Some properties of dynamical degrees with a view towards cubic fourfolds

TL;DR

This paper explores properties of dynamical degrees and spectra to distinguish birational automorphism groups, introduces new tools for their computation, and applies these concepts to cubic fourfolds, revealing specific degree coincidences under generic conditions.

Contribution

It introduces properties like semi-continuity and sub-multiplicativity for dynamical degrees and applies these to cubic fourfolds, showing degree coincidences in generic cases.

Findings

Dynamical degrees can distinguish birational automorphism groups.

Certain properties facilitate computing dynamical degrees in concrete cases.

First and second dynamical degrees coincide for generic compositions of reflections on cubic fourfolds.

Abstract

Dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative, as opposed to only qualitative, ways. We introduce and discuss some properties of those degrees and the Cremona degrees, which facilitate computing or deriving inequalities for them in concrete cases: (generalized) lower semi-continuity, sub-multiplicativity, and an analogue of Picard-Manin/Zariski-Riemann spaces for higher codimension cycles. We also specialize to cubic fourfolds and show that under certain genericity assumptions the first and second dynamical degrees of a composition of reflections in points on the cubic coincide.