| 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 |
| |
