Bounded geometry for PCF-special subvarieties

Abstract For each integer $d\geq 2$ , let $\mathrm {M}_d$ 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 $\mathrm {M}_d$ . A complex-algebraic subvariety $Y \subset \mathrm {M}_d$ 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 $\mathrm {M}_d$ 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 $X \subset \mathrm {M}_d$ of degree $\leq D$ , the Zariski closure $\overline {X \cap \mathrm {PCF}_d}$ 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 $\mathrm {M}_d(\overline {\mathbb {Q}})$ .