| import OAI.AlgebraicGeometry.PlaneCurves.GraphOrder |
| import OAI.AlgebraicGeometry.PlaneCurves.NormalPolynomials |
|
|
| namespace OAI |
|
|
| /-! |
| # Homogeneous multiplicity in normal frames and swapped charts; Homogeneous smooth curve charts |
| -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology |
|
|
| /-- Actual affine chart coordinates of a homogeneous analytic parametrization. -/ |
| def normalizedCurvePoint (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z : ℂ) : ComplexPlane := |
| (X (c.succAbove 0) z / X c z, X (c.succAbove 1) z / X c z) |
|
|
| theorem normalizedCurvePoint_analyticAt |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z₀ : ℂ) |
| (hX : ∀ i, AnalyticAt ℂ (X i) z₀) (hc : X c z₀ ≠ 0) : |
| AnalyticAt ℂ (normalizedCurvePoint X c) z₀ := |
| ((hX (c.succAbove 0)).div (hX c) hc).prod |
| ((hX (c.succAbove 1)).div (hX c) hc) |
|
|
| /-- The actual homogeneous curve equation implies the actual affine chart equation. -/ |
| theorem normalizedCurvePoint_equation |
| (G : MvPolynomial (Fin 3) ℂ) (k : ℕ) (hG : G.IsHomogeneous k) |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z : ℂ) |
| (hc : X c z ≠ 0) (hzero : MvPolynomial.eval (fun a => X a z) G = 0) : |
| planePolynomialEval (Nagata.W27.directChartHom c G) (normalizedCurvePoint X c z) = 0 := by |
| let x : Fin 3 → ℂ := fun a => X a z |
| let v : Fin 2 → ℂ := fun a => x (c.succAbove a) / x c |
| let y : Fin 3 → ℂ := Fin.insertNth (α := fun _ : Fin 3 => ℂ) c 1 v |
| have hy : (fun a => x c * y a) = x := by |
| exact Nagata.Workers.W17.scaled_chart_vector c x hc |
| have hh := Nagata.W16.homogeneous_eval_scale hG y (x c) |
| rw [hy, hzero] at hh |
| have hz : MvPolynomial.eval y G = 0 := |
| (mul_eq_zero.mp hh.symm).resolve_left (pow_ne_zero k hc) |
| have he : planePolynomialEval (Nagata.W27.directChartHom c G) (normalizedCurvePoint X c z) = |
| MvPolynomial.eval v (Nagata.W27.directChartHom c G) := by |
| apply congrArg (fun a : Fin 2 → ℂ => MvPolynomial.eval a (Nagata.W27.directChartHom c G)) |
| funext a |
| fin_cases a <;> rfl |
| rw [he, Nagata.Workers.W17.eval_directChartHom] |
| exact hz |
|
|
| /-- The homogeneous-to-affine equation holds on an actual neighborhood where |
| the chosen analytic homogeneous coordinate stays nonzero. -/ |
| theorem normalizedCurvePoint_equation_locally |
| (G : MvPolynomial (Fin 3) ℂ) (k : ℕ) (hG : G.IsHomogeneous k) |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z₀ : ℂ) |
| (ha : ContinuousAt (X c) z₀) (hc : X c z₀ ≠ 0) |
| (hzero : ∀ᶠ z in 𝓝 z₀, MvPolynomial.eval (fun a => X a z) G = 0) : |
| ∀ᶠ z in 𝓝 z₀, planePolynomialEval (Nagata.W27.directChartHom c G) |
| (normalizedCurvePoint X c z) = 0 := by |
| filter_upwards [ha.eventually_ne hc, hzero] with z hcz hz |
| exact normalizedCurvePoint_equation G k hG X c z hcz hz |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| theorem homogeneous_normal_frame_identity_plane |
| (I : Finset ℕ) (j : ℕ → ℕ) (T : ℕ → MvPolynomial (Fin 3) ℂ) |
| (d k : ℕ) (hhom : ∀ α ∈ I, (T α).IsHomogeneous (d-k*j α)) |
| (hbound : ∀ α ∈ I, k*j α ≤ d) |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z w : ℂ) (hc : X c z ≠ 0) : |
| (∑ α ∈ I, w ^ j α * MvPolynomial.eval (fun a => X a z) (T α)) = |
| (X c z)^d * ∑ α ∈ I, (w/(X c z)^k)^j α * |
| planePolynomialEval (Nagata.W27.directChartHom c (T α)) (normalizedCurvePoint X c z) := by |
| rw [Nagata.Workers.W17.normal_polynomial_frame_identity I j T d k hhom hbound c (fun a => X a z) hc w] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro α hα |
| congr 1 |
| apply congrArg (fun v : Fin 2 → ℂ => MvPolynomial.eval v (Nagata.W27.directChartHom c (T α))) |
| funext a |
| fin_cases a <;> rfl |
|
|
| /-- Central normal order implies actual homogeneous polynomial order on an |
| analytic parametrized curve, with the exact variable degree-d frame factor. -/ |
| theorem homogeneous_normal_order_of_central_order |
| (I : Finset ℕ) (j : ℕ → ℕ) (T : ℕ → MvPolynomial (Fin 3) ℂ) |
| (d k : ℕ) (hhom : ∀ α ∈ I, (T α).IsHomogeneous (d-k*j α)) |
| (hbound : ∀ α ∈ I, k*j α ≤ d) |
| (G : MvPolynomial (Fin 3) ℂ) (hG : G.IsHomogeneous k) |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z₀ w₀ : ℂ) |
| (hX : ∀ a, AnalyticAt ℂ (X a) z₀) (hc : X c z₀ ≠ 0) |
| (hzero : ∀ᶠ z in 𝓝 z₀, MvPolynomial.eval (fun a => X a z) G = 0) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = polynomialNormalCoordinates (Nagata.W27.directChartHom c G)) |
| (hsource : normalizedCurvePoint X c z₀ ∈ e.source) (m : ℕ) |
| (horder : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| q.2 ^ j α * planePolynomialEval (Nagata.W27.directChartHom c (T α)) (e.symm (q.1, 0))) |
| ((normalizedCurvePoint X c z₀).1, w₀ / X c z₀ ^ k) m) : |
| HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| q.2 ^ j α * MvPolynomial.eval (fun a => X a q.1) (T α)) (z₀, w₀) m := by |
| have hp := normalizedCurvePoint_analyticAt X c z₀ hX hc |
| have hcurve := normalizedCurvePoint_equation_locally G k hG X c z₀ (hX c).continuousAt hc hzero |
| have h := parametric_frame_order_of_central_order I j |
| (fun α => Nagata.W27.directChartHom c (T α)) (Nagata.W27.directChartHom c G) |
| e he (normalizedCurvePoint X c) (X c) d k z₀ w₀ hp (hX c) hc hsource hcurve m horder |
| apply h.congr |
| have hne := (hX c).continuousAt.eventually_ne hc |
| have hf : ContinuousAt (Prod.fst : ComplexPlane → ℂ) (z₀, w₀) := continuousAt_fst |
| exact (hf.tendsto.eventually hne).mono fun q hq => by |
| exact (homogeneous_normal_frame_identity_plane I j T d k hhom hbound X c q.1 q.2 hq).symm |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| /-- Swapping actual affine coordinates preserves analyticity of a curve. -/ |
| theorem analyticAt_curve_swap (p : ℂ → ComplexPlane) (z₀ : ℂ) |
| (hp : AnalyticAt ℂ p z₀) : AnalyticAt ℂ (fun z => (p z).swap) z₀ := by |
| exact (((ContinuousLinearMap.snd ℂ ℂ ℂ).analyticAt (p z₀)).comp (f := p) hp).prod |
| (((ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt (p z₀)).comp (f := p) hp) |
|
|
| /-- The alternative tangential coordinate changes neither the homogeneous |
| normal expression nor its variable frame and fiber weights. -/ |
| theorem homogeneous_normal_order_of_swapped_central_order |
| (I : Finset ℕ) (j : ℕ → ℕ) (T : ℕ → MvPolynomial (Fin 3) ℂ) |
| (d k : ℕ) (hhom : ∀ α ∈ I, (T α).IsHomogeneous (d-k*j α)) |
| (hbound : ∀ α ∈ I, k*j α ≤ d) |
| (G : MvPolynomial (Fin 3) ℂ) (hG : G.IsHomogeneous k) |
| (X : Fin 3 → ℂ → ℂ) (c : Fin 3) (z₀ w₀ : ℂ) |
| (hX : ∀ a, AnalyticAt ℂ (X a) z₀) (hc : X c z₀ ≠ 0) |
| (hzero : ∀ᶠ z in 𝓝 z₀, MvPolynomial.eval (fun a => X a z) G = 0) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = polynomialNormalCoordinates |
| (Nagata.Workers.W17.affineSwap (Nagata.W27.directChartHom c G))) |
| (hsource : (normalizedCurvePoint X c z₀).swap ∈ e.source) (m : ℕ) |
| (horder : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| q.2 ^ j α * planePolynomialEval |
| (Nagata.Workers.W17.affineSwap (Nagata.W27.directChartHom c (T α))) (e.symm (q.1, 0))) |
| ((normalizedCurvePoint X c z₀).2, w₀ / X c z₀ ^ k) m) : |
| HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| q.2 ^ j α * MvPolynomial.eval (fun a => X a q.1) (T α)) (z₀, w₀) m := by |
| have hp := analyticAt_curve_swap (normalizedCurvePoint X c) z₀ |
| (normalizedCurvePoint_analyticAt X c z₀ hX hc) |
| have hcurve : ∀ᶠ z in 𝓝 z₀, |
| planePolynomialEval (Nagata.Workers.W17.affineSwap (Nagata.W27.directChartHom c G)) |
| (normalizedCurvePoint X c z).swap = 0 := by |
| simpa only [Nagata.Workers.W17.eval_affineSwap, Prod.swap_swap] using |
| normalizedCurvePoint_equation_locally G k hG X c z₀ (hX c).continuousAt hc hzero |
| have h := parametric_frame_order_of_central_order I j |
| (fun α => Nagata.Workers.W17.affineSwap (Nagata.W27.directChartHom c (T α))) |
| (Nagata.Workers.W17.affineSwap (Nagata.W27.directChartHom c G)) |
| e he (fun z => (normalizedCurvePoint X c z).swap) (X c) d k z₀ w₀ |
| hp (hX c) hc hsource hcurve m horder |
| simp only [Nagata.Workers.W17.eval_affineSwap, Prod.swap_swap] at h |
| apply h.congr |
| have hne := (hX c).continuousAt.eventually_ne hc |
| have hf : ContinuousAt (Prod.fst : ComplexPlane → ℂ) (z₀, w₀) := continuousAt_fst |
| exact (hf.tendsto.eventually hne).mono fun q hq => by |
| exact (homogeneous_normal_frame_identity_plane I j T d k hhom hbound X c q.1 q.2 hq).symm |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
