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The space of rational maps on $$\mathbf P^1$$. (English) Zbl 0966.14031
Introduction: The set of morphisms $$\varphi:\mathbb{P}^1 \to\mathbb{P}^1$$ of degree $$d$$ is parametrized by an affine open subset $$\text{Rat}_d$$ of $$\mathbb{P}^{2d+1}$$. In this paper, we consider the action of $$\text{SL}_2$$ on $$\text{Rat}_d$$ induced by the conjugation action of $$\text{SL}_2$$ on rational maps; that is, $$f\in\text{SL}_2$$ acts on $$\varphi$$ via $$\varphi^f= f^{-1}\circ \varphi\circ f$$. The quotient space $$M_d=\text{Rat}_d/ \text{SL}_2$$ arises very naturally in the study of discrete dynamical systems on $$\mathbb{P}^1$$. We prove that $$M_d$$ exists as an affine integral scheme over $$\mathbb{Z}$$, that $$M_2$$ is isomorphic to $$\mathbb{A}^2_\mathbb{Z}$$ and that the natural completion of $$M_2$$ obtained using geometric invariant theory is isomorphic to $$\mathbb{P}^2_\mathbb{Z}$$. These results, which generalize results of Milnor over $$\mathbb{C}$$, should be useful for studying the arithmetic properties of dynamical systems.

##### MSC:
 14L24 Geometric invariant theory 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 14L30 Group actions on varieties or schemes (quotients) 14E05 Rational and birational maps
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