| Let K = k(C) be the function field of a complete nonsingular curve C over an arbitrary field k. The main result states a morphism ϕ : PNK →PNK 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 P1K and Baker for arbitrary rational maps on P1K . There are two proofs given. The first uses algebraic geometry and more specifically, geometric invariant theory. It is new even in the case of P1K . The second proof, using non-archimedean analysis and dynamics, more directly generalizes proofs of Benedetto and Baker for the N = 1 case. In addition, two applications for the result are given. |
