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.

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.