Moduli spaces of quadratic rational maps with a marked periodic point of small order

TL;DR

This paper investigates the geometric structure of moduli spaces of quadratic rational maps with a marked periodic point, revealing a transition from rationality to general type at a specific period and constructing explicit examples over ield.

Contribution

It provides a detailed analysis of the moduli space surfaces for quadratic maps with periodic points, including explicit descriptions and new examples over ield, especially for period 6.

Findings

Surface is rational over ield for n o 5

Surface is of general type for n=6

Constructs explicit quadratic maps with rational periodic points of order 6

Abstract

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type for $n=6$. An explicit description of the $n=6$ surface lets us find several infinite families of quadratic endomorphisms $f: \mathbb{P}^1 \to \mathbb{P}^1$ defined over $\mathbb{Q}$ with a rational periodic point of order $6$. In one of these families, $f$ also has a rational fixed point, for a total of at least $7$ periodic and $7$ preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over $\mathbb{Q}$ admits rational periodic points of order $n>3$.