File size: 3,128 Bytes
a72760f | 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 | import OAI.NumberTheory.PiExponent.Approximation.LinePullback
import OAI.NumberTheory.PiExponent.Approximation.NumericalMarginRestriction
import OAI.NumberTheory.PiExponent.Approximation.SectionPullback
namespace OAI
namespace PiExponent.ConstantPullbackDegree
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
variable {X Y : Scheme.{0}}
def pullbackFrame_of_range_subset (K : LineBundle Y) (g : X ⟶ Y)
(U : Y.Opens) (h : Set.range g ⊆ (U : Set Y))
(e : K.sheaf.restrict U.ι ≅ structureSheaf U.toScheme) :
(K.pullback g).sheaf ≅ structureSheaf X := by
have hr : Set.range g ⊆ Set.range U.ι := by
rw [Scheme.Opens.range_ι]
exact h
let l := IsOpenImmersion.lift U.ι g hr
have hl : l ≫ U.ι = g := IsOpenImmersion.lift_fac U.ι g hr
exact (Scheme.Modules.pullbackCongr hl.symm).app K.sheaf ≪≫
((Scheme.Modules.pullbackComp l U.ι).app K.sheaf).symm ≪≫
(Scheme.Modules.pullback l).mapIso ((Scheme.Modules.restrictFunctorIsoPullback U.ι).app
K.sheaf).symm ≪≫ (Scheme.Modules.pullback l).mapIso e ≪≫ pullbackUnitIso l
def constantPullbackIso (K : LineBundle Y) (g : X ⟶ Y) (y : Y)
(hg : Set.range g ⊆ ({y} : Set Y)) :
(K.pullback g).sheaf ≅ structureSheaf X := by
apply Classical.choice
obtain ⟨U, hy, ⟨e⟩⟩ := K.locallyRankOne y
refine ⟨pullbackFrame_of_range_subset K g U ?_ e⟩
intro z hz
have hz' : z = y := hg hz
exact hz'.symm ▸ hy
theorem euler_difference_eq_zero (p : X ⟶ Spec (CommRingCat.of ℂ))
(K : LineBundle Y) (g : X ⟶ Y) (y : Y)
(hg : Set.range g ⊆ ({y} : Set Y)) (d : ℕ) :
eulerCharacteristic p d (K.pullback g).sheaf -
eulerCharacteristic p d (structureSheaf X) = 0 := by
rw [eulerCharacteristic_iso p (constantPullbackIso K g y hg) d]
exact sub_self _
theorem euler_difference_eq_zero_of_constant (p : X ⟶ Spec (CommRingCat.of ℂ))
(K : LineBundle Y) (g : X ⟶ Y) (y : Y)
(hg : ∀ x, g x = y) (d : ℕ) :
eulerCharacteristic p d (K.pullback g).sheaf -
eulerCharacteristic p d (structureSheaf X) = 0 :=
euler_difference_eq_zero p K g y (by rintro z ⟨x,rfl⟩; exact hg x) d
theorem curveDegree_eq_zero_of_contracted
(p : X ⟶ Spec (CommRingCat.of ℂ)) (K : LineBundle Y)
(g : X ⟶ Y) (C : NumericalAmpleness.IntegralCurve X) (y : Y)
(hg : Set.range (C.embedding ≫ g) ⊆ ({y} : Set Y)) :
NumericalAmpleness.curveDegree p (K.pullback g) C = 0 := by
have he := eulerCharacteristic_iso (C.embedding ≫ p)
((Scheme.Modules.pullbackComp C.embedding g).app K.sheaf) 1
simp only [CategoryTheory.Functor.comp_obj] at he
unfold NumericalAmpleness.curveDegree
change eulerCharacteristic (C.embedding ≫ p) 1
((Scheme.Modules.pullback C.embedding).obj ((Scheme.Modules.pullback g).obj K.sheaf)) -
eulerCharacteristic (C.embedding ≫ p) 1 (structureSheaf C.scheme) = 0
rw [he]
exact euler_difference_eq_zero (C.embedding ≫ p) K (C.embedding ≫ g) y hg 1
end
end PiExponent.ConstantPullbackDegree
end OAI
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