Moduli Spaces of Degree Two Rational Maps with Portraits Up to Six Points

We consider the moduli space, ${\mathcal{M}}_d^N[\mathcal{P}]$, of degree $d$ rational self maps of ${\mathbb{P}}^N$ with prescribed pre-periodic structure $\mathcal{P}$ which were introduced by Doyle and Silverman. It was shown in arXiv:1305.1054 that, ${\mathcal{M}}_2^1[{\mathcal{P}}_6]$, the moduli space of degree two rational maps with a $6$-cycle is a surface of general type. Here we compute the Kodaira dimension of all the moduli spaces with up to six pre-periodic points and show that $\kappa =-\mathrm{\infty },0,1,2$ are all realized for some $\mathcal{P}$.