  The set of morphisms $\f:\PP^1\to\PP^1$ of degree $d$ is parametrized by an
affine open subset $\Rat_d$ of $\PP^{2d+1}$. We consider the action of~$\SL_2$
on $\Rat_d$ induced by the {\it conjugation action\/} of $\SL_2$ on rational
maps; that is, $f\in\SL_2$ acts on~$\f$ via $\f^f=f^{-1}\circ\f\circ f$. The
quotient space $\M_d=\Rat_d/\SL_2$ arises very naturally in the study of
discrete dynamical systems on~$\PP^1$. We prove that~$\M_d$ exists as an affine
integral scheme over~$\ZZ$, that $\M_2$ is isomorphic to~$\AA^2_\ZZ$, and that
the natural completion of~$\M_2$ obtained using geometric invariant theory is
isomorphic to~$\PP^2_\ZZ$. These results, which generalize results of Milnor
over~$\CC$, should be useful for studying the arithmetic properties of
dynamical systems.
