Irreducibility of periodic curves

In the moduli space of critically marked cubic polynomial, given a positive integer $n$, let $\operatorname{Per}_n(0)$ denote the set corresponding to polynomials having one periodic critical point of period $n$. We show that $\operatorname{Per}_n(0)$ is an irreducible algebraic curve provided that $n$ is prime. This is a joint work with Matthieu Arfeux.