| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.PullbackSectionOpenExact |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.AmpleFromCover |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.MixedSectionOpen |
| import OAI.AlgebraicGeometry.Seshadri.Blowup.BlowupAffineSections |
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| namespace OAI |
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| |
| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace |
| open MaximalSeshadri.Frames MaximalSeshadri.SectionOpens MaximalSeshadri.Projective |
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| variable {X : Scheme.{0}} |
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| theorem LineBundle.twist_affine_cover [IsIntegral X] [CompactSpace X] |
| (P J : LineBundle X) {κ : Type} (t : κ → (O X ⟶ P.sheaf)) |
| (ht : (⨆ i,isoOpen (t i)) = ⊤) |
| (hlocal : ∀ i, ∃ (α : Type) |
| (w : α → (O (isoOpen (t i)).toScheme ⟶ J.sheaf.restrict (isoOpen (t i)).ι)), |
| (⨆ a,isoOpen (w a)) = ⊤ ∧ ∀ a, IsAffineOpen (isoOpen (w a))) : |
| ∃ n : ℕ, 0 < n ∧ ∃ (σ : Type) (_ : Fintype σ) |
| (s : σ → (O X ⟶ ((P.pow n).tensor J).sheaf)), |
| (⨆ a,isoOpen (s a)) = ⊤ ∧ |
| (∀ a, IsAffineOpen (isoOpen (s a))) ∧ |
| (∀ a, (isoOpen (s a) : Set X).Nonempty) := by |
| classical |
| choose α w hw hwa using hlocal |
| have hex (x : X) : ∃ i, x ∈ isoOpen (t i) := by |
| apply Opens.mem_iSup.mp |
| rw [ht]; trivial |
| choose i hi using hex |
| have hex' (x : X) : ∃ a, (⟨x,hi x⟩ : (isoOpen (t (i x))).toScheme) ∈ |
| isoOpen (w (i x) a) := by |
| apply Opens.mem_iSup.mp |
| exact (hw (i x)).symm ▸ (show |
| (⟨x,hi x⟩ : (isoOpen (t (i x))).toScheme) ∈ |
| (⊤ : (isoOpen (t (i x))).toScheme.Opens) from trivial) |
| choose a ha using hex' |
| let V (x : X) : X.Opens := (isoOpen (t (i x))).ι ''ᵁ isoOpen (w (i x) (a x)) |
| have hxV (x : X) : x ∈ V x := ⟨⟨x,hi x⟩,ha x,rfl⟩ |
| obtain ⟨F,hF⟩ := isCompact_univ.elim_finite_subcover (fun x => (V x : Set X)) |
| (fun x => (V x).isOpen) (by intro x _; exact Set.mem_iUnion.mpr ⟨x,hxV x⟩) |
| choose N hN using fun x : F => P.mixed_section_open_extension J (t (i x.val)) |
| ⟨x.val,hi x.val⟩ (w (i x.val) (a x.val)) |
| let n := Finset.univ.sup N |
| choose s hs using fun x : F => hN x n (Finset.le_sup (Finset.mem_univ x)) |
| refine ⟨n+1,by omega,F,inferInstance,s,?_,?_,?_⟩ |
| · apply top_unique |
| intro x _ |
| obtain ⟨y,hy⟩ := Set.mem_iUnion.mp (hF (show x ∈ Set.univ from trivial)) |
| obtain ⟨hyF,hxy⟩ := Set.mem_iUnion.mp hy |
| apply Opens.mem_iSup.mpr |
| refine ⟨⟨y,hyF⟩,?_⟩ |
| rw [show isoOpen (s ⟨y,hyF⟩) = V y from hs ⟨y,hyF⟩] |
| exact hxy |
| · intro x |
| rw [show isoOpen (s x) = V x.val from hs x] |
| exact (hwa (i x.val) (a x.val)).image_of_isOpenImmersion _ |
| · intro x |
| rw [show isoOpen (s x) = V x.val from hs x] |
| exact ⟨x.val,hxV x.val⟩ |
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| variable {B : Scheme.{0}} {I : X.IdealSheafData} {π : B ⟶ X} |
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| theorem IsBlowup.ample_exceptional_twist [IsIntegral B] [CompactSpace B] |
| [CompactSpace X] (hπ : IsBlowup I π) (L : LineBundle X) (hL : L.IsAmple) |
| (J : LineBundle B) (ι : J.sheaf ⟶ O B) (hJ : PresentsPullbackIdeal I π J ι) : |
| ∃ a n : ℕ, 0 < a ∧ 0 < n ∧ |
| ((((L.pow a).pullback π).pow n).tensor J).IsAmple := by |
| classical |
| obtain ⟨a,ha,l,s,hs,hsa,-⟩ := L.ample_common_degree_cover hL |
| let P := (L.pow a).pullback π |
| let t (i : Fin l) : O B ⟶ P.sheaf := pullbackSection π (s i) |
| have ht i : isoOpen (t i) = π ⁻¹ᵁ isoOpen (s i) := pullback_isoOpen_eq (L.pow a) (s i) π |
| have hc : (⨆ i,isoOpen (t i)) = ⊤ := pullback_sections_cover s hs π |
| have hloc (i : Fin l) : ∃ (α : Type) |
| (w : α → (O (isoOpen (t i)).toScheme ⟶ J.sheaf.restrict (isoOpen (t i)).ι)), |
| (⨆ a,isoOpen (w a)) = ⊤ ∧ ∀ a, IsAffineOpen (isoOpen (w a)) := by |
| rw [ht] |
| let U : X.affineOpens := ⟨isoOpen (s i),hsa i⟩ |
| let : IsAffine U.1.toScheme := U.2 |
| exact (hπ.morphismRestrict U.1).affine_section_cover (J.restrict (π ⁻¹ᵁ U.1).ι) |
| (InvertibleLocal.restrictedInclusion J ι (π ⁻¹ᵁ U.1).ι) |
| (presents_morphismRestrict J ι hJ U) |
| obtain ⟨n,hn,σ,hσ,q,hq,hqa,-⟩ := P.twist_affine_cover J t hc hloc |
| exact ⟨a,n,ha,hn,((P.pow n).tensor J).ample_of_affine_section_cover q hq hqa⟩ |
|
|
| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end OAI |
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