| import OAI.NumberTheory.PiExponent.Ampleness.ClosedAmpleRestriction |
| import OAI.NumberTheory.PiExponent.Approximation.SectionZeroSupport |
| import OAI.NumberTheory.PiExponent.Geometry.ReducedComponentStalk |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.SectionOpens |
| open PiExponent.SectionZeroIdeal |
|
|
| def componentZeroToGlobal {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) (C : irreducibleComponents X) |
| (z : (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme) : |
| (zeroIdeal L s).subscheme := by |
| let i := reducedComponentι X C |
| let J := zeroIdeal (L.pullback i) (pullbackSection i s) |
| refine ⟨i (J.subschemeι z), ?_⟩ |
| have hz := z.property |
| change J.subschemeι z ∈ (J.support : Set (reducedComponent X C)) at hz |
| change i (J.subschemeι z) ∈ ((zeroIdeal L s).support : Set X) |
| erw [zeroIdeal_support (L.pullback i) (pullbackSection i s)] at hz |
| erw [zeroIdeal_support L s] |
| change J.subschemeι z ∉ isoOpen (pullbackSection i s) at hz |
| rw [pullback_isoOpen_eq L s i] at hz |
| exact hz |
|
|
| theorem componentZeroToGlobal_mem {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) (C : irreducibleComponents X) |
| (z : (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme) : |
| (zeroIdeal L s).subschemeι (componentZeroToGlobal L s C z) ∈ C.val := |
| reducedComponentι_mem X C _ |
|
|
| def componentZeroPointEquiv {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) (C : irreducibleComponents X) : |
| (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme ≃ |
| {y : (zeroIdeal L s).subscheme // (zeroIdeal L s).subschemeι y ∈ C.val} := by |
| let i := reducedComponentι X C |
| let I := zeroIdeal L s |
| let J := zeroIdeal (L.pullback i) (pullbackSection i s) |
| let g : J.subscheme → {y : I.subscheme // I.subschemeι y ∈ C.val} := |
| fun z => ⟨componentZeroToGlobal L s C z,componentZeroToGlobal_mem L s C z⟩ |
| apply Equiv.ofBijective g |
| constructor |
| · intro z w h |
| apply J.subschemeι.isEmbedding.injective |
| apply i.isEmbedding.injective |
| exact congrArg (fun q => I.subschemeι q.val) h |
| · intro y |
| have hymem : I.subschemeι y.val ∈ Set.range i := by |
| rw [reducedComponentι_range] |
| exact y.property |
| obtain ⟨z,hz⟩ := hymem |
| have hzJ : z ∈ (J.support : Set (reducedComponent X C)) := by |
| erw [zeroIdeal_support (L.pullback i) (pullbackSection i s)] |
| change z ∉ isoOpen (pullbackSection i s) |
| rw [pullback_isoOpen_eq L s i] |
| change i z ∉ isoOpen s |
| rw [hz] |
| have hy := y.val.property |
| change I.subschemeι y.val ∈ (I.support : Set X) at hy |
| erw [zeroIdeal_support L s] at hy |
| exact hy |
| refine ⟨⟨z,hzJ⟩, ?_⟩ |
| apply Subtype.ext |
| apply I.subschemeι.isEmbedding.injective |
| exact hz |
|
|
| instance finite_componentZero {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) (C : irreducibleComponents X) |
| [Finite (zeroIdeal L s).subscheme] : |
| Finite (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme := |
| Finite.of_equiv _ (componentZeroPointEquiv L s C).symm |
|
|
| end |
| end PiExponent.CurveCycle |
|
|
| end OAI |
|
|
