On the Stabilisation of Rational Surface Maps
Abstract
The dynamics of a rational surface map $f : X \dashrightarrow X$ are easier to analyse when $f$ is `algebraically stable'. Here we investigate when and how this condition can be achieved by conjugating $f$ with a birational change of coordinate. We show that if this can be done with a birational morphism, then there is a minimal such conjugacy. For birational $f$ we also show that repeatedly lifting $f$ to its graph gives a stable conjugacy. Finally, we give an example in which $f$ can be birationally conjugated to a stable map, but the conjugacy cannot be achieved solely by blowing up.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2110.00095
 arXiv:
 arXiv:2110.00095
 Bibcode:
 2021arXiv211000095B
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Algebraic Geometry;
 37F10 (primary);
 32H50;
 14E05 (secondary)
 EPrint:
 14 pages. Expanded and improved since v1