| import OAI.NumberTheory.PiExponent.Ampleness.AmpleCurveDegreePositive |
| import OAI.NumberTheory.PiExponent.Cohomology.CurveCycleEuler |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentsDimension |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry |
|
|
| theorem componentEulerDegree_eq_curveDegree {X : Scheme.{0}} |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) (L : LineBundle X) |
| (C : irreducibleComponents X) (hd : topologicalKrullDim (reducedComponent X C) = 1) : |
| componentEulerDegree p L C = NumericalAmpleness.curveDegree p L (componentIntegralCurve C hd) := rfl |
|
|
| theorem componentEulerDegree_nonneg_of_ample {X : Scheme.{0}} [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) (L : LineBundle X) (hL : L.IsAmple) |
| (C : irreducibleComponents X) : 0 ≤ componentEulerDegree p L C := by |
| rcases component_dimension_one_or_zero hd C with hC | hC |
| · rw [componentEulerDegree_eq_curveDegree p L C hC] |
| exact le_of_lt (NumericalAmpleness.curveDegree_pos_of_ample p L hL _) |
| · rw [componentEulerDegree_zero_of_dimension_le_zero p L C hC] |
|
|
| theorem cycleEulerDegree_pos_of_ample {X : Scheme.{0}} [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X = 1) (L : LineBundle X) (hL : L.IsAmple) : |
| 0 < cycleEulerDegree p L := by |
| classical |
| let : Fintype (irreducibleComponents X) := |
| NoetherianSpace.finite_irreducibleComponents.fintype |
| unfold cycleEulerDegree |
| apply Finset.sum_pos' |
| · intro C _ |
| exact mul_nonneg (Int.natCast_nonneg _) (componentEulerDegree_nonneg_of_ample p hd.le L hL C) |
| · obtain ⟨C,hC⟩ := exists_component_dimension_one X hd |
| refine ⟨C,Finset.mem_univ _,mul_pos ?_ ?_⟩ |
| · exact_mod_cast componentMultiplicity_pos C |
| · rw [componentEulerDegree_eq_curveDegree p L C hC] |
| exact NumericalAmpleness.curveDegree_pos_of_ample p L hL _ |
|
|
| theorem cycleEulerDegree_margin {X : Scheme.{0}} [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) (hd : topologicalKrullDim X ≤ 1) |
| (L H : LineBundle X) (ε : ℝ) |
| (hmargin : ∀ C : NumericalAmpleness.IntegralCurve X, |
| ε * (NumericalAmpleness.curveDegree p H C : ℝ) ≤ |
| (NumericalAmpleness.curveDegree p L C : ℝ)) : |
| ε * (cycleEulerDegree p H : ℝ) ≤ (cycleEulerDegree p L : ℝ) := by |
| classical |
| have hc (C : irreducibleComponents X) : |
| ε * (componentEulerDegree p H C : ℝ) ≤ (componentEulerDegree p L C : ℝ) := by |
| rcases component_dimension_one_or_zero hd C with hC | hC |
| · exact hmargin (componentIntegralCurve C hC) |
| · rw [componentEulerDegree_zero_of_dimension_le_zero p H C hC, |
| componentEulerDegree_zero_of_dimension_le_zero p L C hC] |
| simp |
| unfold cycleEulerDegree |
| simp only [Int.cast_sum,Int.cast_mul,Int.cast_natCast] |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro C _ |
| calc |
| ε * ((componentMultiplicity X C : ℝ) * (componentEulerDegree p H C : ℝ)) = |
| (componentMultiplicity X C : ℝ) * (ε * (componentEulerDegree p H C : ℝ)) := by ring |
| _ ≤ (componentMultiplicity X C : ℝ) * (componentEulerDegree p L C : ℝ) := |
| mul_le_mul_of_nonneg_left (hc C) (Nat.cast_nonneg _) |
|
|
| theorem tensor_euler_margin {X : Scheme.{0}} [IsNoetherian X] [Nonempty X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X ≤ 1) (A : LineBundle X) (hA : A.IsAmple) |
| (L H M N : LineBundle X) (ε : ℝ) |
| (hmargin : ∀ C : NumericalAmpleness.IntegralCurve X, |
| ε * (NumericalAmpleness.curveDegree p H C : ℝ) ≤ |
| (NumericalAmpleness.curveDegree p L C : ℝ)) : |
| ε * ((eulerCharacteristic p 1 (H.tensor M).sheaf - eulerCharacteristic p 1 M.sheaf : ℤ) : ℝ) ≤ |
| ((eulerCharacteristic p 1 (L.tensor N).sheaf - eulerCharacteristic p 1 N.sheaf : ℤ) : ℝ) := by |
| rw [tensor_euler_eq_cycle p hd A hA H M,tensor_euler_eq_cycle p hd A hA L N] |
| exact cycleEulerDegree_margin p hd L H ε hmargin |
|
|
| theorem euler_degree_pos_of_ample_curve {X : Scheme.{0}} [IsNoetherian X] [Nonempty X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X = 1) (L : LineBundle X) (hL : L.IsAmple) : |
| 0 < eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X) := by |
| have h := tensor_euler_eq_cycle p hd.le L hL L (L.pow 0) |
| have he := eulerCharacteristic_iso p (moduleTensorRightUnit L.sheaf) 1 |
| change eulerCharacteristic p 1 (L.tensor (L.pow 0)).sheaf = |
| eulerCharacteristic p 1 L.sheaf at he |
| rw [he] at h |
| change eulerCharacteristic p 1 L.sheaf - |
| eulerCharacteristic p 1 (structureSheaf X) = cycleEulerDegree p L at h |
| rw [h] |
| exact cycleEulerDegree_pos_of_ample p hd L hL |
|
|
| end |
| end PiExponent.CurveCycle |
|
|
| end OAI |
|
|
