| import Mathlib |
| import OAI.AlgebraicGeometry.NumericalDimension.ExceptionalFibers |
|
|
| namespace OAI |
|
|
| /-! Relative Negativity. -/ |
|
|
| open AlgebraicGeometry CategoryTheory |
| open scoped TensorProduct nonZeroDivisors |
| open scoped TensorProduct |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace |
|
|
| namespace NumericalDimensionOne |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| lemma nonconstant_curve_in_irreducible_closed_subset |
| (X : ComplexProjectiveVariety) {Z : Set X.scheme} |
| (hZ : IsClosed Z) (hI : IsIrreducible Z) (x : X.scheme) |
| (hx : IsClosed {x}) (hxZ : x ∈ Z) (hZn : Z ≠ {x}) : |
| ∃ C : CurveOn X, Set.range C.morphism ⊆ Z ∧ x ∈ Set.range C.morphism ∧ |
| ¬ Set.range C.morphism ⊆ {x} := by |
| obtain ⟨T,i,hT,hi,hr,_⟩ := integral_closed_closure X.scheme hI.genericPoint |
| let := hT |
| let := hi |
| have hir : Set.range i = Z := hr.trans (hI.closure_genericPoint hZ) |
| obtain ⟨t,ht⟩ : x ∈ Set.range i := hir.symm ▸ hxZ |
| let sT := i ≫ X.structureMap |
| have : IsProper sT := inferInstance |
| have htclosed : IsClosed {t} := by |
| have he : i ⁻¹' {x} = {t} := by |
| ext s |
| simp only [Set.mem_preimage,Set.mem_singleton_iff,← ht] |
| exact i.isClosedEmbedding.injective.eq_iff |
| rw [← he] |
| exact hx.preimage i.continuous |
| have htne : ({t} : Set T) ≠ Set.univ := by |
| intro he |
| apply hZn |
| rw [← hir,← Set.image_univ,← he,Set.image_singleton,ht] |
| obtain ⟨N,j,hNI,_hNt,hsm,hpr,htj,hj⟩ := |
| smooth_proper_curve_through_point_avoiding sT t htclosed {t} htclosed htne rfl |
| let := hNI |
| let := hsm |
| let := hpr |
| let NC : ComplexProjectiveVariety := { |
| scheme := N |
| structureMap := j ≫ sT |
| integral := hNI |
| connected := inferInstance |
| projective := exists_closedImmersion_of_smooth_proper_curve (j ≫ sT) } |
| let C : CurveOn X := { |
| curve := NC |
| smooth := hsm |
| morphism := j ≫ i |
| overComplex := Category.assoc _ _ _ } |
| refine ⟨C,?_,?_,?_⟩ |
| · rintro _ ⟨c,rfl⟩ |
| rw [← hir] |
| exact ⟨j c,rfl⟩ |
| · obtain ⟨c,hc⟩ := htj |
| exact ⟨c,by change i (j c) = x; rw [hc,ht]⟩ |
| · intro hsub |
| apply hj |
| rintro _ ⟨c,rfl⟩ |
| have hc : i (j c) = x := hsub (Set.mem_range_self c) |
| change j c = t |
| exact i.isClosedEmbedding.injective (hc.trans ht.symm) |
|
|
| theorem contracted_curve_through_exceptional_point |
| (X : ComplexProjectiveVariety) (Y : NormalProjectiveVariety) |
| (f : X.scheme ⟶ Y.scheme) [IsDominant f] [IsProper f] |
| (hb : IsBirationalMorphism f) (p : PrimeDivisor X.scheme) |
| (hp : 1 < Order.coheight (f p.1)) (x : X.scheme) |
| (hx : IsClosed {x}) (hxp : x ∈ closure ({p.1} : Set X.scheme)) : |
| ∃ C : CurveOn X, IsCurveContracted f C ∧ x ∈ Set.range C.morphism ∧ |
| ¬ Set.range C.morphism ⊆ {x} := by |
| have : IsNoetherian X.scheme := ⟨⟩ |
| have hy : IsClosed {f x} := by |
| simpa only [Set.image_singleton] using f.isClosedMap {x} hx |
| have hF : IsClosed (f ⁻¹' {f x}) := hy.preimage f.continuous |
| have hnon : ¬ IsOpen ({⟨x,rfl⟩} : Set (f ⁻¹' {f x})) := by |
| intro ho |
| have he := (f.fiberHomeo (f x)).symm.isOpenMap _ ho |
| simp only [Set.image_singleton] at he |
| exact exceptional_fiber_not_isOpen_singleton f hb p hp x hxp he |
| obtain ⟨Z,hZ,hI,hxZ,hZF,hZn⟩ := |
| nonisolated_irreducible_closed_subset (f ⁻¹' {f x}) hF ⟨x,rfl⟩ hnon |
| obtain ⟨C,hCZ,hxC,hCn⟩ := |
| nonconstant_curve_in_irreducible_closed_subset X hZ hI x hx hxZ hZn |
| exact ⟨C,⟨f x,fun c => hZF (hCZ (Set.mem_range_self c))⟩,hxC,hCn⟩ |
| end NumericalDimensionOne |
|
|
| open AlgebraicGeometry CategoryTheory |
| open scoped TensorProduct nonZeroDivisors |
| open scoped TensorProduct |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace |
|
|
| namespace NumericalDimensionOne |
| open AlgebraicGeometry CategoryTheory |
| lemma exists_cartierStalkEquation_of_effective_near |
| {X : Scheme} [IsIntegral X] [IsLocallyNoetherian X] [StalkwiseNormal X] |
| {D : WeilDivisor X} (hD : IsCartierDivisor D) |
| (V : X.Opens) (heff : ∀ p : PrimeDivisor X, p.1 ∈ V → 0 ≤ D p) |
| (x : X) (hx : x ∈ V) : Nonempty (CartierStalkEquation D x) := by |
| obtain ⟨U,_hU,hxU,g,hg,heq⟩ := hD x |
| obtain ⟨W,hW,hxW,hWU⟩ := exists_isAffineOpen_mem_and_subset |
| (show x ∈ U ⊓ V from ⟨hxU,hx⟩) |
| have : Nonempty W := ⟨⟨x,hxW⟩⟩ |
| have hEq : ∀ p : PrimeDivisor X, p.1 ∈ W → D p = X.ord g p.1 := |
| fun p hp => heq p (hWU hp).1 |
| have hsec : IsDivisorSectionOn D W (1 : X.functionField) := |
| Or.inr (fun p hp => by simpa only [order_one,zero_add] using heff p (hWU hp).2) |
| obtain ⟨a,ha⟩ := (cartier_local_section_iff hW hg hEq 1).mp hsec |
| exact ⟨⟨W,hxW,g,hg,hEq,X.presheaf.germ W x hxW a,by |
| rw [X.algebraMap_germ_eq_germToFunctionField hxW a,ha,one_mul]⟩⟩ |
|
|
| lemma cartierCurveDegree_nonneg_of_effective_near |
| {X : ComplexProjectiveVariety} [StalkwiseNormal X.scheme] |
| (D : cartierDivisors (X := X.scheme)) (C : CurveOn X) |
| (V : X.scheme.Opens) (hCV : Set.range C.morphism ⊆ V) |
| (heff : ∀ p : PrimeDivisor X.scheme, p.1 ∈ V → 0 ≤ D.1 p) |
| (hη : C.morphism (genericPoint C.curve.scheme) ∉ divisorSupport D.1) : |
| 0 ≤ cartierCurveDegree D C := by |
| classical |
| let E := pulledCartierRepresentative C D.1 D.2 |
| obtain ⟨t,ht,htord⟩ := (pulledCartierRepresentative_spec C D.1 D.2).exists_effective_local_orders |
| C.morphism hη |
| let F := principalWeilDivisor t + E |
| have hF (q : PrimeDivisor C.curve.scheme) : 0 ≤ F q := by |
| obtain ⟨b⟩ := exists_cartierStalkEquation_of_effective_near D.2 V heff |
| (C.morphism q.1) (hCV (Set.mem_range_self q.1)) |
| obtain ⟨hb,ho⟩ := htord q b |
| rw [show F q = _ from ho,NumericalDimensionOneCurveAux.scheme_order_of_stalk_element q.1 q.2 _ hb] |
| positivity |
| have hdeg : 0 ≤ complexWeilDegree F := Finset.sum_nonneg fun q _ => hF q |
| unfold F complexWeilDegree at hdeg |
| rw [Finsupp.sum_add_index (fun _ _ => rfl) (fun _ _ _ _ => rfl)] at hdeg |
| change 0 ≤ complexWeilDegree (principalWeilDivisor t) + cartierCurveDegree D C at hdeg |
| rwa [principal_degree_zero C.curve C.smooth t,zero_add] at hdeg |
| end NumericalDimensionOne |
|
|
| open AlgebraicGeometry CategoryTheory |
| open scoped TensorProduct nonZeroDivisors |
| open scoped TensorProduct |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace |
|
|
| namespace NumericalDimensionOne |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| lemma cartierCurveDegree_pos_of_nonconstant |
| {X : ComplexProjectiveVariety} [StalkwiseNormal X.scheme] |
| (D : cartierDivisors (X := X.scheme)) (hD : IsAmpleDivisor D.1) |
| (C : CurveOn X) (x : X.scheme) (hx : IsClosed {x}) |
| (hxC : x ∈ Set.range C.morphism) (hn : ¬ Set.range C.morphism ⊆ {x}) : |
| 0 < cartierCurveDegree D C := by |
| let : SmoothOfRelativeDimension 1 C.curve.structureMap := C.smooth |
| have : IsNoetherian C.curve.scheme := ⟨⟩ |
| have hη : C.morphism (genericPoint C.curve.scheme) ≠ x := by |
| intro he |
| apply hn |
| rintro _ ⟨c,rfl⟩ |
| exact ((genericPoint_specializes c).map C.morphism.continuous).mem_closed hx he |
| have hpre : (C.morphism ⁻¹' {x}).Nonempty := hxC |
| obtain ⟨c,hcx,hcc⟩ := (hx.preimage C.morphism.continuous).exists_closed_singleton hpre |
| let p : PrimeDivisor C.curve.scheme := |
| ⟨c,coheight_eq_of_smooth_closed C.curve.structureMap 1 c hcc⟩ |
| exact cartierCurveDegree_positive_of_ample D hD C p (hcx ▸ hη) |
|
|
| theorem projective_cartier_negativity |
| (X : ComplexProjectiveVariety) [StalkwiseNormal X.scheme] |
| (Y : NormalProjectiveVariety) (f : X.scheme ⟶ Y.scheme) |
| [IsDominant f] [IsProper f] (hb : IsBirationalMorphism f) |
| (D : cartierDivisors (X := X.scheme)) |
| (hexc : ∀ p : PrimeDivisor X.scheme, D.1 p ≠ 0 → 1 < Order.coheight (f p.1)) |
| (hanti : ∀ C : CurveOn X, IsCurveContracted f C → cartierCurveDegree D C ≤ 0) : |
| 0 ≤ D.1 := by |
| classical |
| have : IsNoetherian X.scheme := ⟨⟩ |
| let : JacobsonSpace X.scheme := LocallyOfFiniteType.jacobsonSpace X.structureMap |
| intro p₀ |
| by_contra hn₀ |
| have hn : D.1 p₀ < 0 := lt_of_not_ge hn₀ |
| obtain ⟨J,hJ⟩ := exists_ampleDivisor_of_projective X |
| obtain ⟨E,hEa,hEe,_hEz⟩ := exists_effective_ample_avoiding_primes X.structureMap J hJ D.1.support |
| let EC : cartierDivisors (X := X.scheme) := ⟨E,hEa.1⟩ |
| let absD : WeilDivisor X.scheme := D.1.mapRange (fun a : ℤ => |a|) (abs_zero : |(0 : ℤ)| = 0) |
| have habs (p : PrimeDivisor X.scheme) : absD p = |D.1 p| := rfl |
| have hJE : 0 ≤ E + absD := fun p => add_nonneg (hEe p) (abs_nonneg (D.1 p)) |
| obtain ⟨U,hp₀U,b,hb0,hbD⟩ := exists_birational_local_equation_dominating f hb (E+absD) hJE (f p₀.1) |
| let P : cartierDivisors (X := X.scheme) := ⟨principalWeilDivisor b,isCartierDivisor_principal hb0⟩ |
| let B : cartierDivisors (X := X.scheme) := P - EC |
| have hB (p : PrimeDivisor X.scheme) (hp : f p.1 ∈ U) : |D.1 p| ≤ B.1 p := by |
| have h := hbD p hp |
| change E p + |D.1 p| ≤ X.scheme.ord b p.1 at h |
| change |D.1 p| ≤ X.scheme.ord b p.1 - E p |
| linarith |
| have hBn (p : PrimeDivisor X.scheme) (hp : f p.1 ∈ U) : 0 ≤ B.1 p := |
| (abs_nonneg _).trans (hB p hp) |
| have hBp (p : PrimeDivisor X.scheme) (hp : f p.1 ∈ U) (hd : D.1 p < 0) : 0 < B.1 p := by |
| have h := hB p hp |
| rw [abs_of_neg hd] at h |
| linarith |
| let S := D.1.support.filter (fun p => D.1 p < 0 ∧ f p.1 ∈ U) |
| have hS : S.Nonempty := ⟨p₀,Finset.mem_filter.mpr ⟨Finsupp.mem_support_iff.mpr hn.ne,hn,hp₀U⟩⟩ |
| obtain ⟨p,hp,hmax⟩ := S.exists_max_image (fun p => (-D.1 p : ℚ) / B.1 p) hS |
| have hp' := (Finset.mem_filter.mp hp).2 |
| have hbp : 0 < B.1 p := hBp p hp'.2 hp'.1 |
| let A : cartierDivisors (X := X.scheme) := (B.1 p) • D + (-D.1 p) • B |
| have hA (q : PrimeDivisor X.scheme) (hq : f q.1 ∈ U) : 0 ≤ A.1 q := by |
| change 0 ≤ B.1 p * D.1 q + (-D.1 p) * B.1 q |
| by_cases hdq : D.1 q < 0 |
| · have hqS : q ∈ S := Finset.mem_filter.mpr ⟨Finsupp.mem_support_iff.mpr hdq.ne,hdq,hq⟩ |
| have hrat := hmax q hqS |
| have hpQ : (0 : ℚ) < B.1 p := by exact_mod_cast hbp |
| have hqQ : (0 : ℚ) < B.1 q := by exact_mod_cast hBp q hq hdq |
| have hc := (div_le_div_iff₀ hqQ hpQ).mp hrat |
| have hc' : (-D.1 q) * B.1 p ≤ (-D.1 p) * B.1 q := by exact_mod_cast hc |
| nlinarith |
| · exact add_nonneg (mul_nonneg hbp.le (le_of_not_gt hdq)) |
| (mul_nonneg (neg_nonneg.mpr hp'.1.le) (hBn q hq)) |
| have hAp : A.1 p = 0 := by |
| change B.1 p * D.1 p + (-D.1 p) * B.1 p = 0 |
| ring |
| let V : X.scheme.Opens := (f ⁻¹ᵁ U) ⊓ divisorComplement A.1 |
| have hpV : p.1 ∈ V := ⟨hp'.2,by |
| change p.1 ∉ divisorSupport A.1 |
| rw [prime_mem_divisorSupport_iff] |
| exact not_ne_iff.mpr hAp⟩ |
| have hloc : IsLocallyClosed (closure ({p.1} : Set X.scheme) ∩ (V : Set X.scheme)) := |
| isClosed_closure.isLocallyClosed.inter V.isOpen.isLocallyClosed |
| obtain ⟨x,⟨hxp,hxV⟩,hx⟩ := nonempty_inter_closedPoints |
| (show (closure ({p.1} : Set X.scheme) ∩ (V : Set X.scheme)).Nonempty from |
| ⟨p.1,subset_closure (Set.mem_singleton p.1),hpV⟩) hloc |
| obtain ⟨C,hCf,hxC,hCn⟩ := contracted_curve_through_exceptional_point X Y f hb p |
| (hexc p hp'.1.ne) x hx hxp |
| have hEC : 0 < cartierCurveDegree EC C := |
| cartierCurveDegree_pos_of_nonconstant EC hEa C x hx hxC hCn |
| obtain ⟨y,hy⟩ := hCf |
| obtain ⟨c,hcx⟩ := hxC |
| have hCV : Set.range C.morphism ⊆ (f ⁻¹ᵁ U : X.scheme.Opens) := by |
| rintro _ ⟨d,rfl⟩ |
| change f (C.morphism d) ∈ U |
| rw [hy d,← hy c,hcx] |
| exact hxV.1 |
| have hηA : C.morphism (genericPoint C.curve.scheme) ∉ divisorSupport A.1 := by |
| intro hm |
| have hxc := ((genericPoint_specializes c).map C.morphism.continuous).mem_closed |
| (isClosed_divisorSupport A.1) hm |
| rw [hcx] at hxc |
| exact hxV.2 hxc |
| have hAC := cartierCurveDegree_nonneg_of_effective_near A C (f ⁻¹ᵁ U) hCV hA hηA |
| have hDC := hanti C ⟨y,hy⟩ |
| have hPC : cartierCurveDegree P C = 0 := cartierCurveDegree_principal C b hb0 P.2 |
| have hBC : cartierCurveDegree B C = -cartierCurveDegree EC C := by |
| change cartierCurveDegreeHom C (P-EC) = _ |
| rw [map_sub] |
| change cartierCurveDegree P C - cartierCurveDegree EC C = _ |
| rw [hPC,zero_sub] |
| have hAeq : cartierCurveDegree A C = B.1 p * cartierCurveDegree D C + |
| (-D.1 p) * cartierCurveDegree B C := by |
| change cartierCurveDegreeHom C ((B.1 p) • D + (-D.1 p) • B) = _ |
| rw [map_add,map_zsmul,map_zsmul,zsmul_eq_mul,zsmul_eq_mul] |
| rfl |
| rw [hAeq,hBC] at hAC |
| have hDC' : B.1 p * cartierCurveDegree D C ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hbp.le hDC |
| have hEC' : 0 < (-D.1 p) * cartierCurveDegree EC C := mul_pos (neg_pos.mpr hp'.1) hEC |
| nlinarith |
| end NumericalDimensionOne |
|
|
| open AlgebraicGeometry CategoryTheory |
| open scoped TensorProduct nonZeroDivisors |
| open scoped TensorProduct |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace |
|
|
| namespace NumericalDimensionOne |
| open AlgebraicGeometry CategoryTheory |
|
|
| theorem projective_qCartier_negativity |
| (X : ComplexProjectiveVariety) [StalkwiseNormal X.scheme] |
| (Y : NormalProjectiveVariety) (f : X.scheme ⟶ Y.scheme) |
| [IsDominant f] [IsProper f] (hb : IsBirationalMorphism f) |
| (D : QWeilDivisor X.scheme) (hD : IsQCartierDivisor D) |
| (hexc : ∀ p : PrimeDivisor X.scheme, D p ≠ 0 → 1 < Order.coheight (f p.1)) |
| (hanti : ∀ C : CurveOn X, IsCurveContracted f C → qCartierCurveDegree D hD C ≤ 0) : |
| 0 ≤ D := by |
| let M := chooseCartierMultiple hD |
| have hex : ∀ p : PrimeDivisor X.scheme, M.divisor.1 p ≠ 0 → |
| 1 < Order.coheight (f p.1) := by |
| intro p hp |
| apply hexc p |
| have hp' : p ∈ M.divisor.1.support := Finsupp.mem_support_iff.mpr hp |
| rw [M.support_eq] at hp' |
| exact Finsupp.mem_support_iff.mp hp' |
| have hpos := projective_cartier_negativity X Y f hb M.divisor hex (by |
| intro C hC |
| exact M.degree_nonpos_of_qDegree_nonpos hD C (hanti C hC)) |
| intro p |
| exact (M.coeff_nonneg_iff p).mp (hpos p) |
|
|
| theorem higher_dimensional_negativity_missing {n : ℕ} (_hn : 3 ≤ n) |
| (W : ComplexProjectiveVariety) [StalkwiseNormal W.scheme] |
| (_hW : IsSmoothNfold W n) (Y : NormalProjectiveVariety) |
| (q : W.scheme ⟶ Y.scheme) [IsDominant q] [IsProper q] |
| (hbir : IsBirationalMorphism q) |
| (_hbase : q ≫ Y.structureMap = W.structureMap) |
| (D : QWeilDivisor W.scheme) (hD : IsQCartierDivisor D) |
| (hexc : ∀ F : PrimeDivisor W.scheme, D F ≠ 0 → 1 < Order.coheight (q F.1)) |
| (hanti : ∀ C : CurveOn W, IsCurveContracted q C → qCartierCurveDegree D hD C ≤ 0) : |
| 0 ≤ D := by |
| exact projective_qCartier_negativity W Y q hbir D hD hexc hanti |
|
|
| lemma IsCanonicalMMPStep.effective_comparison_on_common_model {n : ℕ} |
| {X Y : CanonicalModel n} {φ : X.scheme.PartialIso Y.scheme} |
| (hstep : IsCanonicalMMPStep X Y φ) |
| (W : ComplexProjectiveVariety) [StalkwiseNormal W.scheme] |
| (hW : IsSmoothNfold W n) |
| (p : W.scheme ⟶ X.scheme) (q : W.scheme ⟶ Y.scheme) |
| [IsDominant p] [IsDominant q] [IsProper p] [IsProper q] |
| (hp : p ≫ X.structureMap = W.structureMap) (hq : q ≫ Y.structureMap = W.structureMap) |
| (θ : W.scheme.PartialIso X.scheme) (hθp : θ.IsOver p (𝟙 X.scheme)) |
| (hθq : (θ.trans φ).IsOver q (𝟙 Y.scheme)) : |
| ∃ P Q : QWeilDivisor W.scheme, |
| IsQCartierPullback p (rationalWeilDivisor X.canonical) P ∧ |
| IsQCartierPullback q (rationalWeilDivisor Y.canonical) Q ∧ |
| 0 ≤ P-Q ∧ |
| (∀ F : PrimeDivisor W.scheme, (P-Q) F ≠ 0 → 1 < Order.coheight (q F.1)) ∧ |
| (∀ F : PrimeDivisor W.scheme, Order.coheight (p F.1) = 1 → |
| 1 < Order.coheight (q F.1) → 0 < (P-Q) F) := by |
| obtain ⟨P,Q,hP,hQ,hD,hex,hstrict,hanti⟩ := |
| hstep.comparison_boundary_on_common_model W hW p q hp hq θ hθp hθq |
| exact ⟨P,Q,hP,hQ,projective_qCartier_negativity W Y.toNormalProjectiveVariety q |
| ⟨θ.trans φ,hθq⟩ (P-Q) hD hex hanti,hex,hstrict⟩ |
| end NumericalDimensionOne |
|
|
| end OAI |
|
|
