| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.CurveFinite |
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| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set |
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| lemma curve_nonclosed_generic {X : Type*} [TopologicalSpace X] [T0Space X] |
| [IrreducibleSpace X] (hd : topologicalKrullDim X ≤ 1) (x : X) |
| (hx : ¬ IsClosed ({x} : Set X)) : IsGenericPoint x Set.univ := by |
| by_contra h |
| have hp : closure ({x} : Set X) ≠ Set.univ := h |
| have hs := proper_irreducible_closed_subsingleton hd isClosed_closure |
| isIrreducible_singleton.closure hp |
| have he : closure ({x} : Set X) = {x} := by |
| apply Set.Subset.antisymm |
| · intro y hy |
| exact hs hy (subset_closure (Set.mem_singleton x)) |
| · exact subset_closure |
| exact hx (he ▸ isClosed_closure) |
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| theorem proper_curve_nonconstant_isFinite {X Y : Scheme} |
| [IsIntegral X] [IsNoetherian X] (f : X ⟶ Y) [IsProper f] |
| (hd : topologicalKrullDim X ≤ 1) (hn : ∃ a b : X, f a ≠ f b) : IsFinite f := by |
| let : LocallyQuasiFinite f := .of_finite_preimage_singleton f (by |
| intro y |
| by_cases hy : IsClosed ({y} : Set Y) |
| · apply proper_closed_finite_of_dimension_one hd (hy.preimage f.continuous) |
| intro he |
| obtain ⟨a,b,hab⟩ := hn |
| have ha : a ∈ f ⁻¹' ({y} : Set Y) := he ▸ Set.mem_univ a |
| have hb : b ∈ f ⁻¹' ({y} : Set Y) := he ▸ Set.mem_univ b |
| exact hab (ha.trans hb.symm) |
| · apply Set.Subsingleton.finite |
| intro a ha b hb |
| apply (curve_nonclosed_generic hd a ?_).eq (curve_nonclosed_generic hd b ?_) |
| · intro hc |
| have hi := f.isClosedMap _ hc |
| rw [Set.image_singleton, ha] at hi |
| exact hy hi |
| · intro hc |
| have hi := f.isClosedMap _ hc |
| rw [Set.image_singleton, hb] at hi |
| exact hy hi) |
| exact .of_isProper_of_locallyQuasiFinite f |
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| end |
| end MaximalSeshadri.Geometry |
| end |
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| end |
| end |
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| end OAI |
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