The space of rational maps on P^1

TL;DR

This paper studies the moduli space of degree d rational maps on the projective line, showing it is an affine scheme and describing its structure for degree 2, with implications for arithmetic dynamics.

Contribution

It proves the existence and structure of the moduli space of rational maps over integers, generalizing Milnor's complex results to an integral setting.

Findings

The moduli space $ ext{M}_d$ exists as an affine integral scheme over $ ext{ZZ}.

For degree 2, $ ext{M}_2$ is isomorphic to $ ext{A}^2_ ext{ZZ}$.

The GIT compactification of $ ext{M}_2$ is isomorphic to $ ext{PP}^2_ ext{ZZ}$.

Abstract

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.