Rational self-maps with a regular iterate on a semiabelian variety

TL;DR

This paper proves that a dominant rational self-map on a semiabelian variety, with a regular iterate and no non-trivial homomorphisms satisfying certain conditions, must itself be regular, with results applicable in characteristic zero and prime characteristic.

Contribution

The paper establishes that under specific conditions, a rational self-map with a regular iterate on a semiabelian variety is necessarily regular, extending understanding of such maps in algebraic geometry.

Findings

If an iterate of a rational self-map is regular, then the map itself is regular.

The results hold in characteristic zero and are extended to prime characteristic.

Examples demonstrate the sharpness of the theorems.

Abstract

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $\Phi\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $\Phi^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $\tau: G\to G_0$ of semiabelian varieties such that $\tau\circ \Phi^{m k}=\tau$ for some $k \geqslant 1$. We show that under these assumptions $\Phi$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.