The singular locus of $V_\alpha$ is exactly $\{(0,0),(\infty,\infty)\}$ for $m\ge2$ and $\alpha\notin\mathbb R$

Proved

CurveSymmetry.family_projective_critical_eq_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

\Sigma_\alpha=\bigl\{(\iota(0),\iota(0)),\ (\iota(\infty),\iota(\infty))\bigr\}=\bigl\{([0:1],[0:1]),\ ([1:0],[1:0])\bigr\}.

This is the singular-point clause of Lemma 4 of the note: the only singular points of $V_\alpha$ are $(0,0)$ and $(\infty,\infty)$ . It holds for every nonreal $\alpha$ , without the condition $|\alpha|=1$ ; that both points are ordinary $m$ -fold points is a separate statement. The proof of Theorem 2 relies on equivalences preserving this pair.

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_eq_pair {m : ℕ} (hm : 2 ≤ m) {α : ℂ}
    (ha : α ≠ star α) :
    familyProjectiveCritical m α =
      {(sphereProjectiveEquiv ((0 : ℂ) : Sphere), sphereProjectiveEquiv ((0 : ℂ) : Sphere)),
        (sphereProjectiveEquiv (∞ : Sphere), sphereProjectiveEquiv (∞ : Sphere))} := by sorry

Source

C. Perassi, Sharp symmetry bounds for real algebraic curves (note), 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)

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).

View graph

The singular locus of VαV_\alphaVα​ is exactly {(0,0),(∞,∞)}\{(0,0),(\infty,\infty)\}{(0,0),(∞,∞)} for m≥2m\ge2m≥2 and α∉R\alpha\notin\mathbb Rα∈/R

The singular locus of $V_\alpha$ is exactly $\{(0,0),(\infty,\infty)\}$ for $m\ge2$ and $\alpha\notin\mathbb R$