Over the Riemann sphere the only singular points of $V_\alpha$ are $(0,0)$ and $(\infty,\infty)$ , for $m\ge2$

Proved

CurveSymmetry.family_projective_critical_pair

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

algebraic-geometrycurve-symmetry-leaneuclidean-geometry

Let $m\ge2$ be an integer and let $\alpha\in\mathbb C$ with $\alpha\ne\bar\alpha$ ; no condition on $|\alpha|$ is imposed. For $x=(x_0,x_1)$ and $y=(y_0,y_1)$ in $\mathbb C^2$ let

F_\alpha(x,y)=\alpha\,x_0^m x_1\,y_1^{m+1}+x_0^{m+1}y_0\,y_1^m+\bar\alpha\,x_1^{m+1}y_0^m\,y_1+x_0\,x_1^m\,y_0^{m+1},

the bihomogenization of bidegree $(m+1,m+1)$ of $P_\alpha(X,Y)=X^m(\alpha+XY)+Y^m(\bar\alpha+XY)$ , and let $V_\alpha=\{F_\alpha=0\}\subset\mathbb P^1\times\mathbb P^1$ . Let $\Sigma_\alpha$ be the set of its singular points by the Jacobian criterion: the points $([x],[y])$ such that $F_\alpha$ , as a function on $\mathbb C^2\times\mathbb C^2$ , vanishes at $(x,y)$ together with its differential. Identify the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ with $\mathbb P^1$ by $\iota(z)=[z:1]$ for $z\in\mathbb C$ and $\iota(\infty)=[1:0]$ .

Then for all $p,q\in\widehat{\mathbb C}$

\bigl(\iota(p),\iota(q)\bigr)\in\Sigma_\alpha\iff(p=0\ \text{and}\ q=0)\ \text{or}\ (p=\infty\ \text{and}\ q=\infty).

Since $\iota$ is a bijection, this decides every point of $\mathbb P^1\times\mathbb P^1$ , boundary points included. It yields the singular-point clause of Lemma 4 and the fact, used in the proof of Theorem 2, that the singular points of $V_\alpha$ form the pair $\{(0,0),(\infty,\infty)\}$ , which every equivalence must preserve.

Formalization Note: $\iota$ is Mathlib's OnePoint.equivProjectivization from OnePoint ℂ to Projectivization ℂ (Fin 2 → ℂ), and the singular locus is defined through HasFDerivAt at chosen representatives.

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
open OnePoint

Formal statement

theorem CurveSymmetry.family_projective_critical_pair {m : ℕ} (hm : 2 ≤ m) {α : ℂ}
    (ha : α ≠ star α) (p q : Sphere) :
    (sphereProjectiveEquiv p, sphereProjectiveEquiv q) ∈ familyProjectiveCritical m α ↔
      (p = (0 : ℂ) ∧ q = (0 : ℂ)) ∨ (p = ∞ ∧ q = ∞) := by sorry

Source

C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for Theorem 2, p. 2; Lemma 4, p. 3, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/FamilyGlobalSingularities.lean (C. Perassi)

View graph

Over the Riemann sphere the only singular points of VαV_\alphaVα​ are (0,0)(0,0)(0,0) and (∞,∞)(\infty,\infty)(∞,∞), for m≥2m\ge2m≥2

Over the Riemann sphere the only singular points of $V_\alpha$ are $(0,0)$ and $(\infty,\infty)$ , for $m\ge2$