Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds

TL;DR

This paper constructs specific birational involutions on projective 3-space and cubic threefolds, revealing complex group structures and automorphisms, thereby advancing understanding of their birational transformation groups.

Contribution

It introduces new families of involutions that do not fit into elementary Sarkisov links, and demonstrates their groups have free product structures and novel automorphisms.

Findings

Constructed birational involutions not fitting into elementary Sarkisov links

Established that the groups of birational transformations have free product structures

Produced automorphisms not generated by inner or field automorphisms

Abstract

We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.