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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | 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
end
end
end
end OAI
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