2014 / December Volume 9 No.4
The Field of Definition for Dynamical Systems on $\mathbb P^N$
| Published Date |
2014 / December
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| Title | The Field of Definition for Dynamical Systems on $\mathbb P^N$ |
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| Pagination | 585-602 |
| Abstract | Let $Hom^N_d$ be the set of morphisms $\phi: \mathbb{P}^{N} \to \mathbb{P}^{N}$ of degree $d$. For $f \in PGL_{N+1}$, let $\phi^f = f^{-1} \circ \phi \circ f$ be the conjugation action and let $M^N_d = Hom_d^N/PGL_{N+1}$ be the moduli space of degree $d$ morphisms of $P^N$. A field of definition for $\xi \in M_d^N$ is a field over which at least one representative $\phi \in \xi$ is defined. The field of moduli for $\xi$ is the fixed field of $G_\xi = \{\sigma \in Gal(\bar{K}/K) col \xi^{\sigma} = \xi\}$. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a field of definition for morphisms with trivial stabilizer group. Additionally, we use classical invariant theory to construct maps whose stabilizer group is non-trivial.
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| AMS Subject Classification |
37P45, 11G99
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| Received |
2014-07-11
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| Accepted |
2014-07-02
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