Bounded geometry for PCF-special subvarieties

For each integer $d\ge 2$, 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 $\le D$. 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 $\le D$, the Zariski closure $\overline{X\cap \mathrm{P}\mathrm{C}\mathrm{F}}$ has at most $N$ irreducible components, each with degree $\le B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.