File size: 3,462 Bytes
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 | import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupGeometry
import OAI.NumberTheory.PiExponent.Ampleness.CurveBlowupImage
namespace OAI
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
namespace PiExponent.AdmissibleCurveImage
open AdmissibleBlowupGeometry CurveBlowupImage
variable {ν Λ D : ℝ} (d : AdmissibleParameters ν Λ D)
theorem center_point_isClosed (x : compactification d)
(hx : x ∈ (centerIdeal d).support) : IsClosed ({x} : Set (compactification d)) := by
change x ∈ ((centerIdeal d).support : Set (compactification d)) at hx
rw [centerIdeal_support d] at hx
obtain ⟨j, z, rfl⟩ := Set.mem_iUnion.mp hx
let q := CompactLogJetIdeal.point (d.curveCenters j) ≫ affineChart d
let : IsClosedImmersion q := CompactJetIdeal.section_isClosedImmersion
(compactificationStructureMap d) q (by
dsimp [q]
rw [Category.assoc, affineChart_over, CompactLogJetIdeal.point_section])
have he : Set.range q = {q z} := by
ext x
constructor
· rintro ⟨z',rfl⟩
exact congrArg q (Subsingleton.elim z' z)
· rintro rfl
exact ⟨z,rfl⟩
rw [← he]
exact q.isClosedEmbedding.isClosed_range
theorem meets_center_complement (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) :
∃ c : C.scheme, (C.embedding ≫ projection d) c ∈ (centerIdeal d).support.compl :=
exists_image_outside_closed_points (C.embedding ≫ projection d)
(centerIdeal d).support (center_point_isClosed d) hn
instance projection_restrict_isIso :
IsIso (projection d ∣_ (centerIdeal d).support.compl) :=
blowup_restrict_isIso (isBlowup d)
abbrev imageCurve (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) :=
CurveBlowupImage.imageCurve (projection d) C hn
def functionFieldIso (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) :
(imageCurve d C hn).scheme.functionField ≅ C.scheme.functionField :=
imageFunctionFieldIso (projection d) C hn (centerIdeal d).support.compl
(meets_center_complement d C hn)
theorem functionFieldIso_generic (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x}) :
Spec.map (functionFieldIso d C hn).hom ≫
(imageCurve d C hn).scheme.fromSpecStalk (genericPoint (imageCurve d C hn).scheme) ≫
(imageCurve d C hn).embedding =
C.scheme.fromSpecStalk (genericPoint C.scheme) ≫ C.embedding ≫ projection d :=
imageFunctionFieldIso_generic (projection d) C hn (centerIdeal d).support.compl
(meets_center_complement d C hn)
theorem image_meets_chart (C : NumericalAmpleness.IntegralCurve (blowup d))
(hn : ¬ ∃ x : compactification d, Set.range (C.embedding ≫ projection d) ⊆ {x})
(hm : ∃ c : C.scheme, (C.embedding ≫ projection d) c ∈ (affineChart d).opensRange) :
∃ c : (imageCurve d C hn).scheme,
(imageCurve d C hn).embedding c ∈ (affineChart d).opensRange := by
obtain ⟨c,hc⟩ := hm
refine ⟨CurveBlowupImage.imageMap (projection d) C hn c, ?_⟩
simpa only [← Scheme.Hom.comp_apply, CurveBlowupImage.imageMap_comp] using hc
end PiExponent.AdmissibleCurveImage
end
end OAI
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