| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.OpenChartDegree |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.ProjectiveBertini MaximalSeshadri.Frames |
|
|
| variable {X Y : Scheme.{0}} |
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|
| theorem pullback_section_zero_iff_local_coefficient |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (I : X.IdealSheafData) [IsIntegral I.subscheme] |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) |
| (U : X.affineOpens) [Nonempty (I.subschemeι ⁻¹ᵁ U.1).toScheme] |
| (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme) : |
| pullbackSection I.subschemeι s = 0 ↔ |
| U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) ∈ I.ideal U := by |
| let : Algebra ℂ Γ(X,U.1) := (openScalars p U.1).toAlgebra |
| let : Algebra ℂ Γ((I.subschemeι ⁻¹ᵁ U.1).toScheme,⊤) := |
| (baseScalars ((I.subschemeι ⁻¹ᵁ U.1).ι ≫ (I.subschemeι ≫ p))).toAlgebra |
| obtain ⟨eF,he⟩ := exists_restricted_pullback_frame I.subschemeι U.1 e s |
| let π := structuralOpenChartMap p I.subschemeι U.1 |
| let f := U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) |
| have hcoeff : π f = coefficient eF |
| (restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι (pullbackSection I.subschemeι s)) := by |
| rw [he] |
| change (I.subschemeι ∣_ U.1).appTop |
| (U.1.topIso.inv (U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)))) = _ |
| rw [U.1.topIso.hom_inv_id_apply] |
| have hker : f ∈ I.ideal U ↔ π f = 0 := by |
| rw [← structuralOpenChartMap_ker p I U] |
| rfl |
| rw [show U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) = f from rfl, |
| hker, hcoeff] |
| constructor |
| · intro h |
| rw [h,restrictSection_zero,coefficient_zero] |
| · intro h |
| apply (L.pullback I.subschemeι).restricted_coefficient_injective |
| (I.subschemeι ⁻¹ᵁ U.1).ι eF |
| change coefficient eF |
| (restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι (pullbackSection I.subschemeι s)) = |
| coefficient eF (restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι |
| (0 : O I.subscheme ⟶ (Scheme.Modules.pullback I.subschemeι).obj L.sheaf)) |
| rw [restrictSection_zero,coefficient_zero] |
| exact h |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
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| end |
| end |
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| end OAI |
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|
