File size: 3,128 Bytes
7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Geometry.SurfaceAffineCover
namespace OAI
section
namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Geometry MaximalSeshadri.Frames
attribute [local instance] MvPolynomial.gradedAlgebra
variable {K σ : Type} [Field K] [Infinite K] [Fintype σ] {X : Scheme}
theorem surface_finite_projection [IsIntegral X] [IsNoetherian X]
(p : X ⟶ Spec (CommRingCat.of K)) [IsProper p] [SmoothOfRelativeDimension 2 p]
(L : LineBundle X) (s : σ → (O X ⟶ L.sheaf))
(hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
[IsClosedImmersion (sectionsMorphism
(p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)] :
∃ t : Fin 3 → (O X ⟶ L.sheaf), ∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤,
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 ⟨v,hv⟩ := surface_three_sections p L s hs
let t : Fin 3 → (O X ⟶ L.sheaf) := fun i => sectionCombination k s (v i)
have ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤ := hv
refine ⟨t,ht,?_⟩
let f := sectionsMorphism k t ht
let : IsProper (X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k)) := by
dsimp only [k]
rw [toSpec_scalarMap]
infer_instance
let : IsProper f := sectionsMorphism_proper k t ht
let : IsAffineHom f := by
apply isAffineHom_of_forall_exists_isAffineOpen
intro y
have hy : y ∈ ⨆ i : Fin 3, Proj.basicOpen (PolyGrade K (Fin 3)) (MvPolynomial.X i) := by
rw [projective_coordinate_cover]
trivial
obtain ⟨i,hi⟩ := Opens.mem_iSup.mp hy
refine ⟨Proj.basicOpen (PolyGrade K (Fin 3)) (MvPolynomial.X i),hi,
Proj.isAffineOpen_basicOpen _ _ (poly_X_mem i) (by decide),?_⟩
rw [show f = sectionsMorphism k t ht from rfl, sectionsMorphism_preimage]
dsimp only [t]
rw [← sectionsMorphism_hyperplane _ s hs]
exact (Proj.isAffineOpen_basicOpen (PolyGrade K σ) _
(linearEquation_homogeneous (v i)) (by decide)).preimage _
exact IsFinite.iff_isProper_and_isAffineHom.mpr ⟨inferInstance,inferInstance⟩
end
end MaximalSeshadri.Projective
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Projective MaximalSeshadri.Frames
theorem Surface.exists_finite_projection (S : Surface) (A : LineBundle S.scheme)
(hA : A.IsAmple) :
∃ n : ℕ, 0 < n ∧ ∃ t : Fin 3 → (O S.scheme ⟶ (A.pow n).sheaf),
∃ ht : (⨆ i, SectionOpens.isoOpen (t i)) = ⊤,
IsFinite (sectionsMorphism (baseScalars S.structureMap) t ht) := by
obtain ⟨n,hn,σ,hσ,s,hs,hclosed⟩ := S.ample_embedding A hA
obtain ⟨t,ht,hf⟩ := @surface_finite_projection ℂ σ _ _ hσ S.scheme _ _
S.structureMap _ _ (A.pow n) s hs hclosed
exact ⟨n,hn,t,ht,hf⟩
end
end MaximalSeshadri.Geometry
end
end OAI
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