Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

Details Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

2008-06-09

arxiv.org/abs/0806.1364v2

Mathematics

Algebraic Geometry

Changes in this version: moved some preliminary material on non-archimedean fields to section 2; clarified the geometric proof

Scientific paper

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 $\phi:{\mathbb P}^N_K\to{\mathbb P}^N_K$ 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 ${\mathcal L}$.

Affiliated with

Also associated with

No associations

Rating

Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-400267