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 ≥ 0, (k ≠ 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 C). We then study a closely related question concerning the irreducibility (over Q) of the set of conjugacy classes of unicritical polynomials, of degree D ≥ 2, with a preperiodic critical point. Our proofs are purely arithmetic; they rely on a result providing sufficient conditions under which irreducibility over C is equivalent to irreducibility over Q, and on a generalized Eisenstein criterion for irreducibility.