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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 51 52 53 54 55 56 57 58 59 60 61 | 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
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