| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentZeroPoints |
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| namespace OAI |
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| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry |
| open PiExponent.SectionZeroIdeal |
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| def sectionZeroIncidenceEquiv {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) : |
| (Σ y : (zeroIdeal L s).subscheme, |
| {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val}) ≃ |
| (Σ C : irreducibleComponents X, |
| (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme) where |
| toFun p := ⟨p.2.val, (componentZeroPointEquiv L s p.2.val).symm ⟨p.1,p.2.property⟩⟩ |
| invFun q := ⟨(componentZeroPointEquiv L s q.1 q.2).val, |
| ⟨q.1,(componentZeroPointEquiv L s q.1 q.2).property⟩⟩ |
| left_inv p := by |
| rcases p with ⟨y,C,hC⟩ |
| dsimp only |
| have h := (componentZeroPointEquiv L s C).apply_symm_apply ⟨y,hC⟩ |
| exact congrArg (fun t : {y : (zeroIdeal L s).subscheme // |
| (zeroIdeal L s).subschemeι y ∈ C.val} => |
| (⟨t.val,⟨C,t.property⟩⟩ : Σ y : (zeroIdeal L s).subscheme, |
| {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val})) h |
| right_inv q := by |
| rcases q with ⟨C,z⟩ |
| dsimp only |
| exact congrArg (Sigma.mk C) ((componentZeroPointEquiv L s C).symm_apply_apply z) |
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| theorem sum_component_zero_incidence {X : Scheme.{0}} (L : LineBundle X) |
| (s : GlobalSections X L.sheaf) [Fintype (irreducibleComponents X)] |
| [Fintype (zeroIdeal L s).subscheme] |
| {R : Type*} [AddCommMonoid R] |
| (w : ∀ C : irreducibleComponents X, |
| (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme → R) : |
| letI : ∀ y : (zeroIdeal L s).subscheme, |
| Fintype {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} := |
| fun _ => Fintype.ofFinite _ |
| letI : ∀ C : irreducibleComponents X, |
| Fintype (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme := |
| fun _ => Fintype.ofFinite _ |
| (∑ y : (zeroIdeal L s).subscheme, |
| ∑ C : {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val}, |
| w C.val ((componentZeroPointEquiv L s C.val).symm ⟨y,C.property⟩)) = |
| ∑ C : irreducibleComponents X, ∑ z, w C z := by |
| classical |
| let : ∀ y : (zeroIdeal L s).subscheme, |
| Fintype {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} := |
| fun _ => Fintype.ofFinite _ |
| let : ∀ C : irreducibleComponents X, |
| Fintype (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme := |
| fun _ => Fintype.ofFinite _ |
| let a : (Σ y : (zeroIdeal L s).subscheme, |
| {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val}) → R := |
| fun p => w p.2.val ((componentZeroPointEquiv L s p.2.val).symm ⟨p.1,p.2.property⟩) |
| let b : (Σ C : irreducibleComponents X, |
| (zeroIdeal (L.pullback (reducedComponentι X C)) |
| (pullbackSection (reducedComponentι X C) s)).subscheme) → R := |
| fun q => w q.1 q.2 |
| exact (Fintype.sum_sigma a).symm.trans |
| ((Fintype.sum_equiv (sectionZeroIncidenceEquiv L s) a b (fun _ => rfl)).trans |
| (Fintype.sum_sigma b)) |
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| end |
| end PiExponent.CurveCycle |
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| end OAI |
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