On irreducibility of prefixed algebraic sets in moduli spaces of prime degree polynomials

Abstract

Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$ polynomials over $\mathbb{C}$, with a marked critical point. Given $k \geq 0,\; p$ an odd prime, we show that the set $\Sigma_{k,1,p}$ of conjugacy classes of degree $p$ polynomials, for which the marked critical point is strictly $(k,1)$-preperiodic, is an irreducible quasi-affine algebraic set. Irreducibility of these sets was conjectured by Milnor, and has been proved for $p=3$ by Buff, Epstein and Koch. We prove that the subspaces of $\Sigma_{k,1,p}$, that arise by varying the ramification index of the marked critical point all the way up to the unicritical case, are all irreducible subvarieties. Finally, using the irreducibility of $\Sigma_{k,1,p}$ we give a new and short proof of the fact that the set of all unicritical points of $\Sigma_{k,1,p}$ form one Galois orbit under the action of absolute Galois group of $\mathbb{Q}$.