openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /SurfacePositiveSquare.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.PullbackSectionOpenExact | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.SurfaceTopGrowth | |
| namespace OAI | |
| section | |
| namespace MaximalSeshadri.Geometry | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Abelian TopologicalSpace | |
| open MaximalSeshadri.Frames MaximalSeshadri.Projective | |
| theorem IntegralCurve.ample_degree_positive (S : Surface) (C : IntegralCurve S) | |
| (L : LineBundle S.scheme) (hL : L.IsAmple) : 0 < curveDegree S L C := by | |
| obtain ⟨m,hm,hp⟩ := C.exists_positive_ambient_power S L hL | |
| rw [curveDegree_pow S L hL] at hp | |
| have hm' : (0 : ℤ) < m := by exact_mod_cast hm | |
| nlinarith | |
| theorem Surface.selfIntersection_pos (S : Surface) (L : LineBundle S.scheme) (hL : L.IsAmple) : | |
| 0 < selfIntersection S L := by | |
| obtain ⟨d,hd,-,N,s,hs,v,hne,hi,hC⟩ := S.coprime_integral_section L hL 1 (by decide) | |
| let := hi | |
| let k := S.structureMap.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom | |
| let C : IntegralCurve S := ⟨(sectionIdeal k s hs v).subscheme, | |
| (sectionIdeal k s hs v).subschemeι,inferInstance,inferInstance,hC⟩ | |
| have hp := C.ample_degree_positive S L hL | |
| have he := generated_curveDegree_eq_power_selfIntersection S L hL d k s hs v hne hC | |
| change curveDegree S L C = (d : ℤ)*selfIntersection S L at he | |
| rw [he] at hp | |
| have hd' : (0 : ℤ) < d := by exact_mod_cast hd | |
| nlinarith | |
| theorem generated_power_curveDegree_eq (S : Surface) (L : LineBundle S.scheme) | |
| (hL : L.IsAmple) (d a : ℕ) {σ : Type} [Fintype σ] | |
| (k : ℂ →+* Γ(S.scheme,⊤)) (s : σ → (O S.scheme ⟶ (L.pow d).sheaf)) | |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) (v : σ → ℂ) | |
| (hne : sectionCombination k s v ≠ 0) | |
| [IsIntegral (sectionIdeal k s hs v).subscheme] | |
| (hd : topologicalKrullDim (sectionIdeal k s hs v).subscheme = 1) : | |
| let C : IntegralCurve S := ⟨(sectionIdeal k s hs v).subscheme, | |
| (sectionIdeal k s hs v).subschemeι,inferInstance,inferInstance,hd⟩ | |
| curveDegree S (L.pow a) C = (a : ℤ)*d*selfIntersection S L := by | |
| dsimp only | |
| rw [curveDegree_pow S L hL,generated_curveDegree_eq_power_selfIntersection S L hL d k s hs v hne hd] | |
| ring | |
| lemma quadratic_lower_eventually_pos (q l c : ℤ) (hq : 0 < q) (B : ℕ) : | |
| ∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N ≤ n → | |
| 0 < (n : ℤ)*((n : ℤ)-1)*q + 2*(n : ℤ)*l + 2*c - 2*B := by | |
| let N := (2*|l|+2*|c|+2*(B : ℤ)+3).toNat | |
| have hN : (N : ℤ) = 2*|l|+2*|c|+2*(B : ℤ)+3 := Int.toNat_of_nonneg (by positivity) | |
| refine ⟨N,?_,fun n hn => ?_⟩ | |
| · have : (0 : ℤ) < N := by rw [hN]; positivity | |
| exact_mod_cast this | |
| · have hn' : (N : ℤ) ≤ n := by exact_mod_cast hn | |
| rw [hN] at hn' | |
| have hn0 : (0 : ℤ) ≤ n := Nat.cast_nonneg n | |
| have hB : (0 : ℤ) ≤ B := Nat.cast_nonneg B | |
| have hq' : 1 ≤ q := hq | |
| have hbase : 0 ≤ (n : ℤ)*((n : ℤ)-1) := mul_nonneg hn0 (by omega) | |
| have hmul := mul_le_mul_of_nonneg_left hq' hbase | |
| have hl := neg_abs_le l | |
| have hc := neg_abs_le c | |
| have hlmul := mul_le_mul_of_nonneg_left hl | |
| (show (0 : ℤ) ≤ 2 * (n : ℤ) by positivity) | |
| have habsl := abs_nonneg l | |
| have habsc := abs_nonneg c | |
| nlinarith | |
| theorem Surface.eventually_nonzero_power_section (S : Surface) (L : LineBundle S.scheme) | |
| (hL : L.IsAmple) : ∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N ≤ n → | |
| ∃ s : O S.scheme ⟶ (L.pow n).sheaf, s ≠ 0 := by | |
| obtain ⟨B,hB⟩ := S.power_H0_quadratic_lower L hL | |
| obtain ⟨N,hN,hpos⟩ := quadratic_lower_eventually_pos (selfIntersection S L) | |
| (eulerCharacteristic S.structureMap 2 L.sheaf - eulerCharacteristic S.structureMap 2 (O S.scheme)) | |
| (eulerCharacteristic S.structureMap 2 (O S.scheme)) (S.selfIntersection_pos L hL) B | |
| refine ⟨N,hN,fun n hn => ?_⟩ | |
| have hd : 0 < cohomologyDimension S.structureMap (L.pow n).sheaf 0 := by | |
| have hp := hpos n hn | |
| have hb := hB n | |
| omega | |
| let := Module.compHom (cohomology (L.pow n).sheaf 0) (baseScalars S.structureMap) | |
| let := S.H0_finite L hL (L.pow n) | |
| let : Nontrivial (cohomology (L.pow n).sheaf 0) := Module.finrank_pos_iff.mp hd | |
| obtain ⟨x,hx⟩ := exists_ne (0 : cohomology (L.pow n).sheaf 0) | |
| refine ⟨Ext.homEquiv₀ x,?_⟩ | |
| intro hs | |
| apply hx | |
| have he := congrArg Ext.mk₀ hs | |
| exact (Ext.mk₀_homEquiv₀_apply x).symm.trans (he.trans (Ext.mk₀_zero _ _)) | |
| end | |
| end MaximalSeshadri.Geometry | |
| end | |
| end OAI | |