### Abstract

Let K = k (C) be the function field of a complete non-singular curve C over an arbitrary field k. The main result of this paper states that a morphism φ : P_{K}^{N} → P_{K}^{N} is isotrivial if and only if it has potential good reduction at all places v of K; this generalizes results of Benedetto for polynomial maps on P_{K}^{1} and Baker for arbitrary rational maps on P_{K}^{1}. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N = 1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of P_{K}^{N} of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf E of rank N + 1 on C decomposes as a direct sum L ⊕ ⋯ ⊕ L of N + 1 copies of the same invertible sheaf L.

Original language | English (US) |
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Pages (from-to) | 3345-3365 |

Number of pages | 21 |

Journal | Journal of Algebra |

Volume | 322 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2009 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

^{N}over function fields.

*Journal of Algebra*,

*322*(9), 3345-3365. https://doi.org/10.1016/j.jalgebra.2008.11.027