# Birational self-maps of threefolds of (un)-bounded genus or gonality

## TL;DR

This paper investigates the complexity of birational self-maps of threefolds by analyzing the genus and gonality of contracted surfaces, establishing conditions for their unboundedness related to specific birational structures.

## Contribution

It characterizes when the genus and gonality of contracted surfaces are unbounded, linking these properties to conic bundles and cubic surface fibrations.

## Key findings

- Genus of contracted surfaces is unbounded iff X is birational to a conic bundle or cubic surface fibration.
- Gonality of contracted surfaces is unbounded iff X is birational to a conic bundle.
- Provides criteria for unbounded complexity of birational self-maps based on the structure of X.

## 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.

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Source: https://tomesphere.com/paper/1905.00940