[Paper Review] Moduli spaces for families of rational maps on P^1

Michelle Manes|ArXiv.org|Feb 11, 2009

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper constructs moduli spaces $M_d(N)$ parameterizing degree-$d$ rational maps on $\mathbb{P}^1$ up to conjugacy, with a point of formal period $N$. It proves $M_2(N)$ is geometrically irreducible for $N > 1$ using explicit normal forms and irreducibility of dynatomic polynomials, but shows that for maps with nontrivial $\mathrm{PGL}_2$-automorphisms, $M_d(pN,\mathfrak{C}_p)$ is reducible for infinitely many $N$, providing a geometric explanation via orbit structure under automorphism actions.

ABSTRACT

Let phi: P^1 --> P^1 be a rational map defined over a field K. We construct the moduli space M_d(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M_2(N) is geometrically irreducible for N>1. Restricting ourselves to maps phi of arbitrary degree d >= 2 such that the composition h^{-1} phi h = phi for some nontrivial h in PGL_2, we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.

Motivation & Objective

• To construct and study the moduli space $M_d(N)$ parameterizing degree-$d$ rational maps on $\mathbb{P}^1$ up to $\mathrm{PGL}_2$-conjugacy with a point of formal period $N$.
• To prove geometric irreducibility of $M_2(N)$ for all $N > 1$ using explicit normal forms and irreducibility of dynatomic polynomials.
• To investigate the geometric structure of moduli spaces for maps with nontrivial automorphisms, particularly $M_d(N,\mathfrak{C}_p)$, and determine when they are reducible.
• To provide a geometric explanation for the reducibility of dynatomic polynomials in automorphism-invariant families, linking it to orbit structure under group actions.

Proposed method

• Construct $M_d(N)$ as a moduli space parameterizing conjugacy classes of degree-$d$ rational maps with a point of formal period $N$, using algebraic geometry and invariant theory.
• Use a normal form for quadratic rational maps $\psi(z) = \frac{z^2 + az}{az + 1}$ to explicitly describe $M_2$, enabling iteration and polynomial parametrization.
• Prove irreducibility of $M_2(N)$ by showing the covering surfaces mapping to it are irreducible, relying on Morton’s result on irreducibility of dynatomic polynomials for $f_c(z) = z^2 + c$.
• Analyze the action of automorphisms $h \in \mathrm{PGL}_2(\overline{K})$ on periodic points, distinguishing cases where $h$ preserves or interchanges orbits of a point of formal period $N$.
• Construct a proper closed subvariety of $M_d(pN,\mathfrak{C}_p)$ of maximal dimension to prove reducibility, showing that the moduli space splits into components based on orbit behavior under automorphism action.
• Use specialization arguments and properties of dynatomic polynomials to prove that $M_d(pN,\mathfrak{C}_p)$ is reducible for all but finitely many $N$ in characteristic 0, and all but finitely many prime $N$ in general fields.

Experimental results

Research questions

• RQ1 Is the moduli space $M_2(N)$ geometrically irreducible for all $N > 1$?
• RQ2 What is the geometric structure of the moduli space $M_d(N,\mathfrak{C}_p)$ for maps with a nontrivial automorphism of prime order $p$?
• RQ3 Why do dynatomic polynomials become reducible for maps with nontrivial automorphisms when $N$ is even, and how does this affect the geometry of the corresponding moduli spaces?
• RQ4 Can the reducibility of $M_d(pN,\mathfrak{C}_p)$ be explained geometrically through the action of automorphisms on periodic orbits?
• RQ5 Does the presence of automorphisms lead to a systematic failure of generic irreducibility in dynamic modular curves?

Key findings

• The moduli space $M_2(N)$ is geometrically irreducible for all $N > 1$, established via explicit normal forms and irreducibility of dynatomic polynomials for the family $f_c(z) = z^2 + c$.
• For maps with a nontrivial $\mathrm{PGL}_2$-automorphism of prime order $p$, the moduli space $M_d(pN,\mathfrak{C}_p)$ is geometrically reducible for all but finitely many integers $N$ when $K$ has characteristic 0.
• The reducibility arises from the action of the automorphism: if $h$ has order dividing $N$, it either preserves or interchanges the orbits of a point of formal period $N$, leading to at least two irreducible components in the moduli space.
• The construction provides a proper closed subvariety of maximal dimension in $M_d(pN,\mathfrak{C}_p)$, proving reducibility without relying on special cases.
• The result contrasts with prior work: unlike previous examples where reducibility occurred for only one map per degree, here it occurs for entire families of maps in every degree $d \geq 2$, with infinitely many $N$ for which reducibility holds.
• For $N$ such that $2^N - 1$ is prime, the curve $M_2(N,\mathfrak{C}_2)$ is irreducible, as shown by specializing the dynatomic polynomial and using irreducibility of $\Phi_N^*(z)$ for $\phi(z) = z^2$.