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 ${\Phi }_n:{\overline{\mathrm{rat}}}_d⇢{\overline{\mathrm{rat}}}_{d^n}$. We characterize the elements of ${\overline{\mathrm{rat}}}_d$ which lie in the indeterminacy locus of ${\Phi }_n$.

49 pages, 3 figures. revised version. added Theorem B, Figure 1 and details in Section 4