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Abstract

The preperiodic dynatomic curve $\mathcal {X}_{n,p}$ is the closure in $\mathbb C^2$ of the set of $(c,z)$ such that $z$ is a preperiodic point of the polynomial $z \mapsto z^d+c$ with preperiod $n$ and period $p$ ($n,p\geq 1$). We prove that each $\mathcal {X}_{n,p}$ has exactly $d-1$ irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of $\mathcal {X}_{n,p}$. We also compute the genus of each component and the Galois group of the defining polynomial of $\mathcal {X}_{n,p}$.