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12dee50 | 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 99 100 101 102 103 104 105 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Cohomology.MixedDivisibleExtension
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite MvPolynomial
open MaximalSeshadri.Frames MaximalSeshadri.TensorPure MaximalSeshadri.Projective
variable {X : Scheme.{0}} {K ι : Type} [CommRing K] [Fintype ι]
theorem LineBundle.mixed_section_embedding [IsIntegral X] [CompactSpace X]
(p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] (L M : LineBundle X)
(s : ι → GlobalSections X L.sheaf)
(hc : (⨆ i, sectionOpen X (s i)) = ⊤)
(haff : ∀ i, IsAffineOpen (sectionOpen X (s i)))
(hne : ∀ i, (sectionOpen X (s i) : Set X).Nonempty)
(d : ∀ i, M.sheaf.restrict (sectionOpen X (s i)).ι ≅ O (sectionOpen X (s i)).toScheme) :
∃ n : ℕ, 0 < n ∧ ∃ σ : Type, ∃ _ : Fintype σ,
∃ t : σ → GlobalSections X ((L.pow n).tensor M).sheaf,
∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤,
IsClosedImmersion (sectionsMorphism
(p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) t ht) := by
classical
let k := p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom
let U (i : ι) := sectionOpen X (s i)
have hfinite (i : ι) : ∃ m : ℕ, ∃ b : Fin m → Γ((U i).toScheme,⊤),
Function.Surjective (eval₂Hom ((U i).ι.appTop.hom.comp k) b) := by
let : IsAffine (U i).toScheme := haff i
obtain ⟨m,b,hb⟩ := affine_coordinate_generators ((U i).ι ≫ p)
refine ⟨m,b,?_⟩
simpa only [Scheme.Hom.comp_appTop,CommRingCat.hom_comp,RingHom.comp_assoc,k] using hb
choose m b hb using hfinite
choose N₀ hN₀ using fun i => L.mixed_frame_extension M (s i) (hne i) (d i)
choose N hN using fun i j => L.divisible_mixed_extension M (s i) (hne i)
(openSectionEquiv M.sheaf (U i) (scalarEnd (b i j) ≫ (d i).inv))
let n := max (Finset.univ.sup N₀)
(Finset.univ.sup (fun v : (i : ι) × Fin (m i) => N v.1 v.2))
have hn₀ (i : ι) : N₀ i ≤ n :=
(Finset.le_sup (f := N₀) (Finset.mem_univ i)).trans (le_max_left _ _)
have hn (i : ι) (j : Fin (m i)) : N i j ≤ n :=
(Finset.le_sup (f := fun v : (i : ι) × Fin (m i) => N v.1 v.2)
(Finset.mem_univ ⟨i,j⟩)).trans (le_max_right _ _)
choose r hr hrv using fun i => hN₀ i n (hn₀ i)
choose t ht htv using fun i j => hN i j n (hn i j)
let P := (L.pow (n+1)).tensor M
let σ := ι ⊕ ((i : ι) × Fin (m i))
let S : σ → GlobalSections X P.sheaf := Sum.elim r (fun v => t v.1 v.2)
have hU (i : ι) : SectionOpens.isoOpen (S (Sum.inl i)) = U i := hr i
have hSn (i : ι) : IsIso (restrictSection (U i).ι (S (Sum.inl i))) :=
isIso_restricted_section _ _ (hU i)
let e (i : ι) : P.sheaf.restrict (U i).ι ≅ O (U i).toScheme :=
(asIso (restrictSection (U i).ι (S (Sum.inl i)))).symm
have hnorm (i : ι) : coefficient (e i) (restrictSection (U i).ι (S (Sum.inl i))) = 1 :=
coefficient_frame (e i)
have hratio (i : ι) (j : Fin (m i)) :
restrictSection (U i).ι (t i j) = scalarEnd (b i j) ≫ restrictSection (U i).ι (r i) := by
apply (openSectionEquiv P.sheaf (U i)).injective
rw [openSectionEquiv_restrict,openSectionEquiv_scalar,openSectionEquiv_restrict]
refine (htv i j).trans ?_
have hw := openSectionEquiv_scalar M.sheaf (U i) (b i j) (d i).inv
have hh := congrArg (pure (L.pow (n+1)).sheaf M.sheaf (U i)
((powerSection (s i) (n+1)).app (U i) (1 : Γ(X,U i)))) hw
exact hh.trans ((pure_smul_right _ _ _ _ _ _).trans
(congrArg (fun z : Γ(P.sheaf,U i) => (U i).topIso.hom (b i j) • z) (hrv i).symm))
have hS : (⨆ v, SectionOpens.isoOpen (S v)) = ⊤ := by
apply top_unique
rw [← hc]
apply iSup_le
intro i
exact (le_of_eq (hU i).symm).trans (le_iSup (fun v : σ => SectionOpens.isoOpen (S v)) (Sum.inl i))
have hproper : IsProper (X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k)) := by
rw [toSpec_scalarMap]
infer_instance
have hgen (i : ι) : Function.Surjective
(eval₂Hom ((U i).ι.appTop.hom.comp k)
(fun v => coefficient (e i) (restrictSection (U i).ι (S v)))) := by
have heq : (fun v => coefficient (e i) (restrictSection (U i).ι (S v))) ∘
(fun j : Fin (m i) => Sum.inr (⟨i,j⟩ : (i : ι) × Fin (m i))) = b i := by
funext j
exact coefficient_ratio (e i) (S (Sum.inl i)) (S (Sum.inr ⟨i,j⟩)) (b i j)
(hnorm i) (hratio i j)
intro z
obtain ⟨q,hq⟩ := hb i z
refine ⟨rename (fun j : Fin (m i) => Sum.inr (⟨i,j⟩ : (i : ι) × Fin (m i))) q,?_⟩
rw [eval₂Hom_rename,heq]
exact hq
have hclosed := framedSectionsMorphism_closed k S U hc e Sum.inl hnorm hU (fun i => haff i) hgen
rw [framedSectionsMorphism_eq_sectionsMorphism k S hS U hc e Sum.inl hnorm] at hclosed
exact ⟨n+1,by omega,σ,inferInstance,S,hS,hclosed⟩
end
end MaximalSeshadri.Geometry
end
end OAI
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