On the dynamical degrees of reflections on cubic fourfolds

TL;DR

This paper calculates the dynamical degrees of specific reflection compositions on smooth cubic fourfolds to explore their implications for the irrationality problem, aiming to identify potential restrictions on dynamical degrees.

Contribution

It provides explicit computations of dynamical degrees for reflections on cubic fourfolds, offering insights into their role in the irrationality problem.

Findings

Computed dynamical degrees for certain reflection compositions.

Identified potential restrictions on dynamical degrees for cubic fourfolds.

Provided evidence contrasting with projective four-space cases.

Abstract

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.