaximalSeshadri.ProjectiveBertini |
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| end |
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| end |
| end |
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| end OAI |
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