| import OAI.NumberTheory.PiExponent.Ampleness.AmpleDivisorDimensionBase |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentMultiplicity |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| theorem reducedComponent_dimension (X : Scheme.{0}) (C : irreducibleComponents X) : |
| topologicalKrullDim (reducedComponent X C) = topologicalKrullDim (C.val : Set X) := by |
| change topologicalKrullDim |
| ((Scheme.IdealSheafData.vanishingIdeal |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).support : Set X) = _ |
| have he : ((Scheme.IdealSheafData.vanishingIdeal |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).support : Set X) = C.val := by |
| simp only [Scheme.IdealSheafData.coe_support_vanishingIdeal] |
| rfl |
| rw [he] |
|
|
| def componentIntegralCurve {X : Scheme.{0}} (C : irreducibleComponents X) |
| (hd : topologicalKrullDim (reducedComponent X C) = 1) : |
| NumericalAmpleness.IntegralCurve X where |
| scheme := reducedComponent X C |
| embedding := reducedComponentι X C |
| closedImmersion := inferInstance |
| integral := inferInstance |
| dimension := hd |
|
|
| theorem exists_component_dimension_one (X : Scheme.{0}) |
| (hd : topologicalKrullDim X = 1) : |
| ∃ C : irreducibleComponents X, topologicalKrullDim (reducedComponent X C) = 1 := by |
| obtain ⟨Z,hZ⟩ := NumericalAmpleness.exists_irreducibleClosed_full_dimension X 1 hd |
| have hZ' : topologicalKrullDim (Z : Set X) = 1 := by simpa using hZ |
| obtain ⟨S,hS,hZS⟩ := exists_mem_irreducibleComponents_subset_of_isIrreducible |
| (Z : Set X) Z.isIrreducible |
| let C : irreducibleComponents X := ⟨S,hS⟩ |
| refine ⟨C, ?_⟩ |
| rw [reducedComponent_dimension] |
| apply le_antisymm |
| · exact (topologicalKrullDim_subspace_le X C.val).trans hd.le |
| · rw [← hZ'] |
| exact (Topology.IsEmbedding.inclusion hZS).isInducing.topologicalKrullDim_le |
|
|
| theorem component_dimension_one_or_zero {X : Scheme.{0}} |
| (hd : topologicalKrullDim X ≤ 1) (C : irreducibleComponents X) : |
| topologicalKrullDim (reducedComponent X C) = 1 ∨ |
| topologicalKrullDim (reducedComponent X C) ≤ 0 := by |
| have hdim := (reducedComponentι X C).isEmbedding.isInducing.topologicalKrullDim_le.trans hd |
| by_cases h : topologicalKrullDim (reducedComponent X C) = 1 |
| · exact Or.inl h |
| · right |
| change topologicalKrullDim (reducedComponent X C) ≤ ((0 : ℕ) : WithBot ℕ∞) |
| rw [← ENat.WithBot.lt_add_one_iff] |
| simpa using lt_of_le_of_ne hdim h |
|
|
| end |
| end PiExponent.CurveCycle |
|
|
| end OAI |
|
|
