| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.SurfaceAffineCover |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.Projective |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Geometry MaximalSeshadri.Frames |
| attribute [local instance] MvPolynomial.gradedAlgebra |
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| variable {K σ : Type} [Field K] [Infinite K] [Fintype σ] {X : Scheme} |
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| theorem surface_three_sections_through [IsIntegral X] [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 p] |
| (L : LineBundle X) (s : σ → (O X ⟶ L.sheaf)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) |
| [IsClosedImmersion (sectionsMorphism (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)] : |
| ∀ x : X, ∃ v : Fin 3 → (σ → K), |
| (⨆ i, SectionOpens.isoOpen (sectionCombination (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s (v i))) = ⊤ ∧ |
| ∀ i, x ∈ SectionOpens.isoOpen (sectionCombination (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s (v i)) := by |
| classical |
| let k := p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
| intro x |
| obtain ⟨v₀,hv₀⟩ := exists_section_avoiding k s hs (fun _ : Unit => x) |
| have hne : sectionCombination k s v₀ ≠ 0 := |
| section_ne_zero_of_mem_isoOpen L _ x (hv₀ ()) |
| let I := sectionIdeal k s hs v₀ |
| let U₀ := SectionOpens.isoOpen (sectionCombination k s v₀) |
| have hsupport : (I.support : Set X) = (U₀ : Set X)ᶜ := sectionIdeal_support_eq k s hs v₀ |
| have hbase : sectionsMorphism k s hs ≫ projectiveToSpec = p := by |
| change _ ≫ projectiveBase = p |
| rw [sectionsMorphism_over, toSpec_scalarMap] |
| have hd : topologicalKrullDim I.subscheme ≤ 1 := |
| sectionIdeal_dimension_le_one p (sectionsMorphism k s hs) hbase L k s hs v₀ hne |
| let : IsLocallyNoetherian I.subscheme := LocallyOfFiniteType.isLocallyNoetherian I.subschemeι |
| let : CompactSpace I.subscheme := QuasiCompact.compactSpace_of_compactSpace I.subschemeι |
| let : IsNoetherian I.subscheme := {} |
| obtain ⟨T,hT,hdetect⟩ := exists_finite_test_set_of_dimension_one I.subscheme hd |
| let : Finite T := hT.to_subtype |
| obtain ⟨v₁,hv₁⟩ := exists_section_avoiding k s hs (fun z : Option T => z.elim x (fun z => I.subschemeι z.1)) |
| let U₁ := SectionOpens.isoOpen (sectionCombination k s v₁) |
| have hfin : ((I.subschemeι ⁻¹ᵁ U₁ : I.subscheme.Opens) : Set I.subscheme)ᶜ.Finite := by |
| apply hdetect _ (I.subschemeι ⁻¹ᵁ U₁).isOpen |
| intro z hz |
| exact hv₁ (some ⟨z,hz⟩) |
| let Z : Set X := I.subschemeι '' (((I.subschemeι ⁻¹ᵁ U₁ : I.subscheme.Opens) : Set I.subscheme)ᶜ) |
| have hZ : Z.Finite := hfin.image _ |
| let : Finite Z := hZ.to_subtype |
| obtain ⟨v₂,hv₂⟩ := exists_section_avoiding k s hs (fun z : Option Z => z.elim x Subtype.val) |
| refine ⟨![v₀,v₁,v₂], ?_, ?_⟩ |
| swap |
| · intro i |
| fin_cases i |
| · exact hv₀ () |
| · exact hv₁ none |
| · exact hv₂ none |
| apply top_unique |
| intro y hy |
| apply Opens.mem_iSup.mpr |
| by_cases hy₀ : y ∈ U₀ |
| · exact ⟨0,hy₀⟩ |
| by_cases hy₁ : y ∈ U₁ |
| · exact ⟨1,hy₁⟩ |
| have hyI : y ∈ I.support := by |
| change y ∈ (I.support : Set X) |
| rw [hsupport] |
| exact hy₀ |
| obtain ⟨z,hz⟩ := (I.range_subschemeι ▸ hyI : y ∈ Set.range I.subschemeι) |
| have hyZ : y ∈ Z := ⟨z, by change I.subschemeι z ∉ U₁; rwa [hz], hz⟩ |
| exact ⟨2,hv₂ (some ⟨y,hyZ⟩)⟩ |
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| theorem surface_finite_projection_through [IsIntegral X] [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] [SmoothOfRelativeDimension 2 p] |
| (L : LineBundle X) (s : σ → (O X ⟶ L.sheaf)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) |
| [IsClosedImmersion (sectionsMorphism (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)] : |
| ∀ x : X, ∃ t : Fin 3 → (O X ⟶ L.sheaf), ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤, |
| IsFinite (sectionsMorphism |
| (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) t ht) ∧ ∀ i, x ∈ SectionOpens.isoOpen (t i) := by |
| intro x |
| let k := p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
| obtain ⟨v,hv,hx⟩ := surface_three_sections_through p L s hs x |
| let t : Fin 3 → (O X ⟶ L.sheaf) := fun i => sectionCombination k s (v i) |
| have ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤ := hv |
| refine ⟨t,ht,?_,hx⟩ |
| let f := sectionsMorphism k t ht |
| let : IsProper (X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k)) := by |
| dsimp only [k] |
| rw [toSpec_scalarMap] |
| infer_instance |
| let : IsProper f := sectionsMorphism_proper k t ht |
| let : IsAffineHom f := by |
| apply isAffineHom_of_forall_exists_isAffineOpen |
| intro y |
| have hy : y ∈ ⨆ i : Fin 3, Proj.basicOpen (PolyGrade K (Fin 3)) (MvPolynomial.X i) := by |
| rw [projective_coordinate_cover] |
| trivial |
| obtain ⟨i,hi⟩ := Opens.mem_iSup.mp hy |
| refine ⟨Proj.basicOpen (PolyGrade K (Fin 3)) (MvPolynomial.X i),hi, |
| Proj.isAffineOpen_basicOpen _ _ (poly_X_mem i) (by decide),?_⟩ |
| rw [show f = sectionsMorphism k t ht from rfl, sectionsMorphism_preimage] |
| dsimp only [t] |
| rw [← sectionsMorphism_hyperplane _ s hs] |
| exact (Proj.isAffineOpen_basicOpen (PolyGrade K σ) _ |
| (linearEquation_homogeneous (v i)) (by decide)).preimage _ |
| exact IsFinite.iff_isProper_and_isAffineHom.mpr ⟨inferInstance,inferInstance⟩ |
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| end |
| end MaximalSeshadri.Projective |
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| end |
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| end OAI |
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