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12dee50 | 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 85 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Intersection.TwoChartIntersection
import OAI.AlgebraicGeometry.Seshadri.Geometry.PencilCharts
import OAI.AlgebraicGeometry.Seshadri.Geometry.PencilOverlap
import OAI.AlgebraicGeometry.Seshadri.Geometry.ChartCoherence
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry.BaseSections
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite
open MaximalSeshadri.Projective MaximalSeshadri.Frames
variable {K : Type} [Field K] {X : Scheme.{0}}
def twoBaseIntersection (k : K →+* Γ(X,⊤)) (M : X.Modules) (U V : X.Opens) :
Submodule K (Sections k M (U ⊓ V)) :=
(res k M (show U ⊓ V ≤ U from inf_le_left)).range ⊓
(res k M (show U ⊓ V ≤ V from inf_le_right)).range
theorem finite_pencil_intersection (k : K →+* Γ(X,⊤)) {M : X.Modules}
(s : Bool → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
[IsFinite (sectionsMorphism k s hs)] (L : LineBundle X) :
let U := SectionOpens.isoOpen (s false)
let V := SectionOpens.isoOpen (s true)
Module.Finite K ↥(twoBaseIntersection k L.sheaf U V) := by
intro U V
let W := U ⊓ V
let a := X.homOfLE (show W ≤ U from inf_le_left)
let b := X.homOfLE (show W ≤ V from inf_le_right)
let kU := U.ι.appTop.hom.comp k
let kV := V.ι.appTop.hom.comp k
let kW := W.ι.appTop.hom.comp k
let u := coefficient (sectionFrame (s false)) (restrictSection U.ι (s true))
let v := coefficient (sectionFrame (s true)) (restrictSection V.ι (s false))
obtain ⟨hU,hu⟩ := finite_pencil_chart k s hs false
obtain ⟨hV,hv⟩ := finite_pencil_chart k s hs true
let : IsAffine U.toScheme := hU
let : IsAffine V.toScheme := hV
have ha : a.opensRange = U.toScheme.basicOpen u := by
rw [Scheme.opensRange_homOfLE, Scheme.Hom.preimage_inf]
simp only [Scheme.Opens.ι_preimage_self, top_inf_eq]
exact pencil_overlap_basic s false
have hb : b.opensRange = V.toScheme.basicOpen v := by
rw [Scheme.opensRange_homOfLE, Scheme.Hom.preimage_inf]
simp only [Scheme.Opens.ι_preimage_self, inf_top_eq]
exact pencil_overlap_basic s true
let e := nestedRestriction L.sheaf (show W ≤ U from inf_le_left)
let d := nestedRestriction L.sheaf (show W ≤ V from inf_le_right)
let F := chartMap kU kW a (nestedBase k (show W ≤ U from inf_le_left))
(L.restrict U.ι).sheaf e
let G := chartMap kV kW b (nestedBase k (show W ≤ V from inf_le_right))
(L.restrict V.ι).sheaf d
let : Module.Finite K ↥(F.range ⊓ G.range) :=
two_chart_intersection_finite kU kV kW a b (nestedBase k inf_le_left)
(nestedBase k inf_le_right) u v ha hb (pencil_overlap_reciprocal s) hu hv
(L.restrict U.ι) (L.restrict V.ι) (L.sheaf.restrict W.ι) e d
let E := chartTop k L.sheaf W
have he : (F.range ⊓ G.range).map E.toLinearMap = twoBaseIntersection k L.sheaf U V := by
rw [Submodule.map_inf E.toLinearMap E.injective]
change F.range.map E.toLinearMap ⊓ G.range.map E.toLinearMap =
(res k L.sheaf (show W ≤ U from inf_le_left)).range ⊓
(res k L.sheaf (show W ≤ V from inf_le_right)).range
exact congrArg₂ (· ⊓ ·)
(chartMap_range_coherent k L.sheaf (show W ≤ U from inf_le_left))
(chartMap_range_coherent k L.sheaf (show W ≤ V from inf_le_right))
let : Module.Finite K ↥((F.range ⊓ G.range).map E.toLinearMap) :=
Module.Finite.equiv (E.submoduleMap (F.range ⊓ G.range))
rw [← he]
infer_instance
end
end MaximalSeshadri.Geometry.BaseSections
end
end
end
end OAI
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