Bounded geometry for PCF-special subvarieties

TL;DR

This paper investigates the structure and finiteness properties of special subvarieties in the moduli space of degree d rational maps, focusing on PCF maps and their algebraic and geometric configurations.

Contribution

It establishes finiteness results for positive-dimensional PCF-special subvarieties and bounds on the complexity of intersections with algebraic subvarieties in the moduli space.

Findings

Finitely many positive-dimensional PCF-special subvarieties with degree ≤ D.

Bounded number of irreducible components in intersections with algebraic subvarieties.

Generalizations to points with small critical height in moduli space.

Abstract

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.