Moduli spaces for families of rational maps on P^1

TL;DR

This paper constructs and analyzes moduli spaces for families of rational maps on the projective line, proving irreducibility for certain cases and reducibility for others, revealing complex geometric structures.

Contribution

It provides an algebraic construction of moduli spaces for rational maps with specified periodic points and proves irreducibility or reducibility results depending on conditions.

Findings

M_2(N) is geometrically irreducible for N>1

Certain moduli spaces are geometrically reducible for infinitely many N

Provides algebraic proofs for irreducibility and reducibility results

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.