| 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}} |
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| 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 |
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| end |
| end |
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| end OAI |
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