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namespace OAI
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
attribute [local instance] MvPolynomial.gradedAlgebra
def complexProjectiveSpace (N : ℕ) : Scheme :=
Proj (MvPolynomial.homogeneousSubmodule (Fin (N + 1)) ℂ)
def constantsInDegreeZero (N : ℕ) :
ℂ →+* (MvPolynomial.homogeneousSubmodule (Fin (N + 1)) ℂ 0) :=
{ toFun := fun c => ⟨MvPolynomial.C c, MvPolynomial.isHomogeneous_C _ c⟩
map_one' := Subtype.ext MvPolynomial.C_1
map_mul' := fun a b => Subtype.ext (MvPolynomial.C_mul (a := a) (a' := b))
map_zero' := Subtype.ext MvPolynomial.C_0
map_add' := fun a b => Subtype.ext (MvPolynomial.C_add (a := a) (a' := b)) }
def projectiveSpaceToSpec (N : ℕ) :
complexProjectiveSpace N ⟶ Spec (CommRingCat.of ℂ) :=
Proj.toSpecZero _ ≫ Spec.map (CommRingCat.ofHom (constantsInDegreeZero N))
structure Surface where
scheme : Scheme
structureMap : scheme ⟶ Spec (CommRingCat.of ℂ)
integral : IsIntegral scheme
smooth : SmoothOfRelativeDimension 2 structureMap
embeddingDimension : ℕ
embedding : scheme ⟶ complexProjectiveSpace embeddingDimension
closedImmersion : IsClosedImmersion embedding
overComplex : embedding ≫ projectiveSpaceToSpec embeddingDimension = structureMap
attribute [instance] Surface.integral Surface.smooth Surface.closedImmersion
variable (X : Scheme)
structure LineBundle where
sheaf : X.Modules
locallyRankOne : ∀ x : X, ∃ U : X.Opens, x ∈ U ∧
Nonempty (sheaf.restrict U.ι ≅ SheafOfModules.unit U.toScheme.ringCatSheaf)
def moduleTensor (M N : X.Modules) : X.Modules :=
(PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).obj
(PresheafOfModulesOfCommRing.Monoidal.tensorObj (R := X.presheaf) M.val N.val)
def modulePow (M : X.Modules) : ℕ → X.Modules
| 0 => SheafOfModules.unit X.ringCatSheaf
| n + 1 => moduleTensor X M (modulePow M n)
def structureSheaf : X.Modules := SheafOfModules.unit X.ringCatSheaf
abbrev GlobalSections (M : X.Modules) := structureSheaf X ⟶ M
def sectionOpen {M : X.Modules} (s : GlobalSections X M) : X.Opens :=
⨆ (U : X.Opens) (_ : IsIso ((Scheme.Modules.restrictFunctor U.ι).map s)), U
def LineBundle.IsAmple (L : LineBundle X) : Prop :=
∀ (x : X) (V : X.Opens), x ∈ V → ∃ n : ℕ, 0 < n ∧
∃ s : GlobalSections X (modulePow X L.sheaf n),
x ∈ sectionOpen X s ∧ sectionOpen X s ≤ V ∧ IsAffineOpen (sectionOpen X s)
end
end MaximalSeshadri.Geometry
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
variable (X : Scheme)
def restrictScalar (U : X.Opens) : Γ(X, ⊤) →+* Γ(X, U) :=
(X.presheaf.map (homOfLE le_top).op).hom
lemma restrictScalar_naturality {U V : X.Opens} (i : U ⟶ V) (r : Γ(X, ⊤)) :
X.presheaf.map i.op (restrictScalar X V r) = restrictScalar X U r := by
change (X.presheaf.map (homOfLE le_top).op ≫ X.presheaf.map i.op) r = _
rw [← X.presheaf.map_comp]
rfl
def sheafHomSMul {M N : X.Modules} (r : Γ(X, ⊤)) (f : M ⟶ N) : M ⟶ N where
val := {
app U := by
letI : Module Γ(X, U.unop) (M.val.obj U) := (M.val.obj U).isModule
letI : Module Γ(X, U.unop) (N.val.obj U) := (N.val.obj U).isModule
let g : M.val.obj U →ₗ[Γ(X, U.unop)] N.val.obj U := (f.val.app U).hom
exact ModuleCat.ofHom ((restrictScalar X U.unop r) • g)
naturality {U V} g := by
let : Module (X.presheaf.obj V) (N.presheaf.obj V) := (N.val.obj V).isModule
ext m
change restrictScalar X V.unop r • f.app V.unop (M.presheaf.map g m) =
N.presheaf.map g (restrictScalar X U.unop r • f.app U.unop m)
have hn := CategoryTheory.congr_fun (f.mapPresheaf.naturality g) m
change f.app V.unop (M.presheaf.map g m) =
N.presheaf.map g (f.app U.unop m) at hn
rw [hn]
calc
_ = X.presheaf.map g (restrictScalar X U.unop r) •
N.presheaf.map g (f.app U.unop m) := by
exact congrArg (fun a : X.presheaf.obj V =>
a • N.presheaf.map g (f.app U.unop m))
(restrictScalar_naturality X g.unop r).symm
_ = _ := (N.map_smul g.unop (restrictScalar X U.unop r) (f.app U.unop m)).symm }
instance sheafHomModule (M N : X.Modules) : Module Γ(X, ⊤) (M ⟶ N) where
smul := sheafHomSMul X
one_smul f := by
ext U m
change restrictScalar X U 1 • f.app U m = f.app U m
simp
mul_smul r s f := by
ext U m
change restrictScalar X U (r * s) • f.app U m =
restrictScalar X U r • (restrictScalar X U s • f.app U m)
simp [mul_smul]
smul_zero r := by
ext U m
change restrictScalar X U r • (0 : Γ(N, U)) = 0
exact smul_zero _
smul_add r f g := by
ext U m
change restrictScalar X U r • (f.app U m + g.app U m) =
restrictScalar X U r • f.app U m + restrictScalar X U r • g.app U m
exact smul_add _ _ _
zero_smul f := by
ext U m
change restrictScalar X U 0 • f.app U m = 0
simp
add_smul r s f := by
ext U m
change restrictScalar X U (r + s) • f.app U m =
restrictScalar X U r • f.app U m + restrictScalar X U s • f.app U m
simp [add_smul]
instance sheafLinear : Linear Γ(X, ⊤) X.Modules where
smul_comp M N P r f g := by
ext U m
change g.app U (restrictScalar X U r • f.app U m) =
restrictScalar X U r • g.app U (f.app U m)
exact g.app_smul _ _
comp_smul M N P f r g := by
ext U m
rfl
end
end MaximalSeshadri.Geometry
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
variable {X : Scheme.{0}}
instance schemeHasExt : HasExt.{1} X.Modules := HasExt.standard X.Modules
def baseScalars (f : X ⟶ Spec (CommRingCat.of ℂ)) : ℂ →+* Γ(X, ⊤) :=
f.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom
abbrev cohomology (M : X.Modules) (n : ℕ) : Type 1 :=
Abelian.Ext.{1} (C := X.Modules) (structureSheaf X) M n
def cohomologyDimension (f : X ⟶ Spec (CommRingCat.of ℂ))
(M : X.Modules) (n : ℕ) : ℕ :=
letI : Module ℂ (cohomology M n) :=
Module.compHom (cohomology M n) (baseScalars f)
Module.finrank ℂ (cohomology M n)
def eulerCharacteristic (f : X ⟶ Spec (CommRingCat.of ℂ)) (d : ℕ)
(M : X.Modules) : ℤ :=
∑ n ∈ Finset.range (d + 1), (-1 : ℤ) ^ n * (cohomologyDimension f M n : ℤ)
def selfIntersection (S : Surface) (L : LineBundle S.scheme) : ℤ :=
eulerCharacteristic S.structureMap 2 (modulePow S.scheme L.sheaf 2) -
2 * eulerCharacteristic S.structureMap 2 L.sheaf +
eulerCharacteristic S.structureMap 2 (SheafOfModules.unit S.scheme.ringCatSheaf)
structure IntegralCurve (S : Surface) where
scheme : Scheme
embedding : scheme ⟶ S.scheme
closedImmersion : IsClosedImmersion embedding
integral : IsIntegral scheme
dimension : topologicalKrullDim scheme = 1
attribute [instance] IntegralCurve.closedImmersion IntegralCurve.integral
def curveDegree (S : Surface) (L : LineBundle S.scheme) (C : IntegralCurve S) : ℤ :=
eulerCharacteristic (C.embedding ≫ S.structureMap) 1
((Scheme.Modules.pullback C.embedding).obj L.sheaf) -
eulerCharacteristic (C.embedding ≫ S.structureMap) 1
(SheafOfModules.unit C.scheme.ringCatSheaf)
end
end MaximalSeshadri.Geometry
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
abbrev complexBase : Scheme := Spec (CommRingCat.of ℂ)
abbrev ComplexPoint (S : Surface) :=
Over.mk (𝟙 complexBase) ⟶ Over.mk S.structureMap
def ComplexPoint.image {S : Surface} (p : ComplexPoint S) : S.scheme :=
p.left (⟨⊥, Ideal.isPrime_bot⟩ : PrimeSpectrum ℂ)
def surfacePower (S : Surface) (r : ℕ) : Over complexBase :=
∏ᶜ (fun _ : Fin r => Over.mk S.structureMap)
def tupleImage (S : Surface) (r : ℕ) (p : Fin r → ComplexPoint S) :
(surfacePower S r).left :=
(Limits.Pi.lift p).left (⟨⊥, Ideal.isPrime_bot⟩ : PrimeSpectrum ℂ)
@[instance_reducible] def tupleZariskiTopology (S : Surface) (r : ℕ) :
TopologicalSpace (Fin r → ComplexPoint S) :=
TopologicalSpace.induced (tupleImage S r) inferInstance
def Configuration (S : Surface) (r : ℕ) :=
{p : Fin r → ComplexPoint S // Function.Injective (fun i => (p i).image)}
instance configurationTopology (S : Surface) (r : ℕ) :
TopologicalSpace (Configuration S r) :=
letI := tupleZariskiTopology S r
inferInstanceAs (TopologicalSpace {p : Fin r → ComplexPoint S //
Function.Injective (fun i => (p i).image)})
def idealOrder {R : Type} [CommRing R] [IsLocalRing R] (I : Ideal R) : ℕ :=
sInf {n : ℕ | ¬ I ≤ IsLocalRing.maximalIdeal R ^ (n + 1)}
def curveMultiplicity (S : Surface) (C : IntegralCurve S) (p : ComplexPoint S) : ℕ := by
classical
exact if h : ∃ q : C.scheme, C.embedding q = p.image then
idealOrder (RingHom.ker (C.embedding.stalkMap h.choose).hom)
else 0
def totalMultiplicity (S : Surface) (r : ℕ) (C : IntegralCurve S)
(p : Configuration S r) : ℕ :=
∑ i : Fin r, curveMultiplicity S C (p.val i)
def seshadriConstant (S : Surface) (L : LineBundle S.scheme) (r : ℕ)
(p : Configuration S r) : ℝ :=
sInf {a : ℝ | ∃ C : IntegralCurve S, 0 < totalMultiplicity S r C p ∧
a = (curveDegree S L C : ℝ) / (totalMultiplicity S r C p : ℝ)}
end
end MaximalSeshadri.Geometry
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
def moduleCurveDegree (S : Surface) (M : S.scheme.Modules) (C : IntegralCurve S) : ℤ :=
eulerCharacteristic (C.embedding ≫ S.structureMap) 1
((Scheme.Modules.pullback C.embedding).obj M) -
eulerCharacteristic (C.embedding ≫ S.structureMap) 1
(structureSheaf C.scheme)
def sectionImageIdeal {X : Scheme} (J : LineBundle X)
(ι : J.sheaf ⟶ structureSheaf X) (U : X.Opens) : Ideal Γ(X, U) := by
letI : Module Γ(X, U) (J.sheaf.val.obj (Opposite.op U)) :=
(J.sheaf.val.obj (Opposite.op U)).isModule
let φ : J.sheaf.val.obj (Opposite.op U) →ₗ[Γ(X, U)] Γ(X, U) :=
(ι.val.app (Opposite.op U)).hom
exact φ.range
def PresentsPullbackIdeal {X Y : Scheme} (I : X.IdealSheafData) (f : Y ⟶ X)
(J : LineBundle Y) (ι : J.sheaf ⟶ structureSheaf Y) : Prop :=
Mono ι ∧ ∀ (U : Y.affineOpens) (V : X.affineOpens)
(e : U.1 ≤ f ⁻¹ᵁ V.1),
sectionImageIdeal J ι U.1 = (I.ideal V).map (f.appLE V.1 U.1 e).hom
def InvertiblePullbackIdeal {X Y : Scheme} (I : X.IdealSheafData) (f : Y ⟶ X) : Prop :=
∃ (J : LineBundle Y) (ι : J.sheaf ⟶ structureSheaf Y), PresentsPullbackIdeal I f J ι
def centreIdeal (S : Surface) (r : ℕ) (p : Configuration S r) : S.scheme.IdealSheafData :=
∏ i : Fin r, (p.val i).left.ker
def IsBlowup {X B : Scheme} (I : X.IdealSheafData) (π : B ⟶ X) : Prop :=
InvertiblePullbackIdeal I π ∧ ∀ (Y : Scheme) (f : Y ⟶ X),
InvertiblePullbackIdeal I f → ∃! h : Y ⟶ B, h ≫ π = f
structure PointBlowup (S : Surface) (r : ℕ) (p : Configuration S r) where
surface : Surface
projection : surface.scheme ⟶ S.scheme
overComplex : projection ≫ S.structureMap = surface.structureMap
isBlowup : IsBlowup (centreIdeal S r p) projection
exceptionalIdeal : Fin r → LineBundle surface.scheme
exceptionalInclusion : ∀ i, (exceptionalIdeal i).sheaf ⟶ structureSheaf surface.scheme
exceptionalPullback : ∀ i, PresentsPullbackIdeal (p.val i).left.ker projection
(exceptionalIdeal i) (exceptionalInclusion i)
def PointBlowup.BoundaryNef {S : Surface} {r : ℕ} {p : Configuration S r}
(B : PointBlowup S r p) (L : LineBundle S.scheme) : Prop :=
∀ C : IntegralCurve B.surface,
0 ≤ (moduleCurveDegree B.surface
((Scheme.Modules.pullback B.projection).obj L.sheaf) C : ℝ) +
Real.sqrt ((selfIntersection S L : ℝ) / r) *
∑ i : Fin r, (curveDegree B.surface (B.exceptionalIdeal i) C : ℝ)
def EventualMaximalSeshadri (S : Surface) (L : LineBundle S.scheme) : Prop :=
∃ r₀ : ℕ, 0 < r₀ ∧ ∀ r : ℕ, r₀ ≤ r →
∃ Z : ℕ → Set (Configuration S r),
(∀ n, IsClosed (Z n) ∧ Z n ≠ Set.univ) ∧
(∃ p : Configuration S r, ∀ n, p ∉ Z n) ∧
∀ p : Configuration S r, (∀ n, p ∉ Z n) →
(∃ B : PointBlowup S r p, B.BoundaryNef L) ∧
seshadriConstant S L r p = Real.sqrt ((selfIntersection S L : ℝ) / r)
end
end MaximalSeshadri.Geometry
end OAI
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