Remarks on the degree growth of birational transformations

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.

2010 Mathematics Subject Classification

14E07, 32H50, 37P05

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This project has been funded by the Swiss National Science Foundation Grant “Birational Geometry” PP00P2 128422 /1 as well as by the Geldner-Stiftung, the FAG Basel, the Janggen Pöhn-Stiftung, and the State Secretariat for Education, Research and Innovation of Switzerland.

Received 21 July 2016

Accepted 16 January 2017

Published 4 June 2018