| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.CurvePencil |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Projective MaximalSeshadri.Frames |
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {K σ : Type} [Field K] [Infinite K] [Fintype σ] {X : Scheme} |
|
|
| omit [Infinite K] in |
| lemma proper_affine_dimension_zero [IsAffine X] |
| (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] : topologicalKrullDim X ≤ 0 := by |
| let : IsFinite p := IsFinite.iff_isProper_and_isAffineHom.mpr ⟨inferInstance,inferInstance⟩ |
| let y : Spec (CommRingCat.of K) := ⟨⊥, Ideal.isPrime_bot⟩ |
| have he : p ⁻¹' ({y} : Set (Spec (CommRingCat.of K))) = Set.univ := by |
| ext x |
| simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_univ, iff_true] |
| exact Subsingleton.elim _ _ |
| have h := p.isDiscrete_preimage_singleton y |
| rw [he] at h |
| let : DiscreteTopology X := isDiscrete_univ_iff.mp h |
| exact topologicalKrullDim_zero_of_discreteTopology X |
|
|
| theorem projective_curve_finite_pencil [IsIntegral X] [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] |
| (hd : topologicalKrullDim X = 1) {M : X.Modules} |
| (s : σ → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) |
| [IsClosedImmersion (sectionsMorphism |
| (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)] : |
| ∃ t : Bool → (O X ⟶ M), ∃ ht : (⨆ b, SectionOpens.isoOpen (t b)) = ⊤, |
| IsFinite (sectionsMorphism |
| (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) t ht) := by |
| let k := p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
| obtain ⟨x⟩ : Nonempty X := inferInstance |
| have hx : x ∈ ⨆ i, SectionOpens.isoOpen (s i) := hs ▸ trivial |
| obtain ⟨i,hi⟩ := Opens.mem_iSup.mp hx |
| have hne : (SectionOpens.isoOpen (s i) : Set X).Nonempty := ⟨x,hi⟩ |
| have hproper : SectionOpens.isoOpen (s i) ≠ ⊤ := by |
| intro he |
| have hA := (Proj.isAffineOpen_basicOpen (PolyGrade K σ) _ (poly_X_mem i) |
| (by decide)).preimage (sectionsMorphism k s hs) |
| rw [sectionsMorphism_preimage, he] at hA |
| let : IsAffine (⊤ : X.Opens).toScheme := hA |
| let : IsAffine X := .of_isIso X.topIso.inv |
| have hdim := proper_affine_dimension_zero p |
| rw [hd] at hdim |
| norm_num at hdim |
| obtain ⟨t,ht,_,hf⟩ := curve_exists_finite_pencil p hd.le s hs i hne hproper |
| exact ⟨t,ht,hf⟩ |
| end |
| end MaximalSeshadri.Geometry |
| end |
|
|
|
|
| end |
| end |
|
|
| end OAI |
|
|
