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 X̄~ 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).