File size: 5,347 Bytes
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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.AlgebraicGeometry.PlaneCurves.MovingConfigurations
import OAI.AlgebraicGeometry.PlaneCurves.NormalSpecialization
namespace OAI
/-!
# Direct and normalized gradients in smooth cubic charts
-/
section
noncomputable section
namespace Nagata.Workers.W25
open scoped BigOperators
/-- The affine partial derivative is the dehomogenized surviving ambient partial. -/
theorem pderiv_directChartHom (c : Fin 3) (G : MvPolynomial (Fin 3) ℂ) (i : Fin 2) :
MvPolynomial.pderiv i (Nagata.W27.directChartHom c G) =
Nagata.W27.directChartHom c (MvPolynomial.pderiv (c.succAbove i) G) := by
classical
let f := Fin.insertNth (α := fun _ : Fin 3 => MvPolynomial (Fin 2) ℂ) c 1 MvPolynomial.X
have hderiv (j : Fin 3) : MvPolynomial.pderiv i (f j) =
if j = c.succAbove i then 1 else 0 := by
by_cases hj : j = c
· subst j
simp [f]
· obtain ⟨k, rfl⟩ := Fin.exists_succAbove_eq hj
simp [f, MvPolynomial.pderiv_X, Pi.single_apply, eq_comm]
change MvPolynomial.pderiv i (MvPolynomial.eval₂ MvPolynomial.C f G) =
MvPolynomial.eval₂ MvPolynomial.C f (MvPolynomial.pderiv (c.succAbove i) G)
rw [Nagata.Workers.W14.pderiv_polynomial_substitution]
simp_rw [hderiv, mul_ite, mul_one, mul_zero]
simp
/-- Exact derivative scaling between a cone representative and its normalized chart. -/
theorem eval_pderiv_directChartHom_scale {G : MvPolynomial (Fin 3) ℂ} {d : ℕ}
(hG : G.IsHomogeneous d) (c : Fin 3) (x : Fin 3 → ℂ) (hc : x c ≠ 0) (i : Fin 2) :
MvPolynomial.eval x (MvPolynomial.pderiv (c.succAbove i) G) =
(x c) ^ (d - 1) * MvPolynomial.eval (fun j => x (c.succAbove j) / x c)
(MvPolynomial.pderiv i (Nagata.W27.directChartHom c G)) := by
rw [pderiv_directChartHom, Nagata.Workers.W17.eval_directChartHom]
have hscale := Nagata.W16.homogeneous_eval_scale (hG.pderiv (i := c.succAbove i))
(Fin.insertNth (α := fun _ : Fin 3 => ℂ) c 1 (fun j => x (c.succAbove j) / x c)) (x c)
have hnorm : (fun a => x c * Fin.insertNth (α := fun _ : Fin 3 => ℂ) c 1
(fun j => x (c.succAbove j) / x c) a) = x :=
Nagata.Workers.W17.scaled_chart_vector c x hc
rw [hnorm] at hscale
exact hscale
/-- Every projective affine chart containing a nonsingular point has a nonzero
ordinary two-coordinate affine gradient. Euler's identity rules out the omitted
ambient partial being the only nonzero derivative. -/
theorem exists_nonzero_pderiv_directChartHom {G : MvPolynomial (Fin 3) ℂ} {d : ℕ}
(hG : G.IsHomogeneous d) (c : Fin 3) (x : Fin 3 → ℂ)
(hc : x c ≠ 0) (hzero : MvPolynomial.eval x G = 0)
(hgrad : ∃ j : Fin 3, MvPolynomial.eval x (MvPolynomial.pderiv j G) ≠ 0) :
∃ i : Fin 2, MvPolynomial.eval (fun j => x (c.succAbove j) / x c)
(MvPolynomial.pderiv i (Nagata.W27.directChartHom c G)) ≠ 0 := by
classical
by_contra h
have haff (i : Fin 2) : MvPolynomial.eval (fun j => x (c.succAbove j) / x c)
(MvPolynomial.pderiv i (Nagata.W27.directChartHom c G)) = 0 := by
by_contra hi
exact h ⟨i, hi⟩
have hamb (i : Fin 2) : MvPolynomial.eval x (MvPolynomial.pderiv (c.succAbove i) G) = 0 := by
rw [eval_pderiv_directChartHom_scale hG c x hc i, haff, mul_zero]
have heuler := congrArg (MvPolynomial.eval x) hG.sum_X_mul_pderiv
simp only [map_sum, map_mul, MvPolynomial.eval_X, map_nsmul, hzero, nsmul_zero] at heuler
have hsum : (∑ j : Fin 3, x j * MvPolynomial.eval x (MvPolynomial.pderiv j G)) =
x c * MvPolynomial.eval x (MvPolynomial.pderiv c G) := by
apply Finset.sum_eq_single c
· intro j _ hj
obtain ⟨i, rfl⟩ := Fin.exists_succAbove_eq hj
rw [hamb, mul_zero]
· simp
have hcderiv : MvPolynomial.eval x (MvPolynomial.pderiv c G) = 0 :=
(mul_eq_zero.mp (hsum.symm.trans heuler)).resolve_left hc
obtain ⟨j, hj⟩ := hgrad
by_cases hjc : j = c
· subst j
exact hj hcderiv
· obtain ⟨i, rfl⟩ := Fin.exists_succAbove_eq hjc
exact hj (hamb i)
end Nagata.Workers.W25
end
end
section
namespace Nagata.Workers.W25
open Nagata.Workers.W28
theorem planePolynomialEval_normalizedCurvePoint (P : MvPolynomial (Fin 2) ℂ)
(X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z : ℂ) :
planePolynomialEval P (normalizedCurvePoint X c z) =
MvPolynomial.eval (fun i => X (c.succAbove i) z / X c z) P := by
apply congrArg (fun v : Fin 2 → ℂ => MvPolynomial.eval v P)
funext i
fin_cases i <;> rfl
/-- Every containing direct chart satisfies the exact smoothness input to the
coordinate-independent normal-polynomial construction. -/
theorem normalizedCurvePoint_gradient_ne_zero
{G : MvPolynomial (Fin 3) ℂ} {d : ℕ} (hG : G.IsHomogeneous d)
(c : Fin 3) (X : Fin 3 → ℂ → ℂ) (z : ℂ) (hc : X c z ≠ 0)
(hzero : MvPolynomial.eval (fun j => X j z) G = 0)
(hgrad : ∃ j : Fin 3, MvPolynomial.eval (fun i => X i z) (MvPolynomial.pderiv j G) ≠ 0) :
polynomialGradient (Nagata.W27.directChartHom c G) (normalizedCurvePoint X c z) ≠ 0 := by
obtain ⟨i, hi⟩ := exists_nonzero_pderiv_directChartHom hG c (fun j => X j z) hc hzero hgrad
apply (polynomialGradient_ne_zero_iff_partials _ _).mpr
fin_cases i
· left
rwa [planePolynomialEval_normalizedCurvePoint]
· right
rwa [planePolynomialEval_normalizedCurvePoint]
end Nagata.Workers.W25
end
end OAI
|