| 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_finite_projection [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)] : |
| ∃ 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) := by |
| let k := p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
| obtain ⟨v,hv⟩ := surface_three_sections p L s hs |
| 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,?_⟩ |
| 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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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Projective MaximalSeshadri.Frames |
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| theorem Surface.exists_finite_projection (S : Surface) (A : LineBundle S.scheme) |
| (hA : A.IsAmple) : |
| ∃ n : ℕ, 0 < n ∧ ∃ t : Fin 3 → (O S.scheme ⟶ (A.pow n).sheaf), |
| ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤, |
| IsFinite (sectionsMorphism (baseScalars S.structureMap) t ht) := by |
| obtain ⟨n,hn,σ,hσ,s,hs,hclosed⟩ := S.ample_embedding A hA |
| obtain ⟨t,ht,hf⟩ := @surface_finite_projection ℂ σ _ _ hσ S.scheme _ _ |
| S.structureMap _ _ (A.pow n) s hs hclosed |
| exact ⟨n,hn,t,ht,hf⟩ |
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| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end OAI |
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