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7f5fab8 | 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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Divisors.SectionChart
import OAI.AlgebraicGeometry.Seshadri.Divisors.MixedSections
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory
open MaximalSeshadri.Frames MaximalSeshadri.Projective
variable {X : Scheme}
lemma normalized_mixed_coefficient {M : X.Modules} (s₀ : O X ⟶ M)
(U : X.Opens) (hU : U ≤ SectionOpens.isoOpen s₀) (n : ℕ)
(s : Fin n → (O X ⟶ M)) :
coefficient (sectionFrameOn (powerSection s₀ n) U
(hU.trans (sectionOpen_le_powerSection s₀ n)))
(restrictSection U.ι (mixedPowerSection n s)) =
∏ i, coefficient (sectionFrameOn s₀ U hU) (restrictSection U.ι (s i)) := by
let e := sectionFrameOn s₀ U hU
let f := sectionFrameOn (powerSection s₀ n) U
(hU.trans (sectionOpen_le_powerSection s₀ n))
let c := frameChange (localPowerFrame U e n) f
have hc : (c : Γ(U.toScheme, ⊤)) * endValue (powerRestrictionUnit U n).hom = 1 := by
have hh := coefficient_change (localPowerFrame U e n) f
(restrictSection U.ι (powerSection s₀ n))
rw [local_powerSection_coefficient U e s₀ n,
sectionFrameOn_normalized (powerSection s₀ n) U
(hU.trans (sectionOpen_le_powerSection s₀ n)),
sectionFrameOn_normalized s₀ U hU, one_pow, mul_one] at hh
exact hh.symm
change coefficient f _ = _
rw [coefficient_change (localPowerFrame U e n) f,
local_mixedPowerSection_coefficient U e n s, ← mul_assoc]
exact (congrArg (fun value => value * ∏ i,
coefficient e (restrictSection U.ι (s i))) hc).trans (one_mul _)
def quarticSection {M : X.Modules} (a b c d : O X ⟶ M) :
O X ⟶ modulePow X M 4 := mixedPowerSection 4 ![a,b,c,d]
lemma quarticSection_pure {M : X.Modules} (a : O X ⟶ M) :
quarticSection a a a a = powerSection a 4 := by
apply (congrArg (mixedPowerSection 4) (show ![a,a,a,a] = fun _ => a from by
funext i; fin_cases i <;> rfl)).trans
exact mixedPowerSection_const 4 a
lemma quarticSection_normalized {M : X.Modules} (s₀ a b c d : O X ⟶ M)
(U : X.Opens) (hU : U ≤ SectionOpens.isoOpen s₀) :
coefficient (sectionFrameOn (powerSection s₀ 4) U
(hU.trans (sectionOpen_le_powerSection s₀ 4)))
(restrictSection U.ι (quarticSection a b c d)) =
coefficient (sectionFrameOn s₀ U hU) (restrictSection U.ι a) *
coefficient (sectionFrameOn s₀ U hU) (restrictSection U.ι b) *
coefficient (sectionFrameOn s₀ U hU) (restrictSection U.ι c) *
coefficient (sectionFrameOn s₀ U hU) (restrictSection U.ι d) := by
refine (normalized_mixed_coefficient s₀ U hU 4 ![a,b,c,d]).trans ?_
simp [Fin.prod_univ_succ, mul_assoc]
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
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