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Birational self-maps of threefolds of (un)-bounded genus or gonality
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 144, Number 2, April 2022
- pp. 575-597
- 10.1353/ajm.2022.0011
- Article
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abstract:
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.