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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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Intersection.GeneralSurfaceEuler
import OAI.AlgebraicGeometry.Seshadri.Cohomology.GeneralSurfaceH2
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Abelian TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective
theorem Surface.positive_square_eventually_effective (S : Surface)
(L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme)
(hq : 0 < selfIntersection S M) (hm : 0 ≤ mixedEuler S L M) :
∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N ≤ n →
∃ s : O S.scheme ⟶ (M.pow n).sheaf, s ≠ 0 := by
obtain ⟨B,C,hB⟩ := S.power_H2_linear_bound L hL M hm
obtain ⟨N,hN,hpos⟩ := quadratic_lower_eventually_pos (selfIntersection S M)
(eulerCharacteristic S.structureMap 2 M.sheaf -
eulerCharacteristic S.structureMap 2 (O S.scheme) - C)
(eulerCharacteristic S.structureMap 2 (O S.scheme)) hq B
refine ⟨N,hN,fun n hn => ?_⟩
have hd : 0 < cohomologyDimension S.structureMap (M.pow n).sheaf 0 := by
have hp := hpos n hn
have he := S.arbitrary_power_euler_quadratic L hL M n
rw [surface_euler_expansion] at he
have hb : (cohomologyDimension S.structureMap (M.pow n).sheaf 2 : ℤ) ≤
B+(C:ℤ)*n := by exact_mod_cast hB n
have hi : (0 : ℤ) ≤ cohomologyDimension S.structureMap (M.pow n).sheaf 1 := Nat.cast_nonneg _
have H : (0 : ℤ) < 2*(cohomologyDimension S.structureMap (M.pow n).sheaf 0 : ℤ) := by
nlinarith
exact_mod_cast (by omega : (0 : ℤ) < cohomologyDimension S.structureMap (M.pow n).sheaf 0)
let := Module.compHom (cohomology (M.pow n).sheaf 0) (baseScalars S.structureMap)
let := S.H0_finite L hL (M.pow n)
let : Nontrivial (cohomology (M.pow n).sheaf 0) := Module.finrank_pos_iff.mp hd
obtain ⟨x,hx⟩ := exists_ne (0 : cohomology (M.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 _ _))
theorem Surface.positive_square_effective_divisible (S : Surface)
(L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme)
(hq : 0 < selfIntersection S M) (hm : 0 ≤ mixedEuler S L M)
(q : ℕ) (hqpos : 0 < q) :
∃ K : ℕ, 0 < K ∧ ∀ k : ℕ, K ≤ k → q ∣ k →
∃ s : O S.scheme ⟶ (M.pow (k/q)).sheaf, s ≠ 0 := by
obtain ⟨N,hN,h⟩ := S.positive_square_eventually_effective L hL M hq hm
refine ⟨q*N,Nat.mul_pos hqpos hN,fun k hk hdiv => h _ ?_⟩
rw [Nat.le_div_iff_mul_le hqpos]
simpa only [Nat.mul_comm] using hk
end
end MaximalSeshadri.Geometry
end
end OAI
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