  We show that the set of conjugacy classes of cubic polynomials with a
prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic
curve. We also establish an analogous result for quadratic rational maps. We
then study a closely related question concerning the irreducibility (over
$\mathbb Q$) of the set of conjugacy classes of unicritical polynomials, of
degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely
algebraic.
