| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Projective.BertiniEquations |
| import OAI.AlgebraicGeometry.Seshadri.Configurations.RationalPoint |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.SectionChart |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.LocalCoordinateAtlas |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.ProjectiveBertini |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Projective MaximalSeshadri.BertiniIntegral MaximalSeshadri.Frames |
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| theorem exists_smooth_integral_cartier_section_of_local_coordinates |
| {K : Type} [Field K] [CharZero K] [IsAlgClosed K] [Uncountable K] |
| {n : ℕ} (X : Scheme) [IsIntegral X] [CompactSpace X] |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] |
| (M : X.Modules) (k : K →+* Γ(X, ⊤)) |
| (s : Option (Fin n) → (O X ⟶ M)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) |
| (h : X ⟶ Proj (PolyGrade K (Option (Fin n)))) |
| (hh : h = sectionsMorphism k s hs) |
| (hbase : h ≫ projectiveToSpec = g) |
| (hlocal : ∀ x : X, ∃ i : Option (Fin n), ∃ V : X.affineOpens, x ∈ V.1 ∧ |
| ∃ φ : PolyChart (R := K) i →+* Γ(X, V.1), Function.Surjective φ ∧ |
| Spec.map (CommRingCat.ofHom φ) ≫ |
| Proj.awayι (PolyGrade K (Option (Fin n))) (MvPolynomial.X i) (poly_X_mem i) (by decide) = |
| V.2.fromSpec ≫ h) |
| (q : MvPolynomial (Option (Fin n)) K) (hq : q ≠ 0) : |
| ∃ t : Option (Fin n) → K, MvPolynomial.aeval t q ≠ 0 ∧ |
| IsIntegral (projectiveHyperplane h t).subscheme ∧ |
| Smooth ((projectiveHyperplane h t).subschemeι ≫ g) ∧ |
| ∀ x : X, ∃ U : X.affineOpens, x ∈ U.1 ∧ ∃ r : Γ(X, U.1), |
| IsRegular r ∧ (projectiveHyperplane h t).ideal U = Ideal.span {r} ∧ |
| ∃ e : M.restrict U.1.ι ≅ O U.1.toScheme, |
| U.1.topIso.hom (coefficient e (restrictSection U.1.ι |
| (sectionCombination k s t))) = r := by |
| classical |
| let A := Classical.choice (finite_atlas_of_local_coordinates g h hbase hlocal) |
| let C := A.chart |
| let : Finite A.ι := A.finite |
| let : ∀ j, Nonempty (C j).U.1 := fun j => (C j).nonempty |
| let : ∀ j, IsDomain Γ(X, (C j).U.1) := fun j => inferInstance |
| let : ∀ j, Algebra K Γ(X, (C j).U.1) := |
| fun j => (openScalars g (C j).U.1).toAlgebra |
| let : ∀ j, Finite (C j).κ := fun j => (C j).finite |
| let : ∀ j, Algebra (MvPolynomial (C j).κ K) Γ(X, (C j).U.1) := |
| fun j => (MvPolynomial.eval₂Hom (openScalars g (C j).U.1) |
| (fun k => -(C j).φ (chartCoordinate (C j).coord ((C j).a k).1))).toAlgebra |
| let : ∀ j, IsScalarTower K (MvPolynomial (C j).κ K) Γ(X, (C j).U.1) := by |
| intro j |
| apply IsScalarTower.of_algebraMap_eq' |
| ext k |
| exact (MvPolynomial.eval₂Hom_C _ _ k).symm |
| let : ∀ j, Algebra.Etale (MvPolynomial (C j).κ K) Γ(X, (C j).U.1) := |
| fun j => (C j).etale |
| let : ∀ j, Algebra.FiniteType K Γ(X, (C j).U.1) := fun j => |
| Algebra.FiniteType.trans (R := K) (S := MvPolynomial (C j).κ K) |
| (A := Γ(X, (C j).U.1)) inferInstance inferInstance |
| let : Fintype A.ι := Fintype.ofFinite _ |
| let ψ (j : A.ι) : Γ(X, (C j).U.1) →ₐ[K] K := |
| Classical.choice (exists_point_finite_type (K := K) (R := Γ(X, (C j).U.1))) |
| let v (j : A.ι) (k : Option (Fin n)) : K := |
| ψ j ((C j).φ (chartCoordinate (C j).coord k)) |
| let P (j : A.ι) : MvPolynomial (Option (Fin n)) K := linearEquation (v j) |
| have hP (j : A.ι) : P j ≠ 0 := |
| linearEquation_ne_zero (v j) (C j).coord (by |
| simp [v, chartCoordinate_self]) |
| let Q := q * ∏ j, P j |
| have hQ : Q ≠ 0 := mul_ne_zero hq (Finset.prod_ne_zero_iff.mpr (fun j _ => hP j)) |
| let δ (j : A.ι) : Fin n ≃ ChartVariables (C j).coord := |
| Equiv.optionSubtype (C j).coord ⟨Equiv.swap none (C j).coord, by simp⟩ |
| let e (j : A.ι) : Option (Fin n) ≃ Option (Fin n) := |
| ((Equiv.optionSubtype (C j).coord).symm (δ j)).1 |
| have hp (j : A.ι) : ∃ a b : (C j).κ, a ≠ b := by |
| let : Fintype (C j).κ := Fintype.ofFinite _ |
| have hcard : Fintype.card (C j).κ = 2 := by |
| rw [← Nat.card_eq_fintype_card] |
| exact (C j).card |
| let : Nontrivial (C j).κ := Fintype.one_lt_card_iff_nontrivial.mp (by omega) |
| exact exists_pair_ne _ |
| choose a b hab using hp |
| have hc : ∀ j, ∀ c, (C j).φ (chartConstants (C j).coord c) = |
| openScalars g (C j).U.1 c := fun j => |
| projective_affine_constants h g hbase (C j).U (spec_openScalars g (C j).U) |
| _ (C j).φ (C j).factor |
| obtain ⟨t, ht, hI, hS, heq⟩ := |
| smooth_integral_hyperplane_of_etale_atlas_with_equations X g h hbase |
| (fun j => (C j).U) A.cover (fun j => spec_openScalars g (C j).U) |
| e (fun j => (C j).φ) (fun j => (C j).factor) (fun j => (C j).κ) |
| (fun j => (δ j).symm ((C j).a (a j))) |
| (fun j => (δ j).symm ((C j).a (b j))) a b hab |
| (by |
| intro j |
| change -(C j).φ (chartCoordinate (C j).coord (δ j ((δ j).symm ((C j).a (a j)))).1) = |
| MvPolynomial.eval₂Hom _ _ (MvPolynomial.X (a j)) |
| rw [Equiv.apply_symm_apply, MvPolynomial.eval₂Hom_X']) |
| (by |
| intro j |
| change -(C j).φ (chartCoordinate (C j).coord (δ j ((δ j).symm ((C j).a (b j)))).1) = |
| MvPolynomial.eval₂Hom _ _ (MvPolynomial.X (b j)) |
| rw [Equiv.apply_symm_apply, MvPolynomial.eval₂Hom_X']) |
| A.base A.overlap A.left A.right Q hQ |
|
|
| have htq : MvPolynomial.aeval t q ≠ 0 := |
| (mul_ne_zero_iff.mp (by simpa only [Q, map_mul] using ht)).1 |
| have htP (j : A.ι) : MvPolynomial.aeval t (P j) ≠ 0 := by |
| have hh := (mul_ne_zero_iff.mp (by simpa only [Q, map_mul] using ht)).2 |
| rw [map_prod] at hh |
| exact Finset.prod_ne_zero_iff.mp hh j (Finset.mem_univ j) |
| refine ⟨t, htq, hI, hS, fun x => ?_⟩ |
| obtain ⟨j, hj⟩ := A.cover x |
| let r : Γ(X, (C j).U.1) := ∑ k, (C j).φ (chartConstants (C j).coord (t k)) * |
| (C j).φ (chartCoordinate (C j).coord k) |
| have hψr : ψ j r = MvPolynomial.aeval t (P j) := by |
| simp only [r, map_sum, map_mul, hc, P, linearEquation, MvPolynomial.aeval_C, |
| MvPolynomial.aeval_X] |
| apply Finset.sum_congr rfl |
| intro k hk |
| have hh : ψ j (openScalars g (C j).U.1 (t k)) = t k := (ψ j).commutes _ |
| rw [hh, mul_comm] |
| rfl |
| have hr : r ≠ 0 := by |
| intro hz |
| apply htP j |
| rw [← hψr, hz, map_zero] |
| obtain ⟨e, he⟩ := sectionsMorphism_affine_equation k s hs (C j).U |
| (C j).coord (C j).φ (by simpa only [← hh] using (C j).factor) |
| exact ⟨(C j).U, hj, r, isRegular_iff_ne_zero.mpr hr, heq j, e, he t⟩ |
|
|
| end |
| end MaximalSeshadri.ProjectiveBertini |
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| end |
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| end |
| end |
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| end OAI |
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