| 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 |
|
|
