Indeterminacy loci of iterate maps in moduli space

Abstract

The moduli space $\mathrm{rat}_d$ of rational maps in one complex variable of degree $d \ge 2$ has a natural compactification by a projective variety $\overline{\mathrm{rat}}_d$ provided by geometric invariant theory. Given $n \ge 2$, the iteration map $\Phi_n : \mathrm{rat}_d \to\mathrm{rat}_{d^n}$, defined by $\Phi_n: [f] \mapsto [f^n]$, extends to a rational map $\Phi_n : \overline{\mathrm{rat}}_d\dashrightarrow \overline{\mathrm{rat}}_{d^n}$. We characterize the elements of $\overline{\mathrm{rat}}_d$ which lie in the indeterminacy locus of $\Phi_n$.