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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.
Reviewer: Reviewer (Berlin)

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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