| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.FiniteEuler |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
| |
| instance Surface.isProper_structureMap (surface : Surface) : IsProper surface.structureMap := by |
| have constants_bijective : Function.Bijective |
| (constantsInDegreeZero surface.embeddingDimension) := by |
| constructor |
| · intro left right equality |
| have polynomial_equality : MvPolynomial.C left = |
| (MvPolynomial.C right : MvPolynomial (Fin (surface.embeddingDimension + 1)) ℂ) := |
| congrArg Subtype.val equality |
| exact MvPolynomial.C_injective _ _ polynomial_equality |
| · intro polynomial |
| have degree_zero : polynomial.val.totalDegree = 0 := |
| (MvPolynomial.totalDegree_zero_iff_isHomogeneous _).mpr polynomial.property |
| refine ⟨polynomial.val.coeff 0, ?_⟩ |
| apply Subtype.ext |
| exact (MvPolynomial.totalDegree_eq_zero_iff_eq_C.mp degree_zero).symm |
| let constantsIso := (RingEquiv.ofBijective |
| (constantsInDegreeZero surface.embeddingDimension) constants_bijective).toCommRingCatIso |
| let : IsIso (CommRingCat.ofHom (constantsInDegreeZero surface.embeddingDimension)) := |
| constantsIso.isIso_hom |
| let polynomialRing := MvPolynomial (Fin (surface.embeddingDimension + 1)) ℂ |
| let grading := MvPolynomial.homogeneousSubmodule (Fin (surface.embeddingDimension + 1)) ℂ |
| let : Algebra ℂ (grading 0) := SetLike.GradeZero.instAlgebra grading |
| let : IsScalarTower ℂ (grading 0) polynomialRing := ⟨fun scalar homogeneous polynomial => by |
| change (scalar • (homogeneous : polynomialRing)) * polynomial = |
| scalar • ((homogeneous : polynomialRing) * polynomial) |
| exact smul_mul_assoc scalar (homogeneous : polynomialRing) polynomial⟩ |
| let : Algebra.FiniteType (grading 0) polynomialRing := |
| Algebra.FiniteType.of_restrictScalars_finiteType ℂ (grading 0) polynomialRing |
| have projective_proper : IsProper (Proj.toSpecZero grading) := |
| Proj.instIsProperToSpecZeroOfFiniteTypeSubtypeMemOfNatNat grading |
| let : IsIso (Spec.map |
| (CommRingCat.ofHom (constantsInDegreeZero surface.embeddingDimension))) := by |
| infer_instance |
| have : IsProper (projectiveSpaceToSpec surface.embeddingDimension) := by |
| unfold projectiveSpaceToSpec |
| exact IsProper.stableUnderComposition.comp_mem _ _ projective_proper inferInstance |
| rw [← surface.overComplex] |
| infer_instance |
| |
| variable {X : Scheme.{0}} [IsIntegral X] |
| |
| theorem proper_baseScalars_surjective (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| [IsProper p] : Function.Surjective (baseScalars p) := by |
| have hi : (baseScalars p).IsIntegral := by |
| apply RingHom.isIntegral_respectsIso.2 |
| (e := (Scheme.ΓSpecIso (CommRingCat.of ℂ)).symm.commRingCatIsoToRingEquiv) |
| exact isIntegral_appTop_of_universallyClosed p |
| exact (IsAlgClosed.ringHom_bijective_of_isIntegral (baseScalars p) hi).2 |
| |
| theorem proper_cohomologyDimension_zero_structure |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] : |
| cohomologyDimension p (structureSheaf X) 0 = 1 := by |
| rw [cohomologyDimension_zero] |
| let := Module.compHom Γ(structureSheaf X,⊤) (baseScalars p) |
| let f : ℂ →ₗ[ℂ] Γ(structureSheaf X,⊤) := |
| { toFun := baseScalars p |
| map_add' := (baseScalars p).map_add |
| map_smul' := by intro a b; exact (baseScalars p).map_mul a b } |
| have hsur : Function.Surjective f := proper_baseScalars_surjective p |
| have hinj : Function.Injective f := (baseScalars p).injective |
| let e := LinearEquiv.ofBijective f ⟨hinj,hsur⟩ |
| exact e.finrank_eq.symm.trans (Module.finrank_self ℂ) |
| |
| theorem proper_structure_cohomology_zero_finite |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] : |
| letI := Module.compHom (cohomology (structureSheaf X) 0) (baseScalars p) |
| Module.Finite ℂ (cohomology (structureSheaf X) 0) := by |
| let := Module.compHom (cohomology (structureSheaf X) 0) (baseScalars p) |
| let := Module.compHom Γ(structureSheaf X,⊤) (baseScalars p) |
| let f : ℂ →ₗ[ℂ] Γ(structureSheaf X,⊤) := |
| { toFun := baseScalars p |
| map_add' := (baseScalars p).map_add |
| map_smul' := by intro a b; exact (baseScalars p).map_mul a b } |
| let : Module.Finite ℂ Γ(structureSheaf X,⊤) := |
| Module.Finite.of_surjective f (proper_baseScalars_surjective p) |
| exact Module.Finite.of_surjective |
| (cohomologyZeroSections p (structureSheaf X)).symm.toLinearMap |
| (cohomologyZeroSections p (structureSheaf X)).symm.surjective |
| |
| end |
| end MaximalSeshadri.Geometry |
| |
| end |
| |
| |
| end |
| end |
| |
| end OAI |
| |
