openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /IntegralSectionDimension.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Configurations.RationalPoint | |
| import OAI.AlgebraicGeometry.Seshadri.Nodal.BranchDimension | |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.SectionDimension | |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.IntegralSectionIdeal | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.EtaleLowerDimension | |
| namespace OAI | |
| 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] [IsAlgClosed K] [Fintype σ] {X : Scheme} | |
| theorem sectionIdeal_dimension_eq_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) | |
| [IsIntegral (sectionIdeal k s hs v).subscheme] : | |
| topologicalKrullDim (sectionIdeal k s hs v).subscheme = 1 := by | |
| apply le_antisymm (sectionIdeal_dimension_le_one g h hbase L k s hs v hv) | |
| let I := sectionIdeal k s hs v | |
| obtain ⟨z⟩ : Nonempty I.subscheme := inferInstance | |
| let y := I.subschemeι z | |
| 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 := C.finite | |
| let : Fintype C.κ := Fintype.ofFinite _ | |
| let : Algebra K Γ(X,C.U.1) := (openScalars g C.U.1).toAlgebra | |
| let : Algebra (MvPolynomial C.κ K) Γ(X,C.U.1) := | |
| (MvPolynomial.eval₂Hom (openScalars g C.U.1) | |
| (fun j => -C.φ (chartCoordinate C.coord (C.a j).1))).toAlgebra | |
| let : IsScalarTower K (MvPolynomial C.κ K) Γ(X,C.U.1) := by | |
| apply IsScalarTower.of_algebraMap_eq' | |
| ext k | |
| exact (MvPolynomial.eval₂Hom_C _ _ k).symm | |
| let : Algebra.Etale (MvPolynomial C.κ K) Γ(X,C.U.1) := C.etale | |
| let : Algebra.FiniteType K Γ(X,C.U.1) := | |
| Algebra.FiniteType.trans (R := K) (S := MvPolynomial C.κ K) | |
| (A := Γ(X,C.U.1)) inferInstance inferInstance | |
| let e := sectionFrameOn (s i) C.U.1 hU | |
| let r := C.U.1.topIso.hom (coefficient e | |
| (restrictSection C.U.1.ι (sectionCombination k s v))) | |
| have hI : I.ideal C.U = Ideal.span {r} := sectionIdeal_on_any_frame k s hs v C.U e | |
| have hp : (Ideal.span {r}).IsPrime := by | |
| rw [← hI] | |
| apply ideal_prime_of_integral_subscheme I C.U y hy | |
| change y ∈ (I.support : Set X) | |
| rw [← I.range_subschemeι] | |
| exact ⟨z,rfl⟩ | |
| let := hp | |
| obtain ⟨f⟩ := exists_point_finite_type (K := K) (R := Γ(X,C.U.1) ⧸ Ideal.span {r}) | |
| have H := MaximalSeshadri.EtaleDimension.etale_principal_dimension_lower K Γ(X,C.U.1) C.κ | |
| (f.comp (Ideal.Quotient.mkₐ K (Ideal.span {r}))) r (by | |
| change f (Ideal.Quotient.mk _ r) = 0 | |
| rw [Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (Set.mem_singleton r)),map_zero]) | |
| have hcard : Fintype.card C.κ = 2 := by rw [← Nat.card_eq_fintype_card]; exact C.card | |
| rw [hcard] at H | |
| have H' : (1 : WithBot ℕ∞) ≤ ringKrullDim (Γ(X,C.U.1) ⧸ Ideal.span {r}) := by | |
| apply ENat.WithBot.add_le_add_one_right_iff.mp | |
| simpa only [Nat.cast_ofNat,one_add_one_eq_two] using H | |
| have H'' := subscheme_dimension_ge_affine_chart I C.U | |
| rw [hI] at H'' | |
| exact H'.trans H'' | |
| end | |
| end MaximalSeshadri.Projective | |
| end | |
| end OAI | |