Remarks on the degree growth of birational transformations

TL;DR

This paper investigates the growth patterns of degrees in iterates of rational maps on algebraic varieties, establishing new constraints, examples, and countability results for degree sequences.

Contribution

It introduces new constraints on degree growth, provides examples of degree sequences, and proves the countability of all such sequences, extending previous results.

Findings

Degree sequences of rational maps are constrained by new conditions.

Examples of possible degree sequences are constructed.

The set of all degree sequences is countable.

Abstract

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.