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 e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2110.00095
- arXiv:
- arXiv:2110.00095
- Bibcode:
- 2021arXiv211000095B
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Algebraic Geometry;
- 37F10 (primary);
- 32H50;
- 14E05 (secondary)
- E-Print:
- 14 pages. Expanded and improved since v1