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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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import Mathlib
import OAI.AlgebraicGeometry.LogKodaira.Descent
import OAI.AlgebraicGeometry.LogKodaira.Blowup
namespace OAI
noncomputable section
open CategoryTheory AlgebraicGeometry
open scoped TensorProduct
universe u v
namespace ReverseLogKodaira
theorem normal_root_zero_test
{R : Type u} {K : Type v}
[CommRing R] [IsDomain R] [Field K] [Algebra R K]
[IsFractionRing R K] [IsIntegrallyClosed R]
{m e : ℕ} (hm : 0 < m) (he : m ∣ e)
{t s : R} (ht : t ≠ 0) (hs : t ^ e ∣ s)
{r : K} (hr : r ^ m = algebraMap R K s) :
∃ c : R, algebraMap R K (t ^ (e / m) * c) = r := by
obtain ⟨c, hc⟩ := hs
have htK : algebraMap R K t ≠ 0 :=
(map_ne_zero_iff (algebraMap R K) (IsFractionRing.injective R K)).mpr ht
have heq : (e / m) * m = e := Nat.div_mul_cancel he
have hpow : (r / algebraMap R K (t ^ (e / m))) ^ m = algebraMap R K c := by
rw [div_pow, hr, hc, map_mul, map_pow, map_pow, ← pow_mul, heq]
exact mul_div_cancel_left₀ _ (pow_ne_zero _ htK)
obtain ⟨d, hd⟩ := IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow
(R := R) hm (hpow ▸ isIntegral_algebraMap)
refine ⟨d, ?_⟩
rw [map_mul, hd]
exact mul_div_cancel₀ _ (by simpa only [map_pow] using pow_ne_zero (e / m) htK)
end ReverseLogKodaira
namespace ReverseLogKodaira.PluriformDescent
open SmoothProjectiveVariety DifferentialBaseChange CanonicalSpecialization
open FiberChartSpecialization
theorem original_fiber_zero_ramified_root_factor
{X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary} {D : Y.ReducedSNCBoundary}
(f : StratumSmoothFibration X Y E D) (m : ℕ) (s : X.RationalPluriform m)
(y : Y.ComplexPoint) (hy : y.point ∈ D.complement)
(hzero : FiberVanishesAt f m s y hy)
(F : FiberModel f.toBoundaryFibration y)
(U : Y.scheme.affineOpens) (hU : U.1 ≤ D.complement) (hyU : y.point ∈ U.1)
(b : Module.Basis (Fin Y.dimension) Γ(Y.scheme, U.1)
(KaehlerDifferential ℂ Γ(Y.scheme, U.1)))
(P : NumeratorChart f.toBoundaryFibration y F U (fiber_dimension_eq f y hy F) m s)
{B K : Type*} [CommRing B] [IsDomain B] [IsIntegrallyClosed B]
[Field K] [Algebra B K] [IsFractionRing B K]
(z : RingHom.ker (complexPointEvaluation y U.1 hyU))
(g : Γ(X.scheme, P.V.1) →+* B)
(h : Blowup.Chart (RingHom.ker (complexPointEvaluation y U.1 hyU)) z →+* B)
(square : g.comp (f.hom.appLE U.1 P.V.1 P.overOpen).hom =
h.comp (Blowup.chartMap (RingHom.ker (complexPointEvaluation y U.1 hyU)) z))
{e : ℕ} (hm : 0 < m) (he : m ∣ e) (τ : B) (hτ : τ ≠ 0)
(ramification : h (Blowup.chartMap
(RingHom.ker (complexPointEvaluation y U.1 hyU)) z z) = τ ^ e) :
letI : Algebra Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) :=
(f.hom.appLE U.1 P.V.1 P.overOpen).hom.toAlgebra
letI : IsScalarTower ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) :=
IsScalarTower.of_algebraMap_eq fun a =>
(appLE_constants f.hom f.over_base U.1 P.V.1 P.overOpen a).symm
letI := P.formallySmooth
∀ r : K,
r ^ m = algebraMap B K (g (pluricanonicalCoordinate
ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1)
Y.dimension F.variety.dimension m b P.relativeBasis P.numerator)) →
∃ c : B, algebraMap B K (τ ^ (e / m) * c) = r := by
let : Algebra Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) :=
(f.hom.appLE U.1 P.V.1 P.overOpen).hom.toAlgebra
let : IsScalarTower ℂ Γ(Y.scheme, U.1) Γ(X.scheme, P.V.1) :=
IsScalarTower.of_algebraMap_eq fun a =>
(appLE_constants f.hom f.over_base U.1 P.V.1 P.overOpen a).symm
let := P.formallySmooth
intro r hr
have hmem := original_coefficient_in_point_ideal_of_fiber_zero
f m s y hy hzero F U hU hyU b P
have hdiv := Blowup.parameter_dvd_image
(RingHom.ker (complexPointEvaluation y U.1 hyU)) z
(f.hom.appLE U.1 P.V.1 P.overOpen).hom g h square hmem
rw [ramification] at hdiv
exact normal_root_zero_test hm he hτ hdiv hr
end ReverseLogKodaira.PluriformDescent
end
end OAI
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