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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 | 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
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