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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 76 77 78 79 80 81 82 83 84 | 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
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