Prefixed curves in moduli space

abstract:We study the geometry of certain algebraic curves in the moduli space of cubic polynomials, and in the moduli space of quadratic rational maps. Given $k\geq 0$, ($k\neq 1$ in the case of quadratic rational maps), we show that the set of conjugacy classes of maps with a prefixed critical point of preperiod $k$, is an algebraic curve that is irreducible (over $\Bbb{C}$). We then study a closely related question concerning the irreducibility (over $\Bbb{Q}$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely arithmetic; they rely on a result providing sufficient conditions under which irreducibility over $\Bbb{C}$ is equivalent to irreducibility over $\Bbb{Q}$, and on a generalized Eisenstein criterion for irreducibility.