| import Mathlib | |
| import OAI.AlgebraicGeometry.NumericalDimension.SurfaceSquare | |
| namespace OAI | |
| /-! Indexed Transfer. -/ | |
| open AlgebraicGeometry CategoryTheory | |
| open scoped TensorProduct nonZeroDivisors | |
| open scoped TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Limits | |
| open Algebra KaehlerDifferential IsLocalRing TensorProduct | |
| open AlgebraicGeometry CategoryTheory TensorProduct | |
| open TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set Topology | |
| open AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization | |
| open scoped IntermediateField.algebraAdjoinAdjoin | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Filter | |
| open Opposite TopCat | |
| open AlgebraicGeometry CategoryTheory TopCat Opposite TopologicalSpace | |
| open CategoryTheory.Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite TopCat | |
| open CategoryTheory TopologicalSpace Opposite | |
| namespace NumericalDimensionOne | |
| open Filter | |
| open scoped ENNReal | |
| section SectionGrowth | |
| variable (X : ComplexProjectiveVariety) [StalkwiseNormal X.scheme] | |
| theorem exponent_le_kappaSigma {n k : ℕ} {D A : WeilDivisor X.scheme} | |
| (hk : k ≤ n) (hA : IsAmpleDivisor A) | |
| (hpos : 0 < divisorSectionLimsup X D A k) : | |
| (k : WithBot ℕ) ≤ kappaSigma X n D := by | |
| classical | |
| exact Finset.le_sup (f := fun j : ℕ => (j : WithBot ℕ)) | |
| (Finset.mem_filter.mpr ⟨Finset.mem_range.mpr (by omega), A, hA, hpos⟩) | |
| theorem divisorSectionLimsup_eq_zero_of_kappaSigma_eq_one | |
| {n k : ℕ} {D A : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) | |
| (hk : 2 ≤ k) (hkn : k ≤ n) (hA : IsAmpleDivisor A) : | |
| divisorSectionLimsup X D A k = 0 := by | |
| by_contra hne | |
| have hpos : 0 < divisorSectionLimsup X D A k := pos_iff_ne_zero.mpr hne | |
| have hle := exponent_le_kappaSigma X hkn hA hpos | |
| rw [hsigma] at hle | |
| have hle' : k ≤ 1 := by exact_mod_cast hle | |
| omega | |
| end SectionGrowth | |
| theorem le_limsup_of_eventually_le_on_multiples | |
| (a : ℕ → ℝ≥0∞) {r : ℕ} (hr : 0 < r) {c : ℝ≥0∞} | |
| (h : ∀ᶠ m in atTop, c ≤ a (r * m)) : | |
| c ≤ limsup a atTop := by | |
| have ht : Tendsto (fun m : ℕ => r * m) atTop atTop := | |
| tendsto_atTop_mono (fun m => Nat.le_mul_of_pos_left m hr) tendsto_id | |
| exact (le_limsup_of_frequently_le h.frequently).trans ht.limsup_comp_le_limsup | |
| theorem positive_limsup_of_quadratic_growth_on_multiples | |
| (a : ℕ → ℝ≥0∞) {r : ℕ} (hr : 0 < r) {c : ℝ≥0∞} (hc : 0 < c) | |
| (h : ∀ᶠ m in atTop, c * (r * m : ℕ) ^ 2 ≤ a (r * m)) : | |
| 0 < limsup (fun m : ℕ => a m / (m : ℝ≥0∞) ^ 2) atTop := by | |
| apply hc.trans_le | |
| apply le_limsup_of_eventually_le_on_multiples _ hr | |
| filter_upwards [h, eventually_gt_atTop 0] with m hm hmpos | |
| have hn : (r * m : ℕ) ≠ 0 := Nat.ne_of_gt (Nat.mul_pos hr hmpos) | |
| apply (ENNReal.le_div_iff_mul_le (Or.inl (by simpa using hn)) | |
| (Or.inl (ENNReal.pow_ne_top (ENNReal.natCast_ne_top _)))).2 | |
| exact hm | |
| section SectionGrowth | |
| variable (X : ComplexProjectiveVariety) [StalkwiseNormal X.scheme] | |
| theorem no_quadratic_growth_on_multiples_of_kappaSigma_eq_one | |
| {n : ℕ} {D A : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) (hn : 2 ≤ n) | |
| (hA : IsAmpleDivisor A) {r : ℕ} (hr : 0 < r) | |
| {c : ℝ≥0∞} (hc : 0 < c) : | |
| ¬ ∀ᶠ m in atTop, | |
| c * (r * m : ℕ) ^ 2 ≤ sectionDimension X ((r * m) • D + A) := by | |
| intro hgrowth | |
| have hp := positive_limsup_of_quadratic_growth_on_multiples | |
| (fun m => sectionDimension X (m • D + A)) hr hc hgrowth | |
| change 0 < divisorSectionLimsup X D A 2 at hp | |
| rw [divisorSectionLimsup_eq_zero_of_kappaSigma_eq_one X hsigma (by decide) hn hA] at hp | |
| exact (lt_irrefl _ hp) | |
| theorem sectionDimension_le_of_injective | |
| (Y : ComplexProjectiveVariety) [StalkwiseNormal Y.scheme] | |
| (D : WeilDivisor X.scheme) (E : WeilDivisor Y.scheme) : | |
| letI := schemeFieldAlgebra (.of ℂ) X.structureMap | |
| letI := schemeFieldAlgebra (.of ℂ) Y.structureMap | |
| ∀ F : divisorSectionsOver (.of ℂ) X.structureMap D →ₗ[ℂ] | |
| divisorSectionsOver (.of ℂ) Y.structureMap E, | |
| Function.Injective F → sectionDimension X D ≤ sectionDimension Y E := by | |
| dsimp only | |
| let := schemeFieldAlgebra (.of ℂ) X.structureMap | |
| let := schemeFieldAlgebra (.of ℂ) Y.structureMap | |
| intro F hF | |
| exact ENat.toENNReal_mono (Cardinal.toENat.monotone' (F.rank_le_of_injective hF)) | |
| theorem sectionDimension_le_of_nonzero_section | |
| {D E : WeilDivisor X.scheme} {s : X.scheme.functionField} | |
| (hs : s ≠ 0) (hsec : IsDivisorSection (E - D) s) : | |
| sectionDimension X D ≤ sectionDimension X E := by | |
| let := schemeFieldAlgebra (.of ℂ) X.structureMap | |
| let F : divisorSectionsOver (.of ℂ) X.structureMap D →ₗ[ℂ] | |
| divisorSectionsOver (.of ℂ) X.structureMap E := { | |
| toFun := fun f => ⟨f.1 * s, by | |
| have hf := divisorSection_mul f.2 hsec | |
| change IsDivisorSection E _ | |
| have hDE : D + (E - D) = E := by abel | |
| simpa only [hDE] using hf⟩ | |
| map_add' := fun f g => Subtype.ext (add_mul _ _ _) | |
| map_smul' := by | |
| intro a f | |
| apply Subtype.ext | |
| change (a • f.1) * s = a • (f.1 * s) | |
| exact smul_mul_assoc a f.1 s } | |
| exact sectionDimension_le_of_injective X X D E F (fun f g h => | |
| Subtype.ext (mul_right_cancel₀ hs (congrArg Subtype.val h))) | |
| theorem exists_fixed_ample_twist_sectionDimension_le | |
| (D P : WeilDivisor X.scheme) : | |
| ∃ A : WeilDivisor X.scheme, IsAmpleDivisor A ∧ | |
| ∀ m : ℕ, sectionDimension X (m • D + P) ≤ sectionDimension X (m • D + A) := by | |
| obtain ⟨B, hB⟩ := exists_ampleDivisor_of_projective X | |
| obtain ⟨A, hA, s, hs, hsec⟩ := exists_ample_twist_dominating B P hB | |
| refine ⟨A, hA, fun m => sectionDimension_le_of_nonzero_section X hs ?_⟩ | |
| simpa only [add_sub_add_left_eq_sub] using hsec | |
| theorem no_quadratic_growth_with_fixed_twist | |
| {n : ℕ} {D : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) (hn : 2 ≤ n) | |
| (P : WeilDivisor X.scheme) {r : ℕ} (hr : 0 < r) | |
| {c : ℝ≥0∞} (hc : 0 < c) : | |
| ¬ ∀ᶠ m in atTop, | |
| c * (r * m : ℕ) ^ 2 ≤ sectionDimension X ((r * m) • D + P) := by | |
| obtain ⟨A, hA, hAP⟩ := exists_fixed_ample_twist_sectionDimension_le X D P | |
| intro hgrowth | |
| apply no_quadratic_growth_on_multiples_of_kappaSigma_eq_one X hsigma hn hA hr hc | |
| filter_upwards [hgrowth] with m hm | |
| exact hm.trans (hAP (r * m)) | |
| end SectionGrowth | |
| end NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory | |
| open scoped TensorProduct nonZeroDivisors | |
| open scoped TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Limits | |
| open Algebra KaehlerDifferential IsLocalRing TensorProduct | |
| open AlgebraicGeometry CategoryTheory TensorProduct | |
| open TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set Topology | |
| open AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization | |
| open scoped IntermediateField.algebraAdjoinAdjoin | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Filter | |
| open Opposite TopCat | |
| open AlgebraicGeometry CategoryTheory TopCat Opposite TopologicalSpace | |
| open CategoryTheory.Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite TopCat | |
| open CategoryTheory TopologicalSpace Opposite | |
| namespace NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory Filter | |
| open scoped ENNReal | |
| section WeilPushforward | |
| variable {W X : Scheme} [IsIntegral W] [IsIntegral X] | |
| [IsLocallyNoetherian X] [StalkwiseNormal X] | |
| (u : W ⟶ X) [IsProper u] (hu : IsBirationalMorphism u) | |
| noncomputable def birationalPrimeLift (p : PrimeDivisor X) : PrimeDivisor W := | |
| Classical.choose (exists_unique_prime_over_of_proper_birational u hu p) | |
| lemma birationalPrimeLift_spec (p : PrimeDivisor X) : | |
| u (birationalPrimeLift u hu p).1 = p.1 ∧ | |
| IsIso (u.stalkMap (birationalPrimeLift u hu p).1) ∧ | |
| ∀ w : W, u w = p.1 → w = (birationalPrimeLift u hu p).1 := | |
| Classical.choose_spec (exists_unique_prime_over_of_proper_birational u hu p) | |
| lemma birationalPrimeLift_injective : Function.Injective (birationalPrimeLift u hu) := by | |
| intro p q hpq | |
| apply Subtype.ext | |
| calc | |
| p.1 = u (birationalPrimeLift u hu p).1 := (birationalPrimeLift_spec u hu p).1.symm | |
| _ = u (birationalPrimeLift u hu q).1 := congrArg (fun v : PrimeDivisor W => u v.1) hpq | |
| _ = q.1 := (birationalPrimeLift_spec u hu q).1 | |
| noncomputable def birationalWeilPushforward : WeilDivisor W →+ WeilDivisor X := | |
| Finsupp.comapDomain.addMonoidHom (birationalPrimeLift_injective u hu) | |
| @[simp] lemma birationalWeilPushforward_apply (P : WeilDivisor W) (p : PrimeDivisor X) : | |
| birationalWeilPushforward u hu P p = P (birationalPrimeLift u hu p) := rfl | |
| theorem birationalWeilPushforward_single_of_contracted | |
| (q : PrimeDivisor W) (hq : 1 < Order.coheight (u q.1)) (a : ℤ) : | |
| birationalWeilPushforward u hu (Finsupp.single q a) = 0 := by | |
| classical | |
| ext p | |
| rw [birationalWeilPushforward_apply, Finsupp.single_apply, Finsupp.zero_apply] | |
| apply ite_eq_right | |
| intro h | |
| have he : u q.1 = p.1 := by | |
| rw [h] | |
| exact (birationalPrimeLift_spec u hu p).1 | |
| rw [he, p.2] at hq | |
| exact (lt_irrefl _ hq) | |
| variable [IsLocallyNoetherian W] [IsDominant u] | |
| theorem birationalWeilPushforward_cartierPullback | |
| {D : WeilDivisor X} {P : WeilDivisor W} (hP : IsCartierPullback u D P) : | |
| birationalWeilPushforward u hu P = D := by | |
| ext p | |
| let q := birationalPrimeLift u hu p | |
| have hq : u q.1 = p.1 := (birationalPrimeLift_spec u hu p).1 | |
| let : IsIso (u.stalkMap q.1) := (birationalPrimeLift_spec u hu p).2.1 | |
| have hp : Order.coheight (u q.1) = 1 := by rw [hq]; exact p.2 | |
| have he : (⟨u q.1, hp⟩ : PrimeDivisor X) = p := Subtype.ext hq | |
| change P q = D p | |
| simpa only [he] using IsCartierPullback.coeff_eq_of_stalk_iso u q hp hP | |
| theorem divisorSection_birational_descent | |
| {P : WeilDivisor W} {s : X.functionField} | |
| (hsec : IsDivisorSection P (dominantFunctionFieldMap u s)) : | |
| IsDivisorSection (birationalWeilPushforward u hu P) s := by | |
| by_cases hs : s = 0 | |
| · exact Or.inl hs | |
| rcases hsec with hzero | hsec | |
| · exact ((map_ne_zero (dominantFunctionFieldMap u)).mpr hs hzero).elim | |
| right | |
| intro p | |
| let q := birationalPrimeLift u hu p | |
| have hq : u q.1 = p.1 := (birationalPrimeLift_spec u hu p).1 | |
| let : IsIso (u.stalkMap q.1) := (birationalPrimeLift_spec u hu p).2.1 | |
| have hp : Order.coheight (u q.1) = 1 := by rw [hq]; exact p.2 | |
| have horder := order_dominantFunctionFieldMap_of_stalk_iso u q hp s hs | |
| have h := hsec q | |
| rw [horder, hq] at h | |
| exact h | |
| end WeilPushforward | |
| section ProjectiveDescent | |
| variable (W X : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (hcomplex : u ≫ X.structureMap = W.structureMap) | |
| include hu hcomplex | |
| theorem birational_sections_linear_injection (P : WeilDivisor W.scheme) : | |
| letI := schemeFieldAlgebra (.of ℂ) W.structureMap | |
| letI := schemeFieldAlgebra (.of ℂ) X.structureMap | |
| ∃ F : divisorSectionsOver (.of ℂ) W.structureMap P →ₗ[ℂ] | |
| divisorSectionsOver (.of ℂ) X.structureMap (birationalWeilPushforward u hu P), | |
| Function.Injective F ∧ ∀ r, dominantFunctionFieldMap u (F r).1 = r.1 := by | |
| dsimp only | |
| let := schemeFieldAlgebra (.of ℂ) W.structureMap | |
| let := schemeFieldAlgebra (.of ℂ) X.structureMap | |
| obtain ⟨φ, hφ⟩ := hu | |
| let e := partialIsoFunctionFieldEquiv φ | |
| have he (s : W.scheme.functionField) : dominantFunctionFieldMap u (e s) = s := by | |
| apply e.injective | |
| exact partialIso_dominantFunctionFieldMap u φ hφ _ | |
| let F : divisorSectionsOver (.of ℂ) W.structureMap P →ₗ[ℂ] | |
| divisorSectionsOver (.of ℂ) X.structureMap (birationalWeilPushforward u ⟨φ, hφ⟩ P) := { | |
| toFun := fun s => ⟨e s.1, divisorSection_birational_descent u ⟨φ, hφ⟩ (by | |
| rw [he] | |
| exact s.2)⟩ | |
| map_add' := fun _ _ => Subtype.ext (map_add e _ _) | |
| map_smul' := by | |
| intro a s | |
| apply Subtype.ext | |
| apply (dominantFunctionFieldMap u).injective | |
| change dominantFunctionFieldMap u (e (schemeFieldScalar (.of ℂ) W.structureMap a * s.1)) = | |
| dominantFunctionFieldMap u (schemeFieldScalar (.of ℂ) X.structureMap a * e s.1) | |
| rw [he, map_mul, dominantFunctionFieldMap_scalar _ _ _ _ hcomplex, he] } | |
| refine ⟨F, ?_, fun r => he r.1⟩ | |
| intro r s h | |
| exact Subtype.ext (e.injective (congrArg Subtype.val h)) | |
| theorem sectionDimension_le_birationalWeilPushforward (P : WeilDivisor W.scheme) : | |
| sectionDimension W P ≤ sectionDimension X (birationalWeilPushforward u hu P) := by | |
| obtain ⟨F, hF, _⟩ := birational_sections_linear_injection W X u hu hcomplex P | |
| exact sectionDimension_le_of_injective W X _ _ F hF | |
| theorem exists_fixed_ample_twist_on_birational_model | |
| {D : WeilDivisor X.scheme} {L : WeilDivisor W.scheme} | |
| (hL : IsCartierPullback u D L) (P : WeilDivisor W.scheme) : | |
| ∃ A : WeilDivisor X.scheme, IsAmpleDivisor A ∧ ∀ m : ℕ, | |
| sectionDimension W (m • L + P) ≤ sectionDimension X (m • D + A) := by | |
| obtain ⟨A, hA, hAP⟩ := exists_fixed_ample_twist_sectionDimension_le X D | |
| (birationalWeilPushforward u hu P) | |
| refine ⟨A, hA, fun m => ?_⟩ | |
| have hdes := sectionDimension_le_birationalWeilPushforward W X u hu hcomplex (m • L + P) | |
| rw [map_add, map_nsmul, birationalWeilPushforward_cartierPullback u hu hL] at hdes | |
| exact hdes.trans (hAP m) | |
| theorem no_quadratic_growth_on_birational_model | |
| {n : ℕ} {D : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) (hn : 2 ≤ n) | |
| {L : WeilDivisor W.scheme} (hL : IsCartierPullback u D L) | |
| (P : WeilDivisor W.scheme) {r : ℕ} (hr : 0 < r) | |
| {c : ℝ≥0∞} (hc : 0 < c) : | |
| ¬ ∀ᶠ m in Filter.atTop, | |
| c * (r * m : ℕ) ^ 2 ≤ sectionDimension W ((r * m) • L + P) := by | |
| obtain ⟨A, hA, hAP⟩ := exists_fixed_ample_twist_on_birational_model W X u hu hcomplex hL P | |
| intro hgrowth | |
| apply no_quadratic_growth_on_multiples_of_kappaSigma_eq_one X hsigma hn hA hr hc | |
| filter_upwards [hgrowth] with m hm | |
| exact hm.trans (hAP (r * m)) | |
| end ProjectiveDescent | |
| end NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory | |
| open scoped TensorProduct nonZeroDivisors | |
| open scoped TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Limits | |
| open Algebra KaehlerDifferential IsLocalRing TensorProduct | |
| open AlgebraicGeometry CategoryTheory TensorProduct | |
| open TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set Topology | |
| open AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization | |
| open scoped IntermediateField.algebraAdjoinAdjoin | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Filter | |
| open Opposite TopCat | |
| open AlgebraicGeometry CategoryTheory TopCat Opposite TopologicalSpace | |
| open CategoryTheory.Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite TopCat | |
| open CategoryTheory TopologicalSpace Opposite | |
| namespace NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory Filter | |
| open scoped ENNReal | |
| theorem sectionDimension_mono (X : ComplexProjectiveVariety) [StalkwiseNormal X.scheme] | |
| {D E : WeilDivisor X.scheme} (hDE : D ≤ E) : | |
| sectionDimension X D ≤ sectionDimension X E := by | |
| apply sectionDimension_le_of_nonzero_section X (s := 1) one_ne_zero | |
| right | |
| intro p | |
| simpa only [order_one, zero_add, Finsupp.sub_apply] using sub_nonneg.mpr (hDE p) | |
| theorem sectionDimension_le_of_cartierPullback | |
| (W V : ComplexProjectiveVariety) [StalkwiseNormal W.scheme] [StalkwiseNormal V.scheme] | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hq : q ≫ V.structureMap = W.structureMap) | |
| {D : WeilDivisor V.scheme} {P : WeilDivisor W.scheme} | |
| (hP : IsCartierPullback q D P) : sectionDimension V D ≤ sectionDimension W P := by | |
| obtain ⟨F, hF, _⟩ := dominant_sections_linear_injection W V q hq D P hP | |
| exact sectionDimension_le_of_injective V W D P F hF | |
| theorem exists_fixed_ample_twist_of_effective_comparison | |
| (W X V : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] [StalkwiseNormal V.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (huC : u ≫ X.structureMap = W.structureMap) | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hqC : q ≫ V.structureMap = W.structureMap) | |
| {D : WeilDivisor X.scheme} {L P : WeilDivisor V.scheme} | |
| {K M Q : WeilDivisor W.scheme} | |
| (hK : IsCartierPullback u D K) (hM : IsCartierPullback q L M) | |
| (hQ : IsCartierPullback q P Q) (hcomparison : M ≤ K) : | |
| ∃ A : WeilDivisor X.scheme, IsAmpleDivisor A ∧ ∀ m : ℕ, | |
| sectionDimension V (m • L + P) ≤ sectionDimension X (m • D + A) := by | |
| obtain ⟨A, hA, hAW⟩ := exists_fixed_ample_twist_on_birational_model W X u hu huC hK Q | |
| refine ⟨A, hA, fun m => ?_⟩ | |
| calc | |
| sectionDimension V (m • L + P) ≤ sectionDimension W (m • M + Q) := | |
| sectionDimension_le_of_cartierPullback W V q hqC ((hM.nsmul m).add hQ) | |
| _ ≤ sectionDimension W (m • K + Q) := | |
| sectionDimension_mono W (by | |
| simpa only [add_comm Q] using add_le_add_right (nsmul_le_nsmul_right hcomparison m) Q) | |
| _ ≤ sectionDimension X (m • D + A) := hAW m | |
| theorem no_quadratic_growth_under_effective_comparison | |
| (W X V : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] [StalkwiseNormal V.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (huC : u ≫ X.structureMap = W.structureMap) | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hqC : q ≫ V.structureMap = W.structureMap) | |
| {n : ℕ} {D : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) (hn : 2 ≤ n) | |
| {L P : WeilDivisor V.scheme} {K M Q : WeilDivisor W.scheme} | |
| (hK : IsCartierPullback u D K) (hM : IsCartierPullback q L M) | |
| (hQ : IsCartierPullback q P Q) (hcomparison : M ≤ K) | |
| {r : ℕ} (hr : 0 < r) {c : ℝ≥0∞} (hc : 0 < c) : | |
| ¬ ∀ᶠ m in atTop, | |
| c * (r * m : ℕ) ^ 2 ≤ sectionDimension V ((r * m) • L + P) := by | |
| obtain ⟨A, hA, hAX⟩ := exists_fixed_ample_twist_of_effective_comparison W X V | |
| u hu huC q hqC hK hM hQ hcomparison | |
| intro hgrowth | |
| apply no_quadratic_growth_on_multiples_of_kappaSigma_eq_one X hsigma hn hA hr hc | |
| filter_upwards [hgrowth] with m hm | |
| exact hm.trans (hAX (r * m)) | |
| end NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory | |
| open scoped TensorProduct nonZeroDivisors | |
| open scoped TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Limits | |
| open Algebra KaehlerDifferential IsLocalRing TensorProduct | |
| open AlgebraicGeometry CategoryTheory TensorProduct | |
| open TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set Topology | |
| open AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization | |
| open scoped IntermediateField.algebraAdjoinAdjoin | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Filter | |
| open Opposite TopCat | |
| open AlgebraicGeometry CategoryTheory TopCat Opposite TopologicalSpace | |
| open CategoryTheory.Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite TopCat | |
| open CategoryTheory TopologicalSpace Opposite | |
| namespace NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory Filter | |
| open scoped ENNReal | |
| theorem exists_fixed_ample_twist_of_indexed_effective_comparison | |
| (W X V : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] [StalkwiseNormal V.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (huC : u ≫ X.structureMap = W.structureMap) | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hqC : q ≫ V.structureMap = W.structureMap) | |
| {d : ℕ} {D : WeilDivisor X.scheme} {L P : WeilDivisor V.scheme} | |
| {K M Q : WeilDivisor W.scheme} | |
| (hK : IsCartierPullback u (d • D) K) (hM : IsCartierPullback q L M) | |
| (hQ : IsCartierPullback q P Q) (hcomparison : M ≤ K) : | |
| ∃ A : WeilDivisor X.scheme, IsAmpleDivisor A ∧ ∀ m : ℕ, | |
| sectionDimension V (m • L + P) ≤ sectionDimension X ((d * m) • D + A) := by | |
| obtain ⟨A, hA, hAV⟩ := exists_fixed_ample_twist_of_effective_comparison W X V | |
| u hu huC q hqC hK hM hQ hcomparison | |
| refine ⟨A, hA, fun m => ?_⟩ | |
| simpa only [smul_smul, Nat.mul_comm m d] using hAV m | |
| theorem no_quadratic_growth_under_indexed_effective_comparison | |
| (W X V : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] [StalkwiseNormal V.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (huC : u ≫ X.structureMap = W.structureMap) | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hqC : q ≫ V.structureMap = W.structureMap) | |
| {n d : ℕ} (hd : 0 < d) {D : WeilDivisor X.scheme} | |
| (hsigma : kappaSigma X n D = (1 : WithBot ℕ)) (hn : 2 ≤ n) | |
| {L P : WeilDivisor V.scheme} {K M Q : WeilDivisor W.scheme} | |
| (hK : IsCartierPullback u (d • D) K) (hM : IsCartierPullback q L M) | |
| (hQ : IsCartierPullback q P Q) (hcomparison : M ≤ K) | |
| {r : ℕ} (hr : 0 < r) {c : ℝ≥0∞} (hc : 0 < c) : | |
| ¬ ∀ᶠ m in atTop, | |
| c * (r * m : ℕ) ^ 2 ≤ sectionDimension V ((r * m) • L + P) := by | |
| obtain ⟨A, hA, hAX⟩ := exists_fixed_ample_twist_of_indexed_effective_comparison W X V | |
| u hu huC q hqC hK hM hQ hcomparison | |
| have hd0 : (d : ℝ≥0∞) ^ 2 ≠ 0 := by simp [Nat.ne_of_gt hd] | |
| have hdt : (d : ℝ≥0∞) ^ 2 ≠ ⊤ := ENNReal.pow_ne_top (ENNReal.natCast_ne_top _) | |
| have hpos : 0 < c / (d : ℝ≥0∞) ^ 2 := ENNReal.div_pos (ne_of_gt hc) hdt | |
| intro hgrowth | |
| apply no_quadratic_growth_on_multiples_of_kappaSigma_eq_one X hsigma hn hA | |
| (Nat.mul_pos hd hr) hpos | |
| filter_upwards [hgrowth] with m hm | |
| have hscale : c / (d : ℝ≥0∞) ^ 2 * ((d * r) * m : ℕ) ^ 2 = | |
| c * (r * m : ℕ) ^ 2 := by | |
| rw [Nat.mul_assoc, Nat.cast_mul, mul_pow, ← mul_assoc, | |
| ENNReal.div_mul_cancel hd0 hdt] | |
| rw [hscale, Nat.mul_assoc] | |
| exact hm.trans (hAX (r * m)) | |
| end NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory | |
| open scoped TensorProduct nonZeroDivisors | |
| open scoped TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open CategoryTheory Opposite AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Limits | |
| open Algebra KaehlerDifferential IsLocalRing TensorProduct | |
| open AlgebraicGeometry CategoryTheory TensorProduct | |
| open TensorProduct | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Set Topology | |
| open AlgebraicGeometry TopologicalSpace | |
| open AlgebraicGeometry CategoryTheory HomogeneousLocalization | |
| open scoped IntermediateField.algebraAdjoinAdjoin | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Filter | |
| open Opposite TopCat | |
| open AlgebraicGeometry CategoryTheory TopCat Opposite TopologicalSpace | |
| open CategoryTheory.Limits | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite TopCat | |
| open CategoryTheory TopologicalSpace Opposite | |
| namespace NumericalDimensionOne | |
| open AlgebraicGeometry CategoryTheory Filter | |
| open scoped ENNReal | |
| theorem exists_indexed_ample_twist_of_rational_comparison | |
| (W X V : ComplexProjectiveVariety) | |
| [StalkwiseNormal W.scheme] [StalkwiseNormal X.scheme] [StalkwiseNormal V.scheme] | |
| (u : W.scheme ⟶ X.scheme) [IsProper u] [IsDominant u] | |
| (hu : IsBirationalMorphism u) (huC : u ≫ X.structureMap = W.structureMap) | |
| (q : W.scheme ⟶ V.scheme) [IsDominant q] | |
| (hqC : q ≫ V.structureMap = W.structureMap) | |
| {D : WeilDivisor X.scheme} (hD : IsCartierDivisor D) | |
| {L : QWeilDivisor V.scheme} {K M : QWeilDivisor W.scheme} | |
| (hK : IsQCartierPullback u (rationalWeilDivisor D) K) | |
| (hM : IsQCartierPullback q L M) (hcomparison : M ≤ K) | |
| (P : WeilDivisor V.scheme) (hP : IsCartierDivisor P) : | |
| ∃ d : ℕ, 0 < d ∧ ∃ L' : WeilDivisor V.scheme, | |
| IsCartierDivisor L' ∧ (d : ℚ) • L = rationalWeilDivisor L' ∧ | |
| ∃ A : WeilDivisor X.scheme, IsAmpleDivisor A ∧ ∀ m : ℕ, | |
| sectionDimension V (m • L' + P) ≤ sectionDimension X ((d * m) • D + A) := by | |
| obtain ⟨K', hK'⟩ := exists_cartierPullback u D hD | |
| have heK : K = rationalWeilDivisor K' := hK.eq_integral hK' | |
| obtain ⟨d, hd, L', M', hL', heL, heM, hM'⟩ := hM | |
| obtain ⟨Q, hQ⟩ := exists_cartierPullback q P hP | |
| have hle : M' ≤ d • K' := by | |
| intro p | |
| have hcoeff := DFunLike.congr_fun heM p | |
| change (d : ℚ) * M p = (M' p : ℚ) at hcoeff | |
| have hp := hcomparison p | |
| rw [heK] at hp | |
| change M p ≤ (K' p : ℚ) at hp | |
| have hmul := mul_le_mul_of_nonneg_left hp (Nat.cast_nonneg d : (0 : ℚ) ≤ d) | |
| rw [hcoeff] at hmul | |
| change M' p ≤ (d : ℤ) * K' p | |
| exact_mod_cast hmul | |
| obtain ⟨A, hA, hAX⟩ := exists_fixed_ample_twist_of_indexed_effective_comparison W X V | |
| u hu huC q hqC (hK'.nsmul d) hM' hQ hle | |
| exact ⟨d, hd, L', hL', heL, A, hA, hAX⟩ | |
| end NumericalDimensionOne | |
| end OAI | |