| import OAI.NumberTheory.PiExponent.Ampleness.NakaiSectionLifting |
| import OAI.NumberTheory.PiExponent.Approximation.ClosedThickeningSections |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace PiExponentSeshadri.Geometry |
|
|
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian |
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| variable {X Y : Scheme.{0}} |
| local instance (Z : Scheme.{0}) : HasExt.{1} Z.Modules := HasExt.standard _ |
|
|
| theorem closed_line_unit_epi (f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X) : |
| Epi ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf) := |
| PiExponentSeshadri.LineClosedUnit.epi f M |
|
|
| theorem pullbackSection_surjective_of_kernel_cohomology_zero |
| (f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X) |
| (hzero : ∀ z : cohomology |
| (kernel ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf)) 1, |
| z = 0) : |
| Function.Surjective (fun s : GlobalSections X M.sheaf => pullbackSection f s) := by |
| let u := (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf |
| have : Epi u := closed_line_unit_epi f M |
| let S := ShortComplex.mk (kernel.ι u) u (kernel.condition u) |
| have hS : S.ShortExact := { exact := ShortComplex.exact_kernel u } |
| exact PiExponent.NumericalAmpleness.pullbackSection_surjective_of_unit_surjective f M |
| (globalSections_surjective_of_ext_one_zero S hS hzero) |
|
|
| end PiExponentSeshadri.Geometry |
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|
| end |
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| end OAI |
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|
