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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.NumberTheory.PiExponent.Geometry.CurveComponentZeroPoints
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry
open PiExponent.SectionZeroIdeal
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)
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))
end
end PiExponent.CurveCycle
end OAI
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