| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Configurations.RationalPoint | |
| import OAI.AlgebraicGeometry.Seshadri.Configurations.GlobalCenter | |
| import OAI.AlgebraicGeometry.Seshadri.Projective.QuarticReindex | |
| import OAI.AlgebraicGeometry.Seshadri.Configurations.SmoothRationalPoints | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.Projective | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open MaximalSeshadri.Frames MaximalSeshadri.Geometry MaximalSeshadri.ProjectiveBertini | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {K σ : Type} [Field K] [Fintype σ] {X : Scheme} | |
| omit [Fintype σ] in | |
| lemma embedding_reindex_nonempty [IsIntegral X] {M : X.Modules} | |
| (k : K →+* Γ(X, ⊤)) (s : Option σ → (O X ⟶ M)) | |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) | |
| [IsClosedImmersion (sectionsMorphism k s hs)] : | |
| ∃ t : Option σ → (O X ⟶ M), ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤, | |
| IsClosedImmersion (sectionsMorphism k t ht) ∧ | |
| (SectionOpens.isoOpen (t none) : Set X).Nonempty := by | |
| classical | |
| let x : X := Classical.ofNonempty | |
| obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (hs.ge (Set.mem_univ x)) | |
| let e : Option σ ≃ Option σ := Equiv.swap none i | |
| let ht := sections_cover_reindex s hs e | |
| refine ⟨s ∘ e, ht, sectionsMorphism_reindex_closed k s hs e ht, x, ?_⟩ | |
| change x ∈ SectionOpens.isoOpen (s (e none)) | |
| simpa only [e, Equiv.swap_apply_left] using hi | |
| lemma exists_centered_embedding [IsIntegral X] [IsAlgClosed K] {M : X.Modules} | |
| (k : K →+* Γ(X, ⊤)) (s : Option σ → (O X ⟶ M)) | |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) | |
| [IsClosedImmersion (sectionsMorphism k s hs)] : | |
| ∃ t : Option σ → (O X ⟶ M), ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤, | |
| IsClosedImmersion (sectionsMorphism k t ht) ∧ | |
| ∃ y : X, ∀ x : X, x ∉ centeredOpen t ↔ x = y := by | |
| obtain ⟨s, hs, hcl, h₀⟩ := embedding_reindex_nonempty k s hs | |
| have := hcl | |
| let U := SectionOpens.isoOpen (s none) | |
| let kU := U.ι.appTop.hom.comp k | |
| let a := fun i => coefficient (sectionFrame (s none)) (restrictSection U.ι (s i)) | |
| obtain ⟨hUa, hgen⟩ := sectionsMorphism_chart_generators k s hs none | |
| have : IsAffine U.toScheme := hUa | |
| let : Nonempty U := ⟨⟨h₀.choose, h₀.choose_spec⟩⟩ | |
| let : Algebra K Γ(U.toScheme, ⊤) := kU.toAlgebra | |
| have ha : Function.Surjective (MvPolynomial.aeval (R := K) a) := hgen | |
| have : Algebra.FiniteType K Γ(U.toScheme, ⊤) := Algebra.FiniteType.of_surjective | |
| (MvPolynomial.aeval (R := K) a) ha | |
| obtain ⟨p⟩ := exists_point_finite_type (K := K) (R := Γ(U.toScheme, ⊤)) | |
| let t := shiftedSections k s (fun i => -p (a (some i))) | |
| let ht := shiftedSections_cover k s hs (fun i => -p (a (some i))) | |
| have hp : p.toRingHom.comp kU = RingHom.id K := p.comp_algebraMap | |
| obtain ⟨y, hy⟩ := centeredOpen_complement_singleton k s hs p.toRingHom hp | |
| exact ⟨t, ht, shiftedSections_closed k s hs h₀ _, y, hy⟩ | |
| omit [Fintype σ] in | |
| lemma centeredOpen_nonempty_of_singleton [IsIntegral X] [IsAlgClosed K] [Uncountable K] | |
| (g : X ⟶ Spec (CommRingCat.of K)) [SmoothOfRelativeDimension 2 g] | |
| {M : X.Modules} (s : Option σ → (O X ⟶ M)) (y : X) | |
| (hy : ∀ x : X, x ∉ centeredOpen s ↔ x = y) : | |
| (centeredOpen s : Set X).Nonempty := by | |
| by_contra h | |
| apply smooth_scheme_uncountable_rational_points g 2 (by decide) | |
| apply (Set.countable_singleton y).mono | |
| rintro x ⟨p, rfl⟩ | |
| exact (hy _).mp (fun hx => h ⟨_, hx⟩) | |
| end | |
| end MaximalSeshadri.Projective | |
| end | |
| end | |
| end | |
| end OAI | |