2008/06/09 by Clayton Petsche, Petsche, Clayton, Lucien Szpiro +3 · 1 citation
Computer Science · Mathematics · #14G99 #14H05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0806.1364
openalex publication_date 2008/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K=k(C) be the function field of a complete nonsingular curve C over an arbitrary field k. The main result of this paper states that a morphism ϕ:\mathbb PNK→\mathbb 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 \mathbb P1K and Baker for arbitrary rational maps on \mathbb P1K. 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 \mathbb PNK 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 \mathcal E of rank N+1 on C decomposes as a direct sum \mathcal L⊕...⊕\mathcal L of N+1 copies of the same invertible sheaf \mathcal L.