Indeterminacy Loci of Iterate Maps

TL;DR

This paper investigates the indeterminacy loci of iterate maps in the moduli space of complex rational maps, providing conditions that identify when a map belongs to these loci, thus addressing a question posed by Laura DeMarco.

Contribution

It offers new criteria to determine when a rational map lies in the indeterminacy locus of its iterate map, advancing understanding of the structure of these loci.

Findings

Identifies conditions implying a map is in the indeterminacy locus

Provides partial answers to DeMarco's question

Enhances understanding of iterate map indeterminacy loci

Abstract

We consider the indeterminacy locus $I(\Phi_n)$ of the iterate map $\Phi_n:\overline{M}_d-rightarrow\overline{M}_{d^n}$, where $\overline{M}_d$ is the GIT compactification of the moduli space $M_d$ of degree $d$ complex rational maps. We give natural conditions on $f$ that imply $[f]\in I(\Phi_n)$. These provide partial answers to a question of Laura DeMarco.