| 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 | |