| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.SurfaceTwistEuler |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Frames MaximalSeshadri.Projective |
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| theorem generated_curveDegree_eq_mixedEuler (S : Surface) |
| (L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme) (d : 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 M C = (d : ℤ)*mixedEuler S L M := by |
| let C : IntegralCurve S := ⟨(sectionIdeal k s hs v).subscheme, |
| (sectionIdeal k s hs v).subschemeι,inferInstance,inferInstance,hd⟩ |
| have h := generated_twist_euler_difference S L hL L M d k s hs v hne hd 0 |
| have h' := generated_power_euler_difference S L hL L d k s hs v hne hd 0 |
| have h₀ := S.twist_euler L hL M 0 |
| have h₁ := S.twist_euler L hL M d |
| simp only [Nat.zero_add,Nat.cast_zero,zero_mul,add_zero] at h h' h₀ |
| change curveDegree S M C = _ |
| linarith |
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| theorem mixedEuler_pow_right (S : Surface) (L : LineBundle S.scheme) (hL : L.IsAmple) |
| (M : LineBundle S.scheme) (n : ℕ) : |
| mixedEuler S L (M.pow n) = (n : ℤ)*mixedEuler S L M := by |
| classical |
| 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 h₁ := generated_curveDegree_eq_mixedEuler S L hL M d k s hs v hne hC |
| have h₂ := generated_curveDegree_eq_mixedEuler S L hL (M.pow n) d k s hs v hne hC |
| change curveDegree S M C = (d : ℤ)*mixedEuler S L M at h₁ |
| change curveDegree S (M.pow n) C = (d : ℤ)*mixedEuler S L (M.pow n) at h₂ |
| rw [curveDegree_pow S L hL,h₁] at h₂ |
| have hd' : (d : ℤ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hd |
| apply mul_left_cancel₀ hd' |
| rw [← h₂] |
| ring |
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| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end OAI |
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