| import OAI.NumberTheory.PiExponent.Approximation.RegularComponentSection |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentLocalLengths |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentZeroPoints |
| import OAI.NumberTheory.PiExponent.Geometry.CurveCutQuotient |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames |
| open PiExponent.SectionZeroIdeal PiExponent.SectionZeroStalk |
|
|
| theorem componentLocalCutLength_eq_zero_stalk {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) |
| (U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme) |
| (hy : (zeroIdeal L s).subschemeι (componentZeroToGlobal L s C z) ∈ U.1) : |
| componentLocalCutLength X ((zeroIdeal L s).subschemeι (componentZeroToGlobal L s C z)) |
| ⟨C,componentZeroToGlobal_mem L s C z⟩ |
| (sectionGerm L s U e _ hy) = |
| Module.length ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk z) |
| ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk z) := by |
| let i := reducedComponentι X C |
| let J := zeroIdeal (L.pullback i) (pullbackSection i s) |
| let y := J.subschemeι z |
| let a := sectionGerm L s U e (i y) hy |
| obtain ⟨eF,heF⟩ := exists_pullback_frame_sectionGerm i L s U e |
| let V : (reducedComponent X C).affineOpens := ⟨i ⁻¹ᵁ U.1,U.2.preimage i⟩ |
| let b := sectionGerm (L.pullback i) (pullbackSection i s) V eF y hy |
| have hb : b = i.stalkMap y a := heF y hy |
| have hcut := reducedComponent_cut_length X C y a |
| change componentLocalCutLength X (i y) ⟨C,reducedComponentι_mem X C y⟩ a = |
| Module.length (J.subscheme.presheaf.stalk z) (J.subscheme.presheaf.stalk z) |
| change Module.length (X.presheaf.stalk (i y)) |
| ((X.presheaf.stalk (i y)) ⧸ ((reducedComponentStalkPrime X C y).val ⊔ Ideal.span {a})) = _ |
| rw [hcut,← hb] |
| have hsc : Module.length ((reducedComponent X C).presheaf.stalk y) |
| (((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b}) = |
| Module.length (((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b}) |
| (((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b}) := |
| Module.length_eq_of_surjective (M := ((reducedComponent X C).presheaf.stalk y) ⧸ |
| Ideal.span {b}) Ideal.Quotient.mk_surjective |
| exact hsc.trans (intrinsic_length_eq_of_ringEquiv |
| (zeroStalkQuotientEquiv (L.pullback i) (pullbackSection i s) V eF z hy)) |
|
|
| theorem componentLocalCutLength_eq_zero_stalk_at {X : Scheme.{0}} |
| (L : LineBundle X) (s : GlobalSections X L.sheaf) (C : irreducibleComponents X) |
| (t : {y : (zeroIdeal L s).subscheme |
| (U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme) |
| (hy : (zeroIdeal L s).subschemeι t.val ∈ U.1) : |
| componentLocalCutLength X ((zeroIdeal L s).subschemeι t.val) ⟨C,t.property⟩ |
| (sectionGerm L s U e _ hy) = |
| Module.length ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk |
| ((componentZeroPointEquiv L s C).symm t)) |
| ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk |
| ((componentZeroPointEquiv L s C).symm t)) := by |
| obtain ⟨z,rfl⟩ := (componentZeroPointEquiv L s C).surjective t |
| exact (componentLocalCutLength_eq_zero_stalk L s C z U e hy).trans |
| (congrArg (fun w : (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme => |
| Module.length ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk w) |
| ((zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk w)) |
| ((componentZeroPointEquiv L s C).symm_apply_apply z)).symm |
|
|
| end |
| end PiExponent.CurveCycle |
|
|
| end OAI |
|
|
