Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields
| dc.creator | Petsche, Clayton | |
| dc.creator | Szpiro, Lucien | |
| dc.creator | Tepper, Michael | |
| dc.date | 2008-06-09 | |
| dc.date | 2008-11-20 | |
| dc.date.accessioned | 2026-07-07T10:19:19Z | |
| dc.date.available | 2026-07-07T10:19:19Z | |
| dc.description | 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 $ϕ:{\mathbb P}^N_K\to{\mathbb P}^N_K$ 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 ${\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\mathbb P}^1_K$. 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 ${\mathbb P}^N_K$ 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 ${\mathcal L}\oplus...\oplus{\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\mathcal L}$. | |
| dc.description | Changes in this version: moved some preliminary material on non-archimedean fields to section 2; clarified the geometric proof of Theorem 1; replaced our proof of Prop. 2(c)--which had a gap in it--with a reference to the proof by Fakhruddin; corrected several small errors and typos, and added some new references | |
| dc.identifier | https://arxiv.org/abs/0806.1364 | |
| dc.identifier | http://arxiv.org/abs/0806.1364 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174469 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 14G99; 14H05 | |
| dc.title | Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields | |
| dc.type | text |