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b797a5b | 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 | import OAI.NumberTheory.PiExponent.Ampleness.CurveBlowupImageBase
namespace OAI
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
namespace PiExponent.CurveBlowupImage
variable {C X : Scheme.{0}}
theorem nonconstant_of_closedImmersion_on_open (f : C ⟶ X)
(hdim : topologicalKrullDim C = 1) (U : X.Opens)
[IsClosedImmersion (f ∣_ U)] (hU : Nonempty (f ⁻¹ᵁ U)) :
¬ ∃ x : X, Set.range f ⊆ {x} := by
rintro ⟨x, hx⟩
have he : ∀ c : C, f c = x := fun c => hx ⟨c, rfl⟩
obtain ⟨q⟩ := hU
have hxU : x ∈ U := (he q.1) ▸ q.2
let : Subsingleton C := ⟨fun a b => by
have ha : f a ∈ U := (he a).symm ▸ hxU
have hb : f b ∈ U := (he b).symm ▸ hxU
have hab : (f ∣_ U) (⟨a,ha⟩ : (f ⁻¹ᵁ U).toScheme) =
(f ∣_ U) (⟨b,hb⟩ : (f ⁻¹ᵁ U).toScheme) := by
apply Subtype.ext
exact (morphismRestrict_base_coe f U ⟨a, ha⟩).trans
((he a).trans ((he b).symm.trans (morphismRestrict_base_coe f U ⟨b, hb⟩).symm))
exact congrArg Subtype.val ((f ∣_ U).isEmbedding.injective hab)⟩
have hh := topologicalKrullDim_zero_of_discreteTopology C
rw [hdim] at hh
norm_num at hh
theorem exists_image_outside_closed_points [IsIntegral C] (f : C ⟶ X)
(Z : Set X) (hZ : ∀ x ∈ Z, IsClosed ({x} : Set X))
(hn : ¬ ∃ x : X, Set.range f ⊆ {x}) : ∃ c : C, f c ∉ Z := by
by_contra h
push Not at h
have hc := hZ (f (genericPoint C)) (h (genericPoint C))
apply hn
refine ⟨f (genericPoint C), ?_⟩
rintro _ ⟨c,rfl⟩
have hh := ((genericPoint_spec C).specializes (Set.mem_univ c)).map f.continuous
have hm := hh.mem_closure
rwa [hc.closure_eq] at hm
end PiExponent.CurveBlowupImage
end
end OAI
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