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 : PKN → PKN 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 PK1 and Baker for arbitrary rational maps on PK1. 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 PKN 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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory