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12dee50 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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
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