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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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Intersection.FiniteIntersection
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
variable {K : Type} [Field K] {X : Scheme}
theorem finite_schematic_intersection (g : X ⟶ Spec (CommRingCat.of K)) [IsProper g]
(I J : X.IdealSheafData) [IsIntegral I.subscheme]
(hd : topologicalKrullDim I.subscheme ≤ 1) (hJ : ¬ J ≤ I) :
Finite (I ⊔ J).subscheme := by
classical
let Q := J.comap I.subschemeι
have hQ : Q ≠ ⊥ := by
intro he
apply hJ
have h := Scheme.IdealSheafData.le_map_comap J I.subschemeι
simpa only [show J.comap I.subschemeι = ⊥ from he,
Scheme.IdealSheafData.map_bot, Scheme.IdealSheafData.ker_subschemeι] using h
let : Finite Q.subscheme := finite_subscheme_of_curve (I.subschemeι ≫ g) hd Q hQ
have hfin : (Q.support : Set I.subscheme).Finite := by
rw [← Q.range_subschemeι]
exact Set.finite_range Q.subschemeι
let u : (I ⊔ J).subscheme ⟶ I.subscheme :=
Scheme.IdealSheafData.inclusion le_sup_left
have hu : u ≫ I.subschemeι = (I ⊔ J).subschemeι :=
Scheme.IdealSheafData.inclusion_subschemeι le_sup_left
have hm (z : (I ⊔ J).subscheme) : u z ∈ Q.support := by
rw [Scheme.IdealSheafData.support_comap]
change I.subschemeι (u z) ∈ J.support
rw [← Scheme.Hom.comp_apply, hu]
have hh : (I ⊔ J).subschemeι z ∈ (I ⊔ J).support := by
change (I ⊔ J).subschemeι z ∈ ((I ⊔ J).support : Set X)
rw [← Scheme.IdealSheafData.range_subschemeι]
exact ⟨z,rfl⟩
rw [Scheme.IdealSheafData.support_sup] at hh
exact hh.2
let : Finite (Q.support : Set I.subscheme) := hfin.to_subtype
apply Finite.of_injective (fun z => (⟨u z,hm z⟩ : (Q.support : Set I.subscheme)))
intro a b hab
apply (I ⊔ J).subschemeι.isEmbedding.injective
have h := congrArg (fun z : (Q.support : Set I.subscheme) => I.subschemeι z.1) hab
simpa only [← Scheme.Hom.comp_apply, hu] using h
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
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