| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Projective.FiniteAtlas |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.CurveFinite |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.PrincipalSectionIdeal |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.CartierSection |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.EtaleDimension |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.LocalDimension |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Projective |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Geometry MaximalSeshadri.ProjectiveBertini MaximalSeshadri.Frames |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
| variable {K σ τ : Type} [Field K] [Fintype σ] {X : Scheme} |
|
|
| lemma EtaleProjectiveChart.dimension_le [IsIntegral X] |
| (g : X ⟶ Spec (CommRingCat.of K)) (h : X ⟶ Proj (PolyGrade K τ)) |
| (C : EtaleProjectiveChart g h) : ringKrullDim Γ(X,C.U.1) ≤ 2 := by |
| let := C.finite |
| let φ := MvPolynomial.eval₂Hom (openScalars g C.U.1) |
| (fun k => -C.φ (chartCoordinate C.coord (C.a k).1)) |
| let : Algebra (MvPolynomial C.κ K) Γ(X,C.U.1) := φ.toAlgebra |
| let : Algebra.Etale (MvPolynomial C.κ K) Γ(X,C.U.1) := C.etale |
| have H := MaximalSeshadri.EtaleDimension.etale_polynomial_dimension_le K Γ(X,C.U.1) C.κ |
| simpa only [C.card, Nat.cast_ofNat] using H |
|
|
| theorem sectionIdeal_dimension_le_one [IsIntegral X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (h : X ⟶ Proj (PolyGrade K τ)) [IsClosedImmersion h] |
| (hbase : h ≫ projectiveToSpec = g) |
| (L : LineBundle X) (k : K →+* Γ(X,⊤)) (s : σ → (O X ⟶ L.sheaf)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) (v : σ → K) |
| (hv : sectionCombination k s v ≠ 0) : |
| topologicalKrullDim (sectionIdeal k s hs v).subscheme ≤ 1 := by |
| apply subscheme_dimension_le_of_affine_neighborhoods |
| intro x |
| let y := (sectionIdeal k s hs v).subschemeι x |
| obtain ⟨i,hi⟩ := Opens.mem_iSup.mp (hs.ge (show y ∈ (⊤ : X.Opens) from trivial)) |
| obtain ⟨C,hy,hU⟩ := exists_etale_projective_neighborhood g h hbase |
| (SectionOpens.isoOpen (s i)) y hi |
| let := C.nonempty |
| let e := sectionFrameOn (s i) C.U.1 hU |
| let f := C.U.1.topIso.hom (coefficient e |
| (restrictSection C.U.1.ι (sectionCombination k s v))) |
| have hf : f ≠ 0 := (isRegular_iff_ne_zero).mp (L.section_coefficient_regular _ hv C.U e) |
| refine ⟨C.U,hy,?_⟩ |
| rw [sectionIdeal_on_any_frame k s hs v C.U e] |
| apply ENat.WithBot.add_le_add_one_right_iff.mp |
| have H := (ringKrullDim_quotient_succ_le_of_nonZeroDivisor |
| (mem_nonZeroDivisors_iff_ne_zero.mpr hf)).trans (EtaleProjectiveChart.dimension_le g h C) |
| exact H |
|
|
| theorem sectionIdeal_proper_closed_finite [IsIntegral X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (h : X ⟶ Proj (PolyGrade K τ)) [IsClosedImmersion h] |
| (hbase : h ≫ projectiveToSpec = g) |
| (L : LineBundle X) (k : K →+* Γ(X,⊤)) (s : σ → (O X ⟶ L.sheaf)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) (v : σ → K) |
| (hv : sectionCombination k s v ≠ 0) |
| [IsIntegral (sectionIdeal k s hs v).subscheme] |
| [IsNoetherian (sectionIdeal k s hs v).subscheme] |
| {Z : Set (sectionIdeal k s hs v).subscheme} (hZ : IsClosed Z) (hp : Z ≠ Set.univ) : |
| Z.Finite := |
| proper_closed_finite_of_dimension_one |
| (sectionIdeal_dimension_le_one g h hbase L k s hs v hv) hZ hp |
|
|
| end |
| end MaximalSeshadri.Projective |
| end |
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| end |
| end |
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| end OAI |
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|
