Bounded geometry for PCF-special subvarieties

For each integer d≥ 2 , let Md denote the moduli space of maps f: P¹→ P¹ of degree d . We study the geometric configurations of subsets of postcritically finite (or PCF) maps in Md . A complex-algebraic subvariety Y ⊂ Md 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 Md with degree ≤ D . In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety X ⊂ Md of degree ≤ D , the Zariski closure X ∩ PCFd̄ has at most N irreducible components, each with degree ≤ B . We also prove generalizations of these results for points with small critical height in Md(Q̄) .