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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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Intersection.SurfaceTwistEuler
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective
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
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
end
end MaximalSeshadri.Geometry
end
end OAI
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