openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /PositiveSquareEffectivity.lean
| 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 | |