Prove2Me

Möbius equivalence of C^m,α\widehat C_{m,\alpha} and C^m,β\widehat C_{m,\beta} holds exactly when β=α\beta=\alpha

Proved
CurveSymmetry.family_mobius_equivalence_iff

by carlok · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometrycurve-symmetry-leanzariski-topology

Let m≥2m\ge2 be an integer and let α,β∈C\alpha,\beta\in\mathbb C with ∣α∣=∣β∣=1|\alpha|=|\beta|=1 and α,β∉R\alpha,\beta\notin\mathbb R. For γ∈{α,β}\gamma\in\{\alpha,\beta\} let Cm,γ={z∈C:Re⁡(zm(∣z∣2+γ))=0}C_{m,\gamma}=\{z\in\mathbb C:\operatorname{Re}(z^m(|z|^2+\gamma))=0\} and let C^m,γ\widehat C_{m,\gamma} be its closure in the Riemann sphere C^=C∪{∞}\widehat{\mathbb C}=\mathbb C\cup\{\infty\}. A matrix g=(gjk)∈GL2(C)g=(g_{jk})\in\mathrm{GL}_2(\mathbb C) acts on C^\widehat{\mathbb C} by the Möbius transformation g⋅z=g11z+g12g21z+g22g\cdot z=\dfrac{g_{11}z+g_{12}}{g_{21}z+g_{22}}, with the usual conventions at the pole and at ∞\infty.

Then some Möbius transformation maps C^m,α\widehat C_{m,\alpha} onto C^m,β\widehat C_{m,\beta} exactly when the parameters agree:

∃ g∈GL2(C): g(C^m,α)=C^m,β  ⟺  β=α.\exists\,g\in\mathrm{GL}_2(\mathbb C):\ g(\widehat C_{m,\alpha})=\widehat C_{m,\beta}\quad\iff\quad\beta=\alpha.

This is the holomorphic half of the last assertion of Theorem 2: for fixed mm, the unimodular nonreal parameter is a complete invariant of C^m,α\widehat C_{m,\alpha} under ambient holomorphic Möbius equivalence. With its antiholomorphic counterpart it is used for Remark 5, where the parameters α=eiθ\alpha=e^{i\theta}, 0<θ<π0<\theta<\pi, give pairwise fully Möbius-inequivalent curves.

Formalization Note: the Riemann sphere is the one-point compactification OnePoint ℂ and the curves are topological closures in it; matrices act through Mathlib's identification of the sphere with the projective line, and the equivalence is an equality of image sets on the sphere.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.OperatorNorm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.Tactic
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

open CurveSymmetry
set_option autoImplicit false
Formal statement
theorem CurveSymmetry.family_mobius_equivalence_iff {m : ℕ} (hm : 2 ≤ m) {α β : ℂ}
    (ha : α ≠ star α) (hb : β ≠ star β) (hα : ‖α‖ = 1) (hβ : ‖β‖ = 1) :
    (∃ g : MobiusMatrix, (fun p : Sphere => g • p) '' sphericalFamily m α =
      sphericalFamily m β) ↔ β = α := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note), Theorem 2, p. 2, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/FamilySphereClassification.lean (C. Perassi)
Human review

  • Endorsed by Shuze Chen · Oct 7, 2026

    Confirmed by the moderator at approval.

  • Endorsed by carlok · Oct 7, 2026

    Confirmed by the mission captain (proposal self-audit).

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