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7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.SectionOpens MaximalSeshadri.Projective
variable {X : Scheme.{0}}
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⟩
variable {B : Scheme.{0}} {I : X.IdealSheafData} {π : B ⟶ X}
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
end
end OAI
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