On the degree growth of iterated birational maps

TL;DR

This paper constructs a family of birational maps on two-dimensional varieties exhibiting cubic degree growth, surpassing known bounds due to singularities, with detailed calculations provided.

Contribution

It introduces a new example of birational maps with cubic degree growth, extending the understanding of degree dynamics on singular varieties.

Findings

Degree growth is cubic for the constructed maps

Singularities enable growth beyond traditional bounds

Detailed calculations confirm the growth behavior

Abstract

We construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such maps acting on two dimensional compact K\"ahler varieties. The example we construct goes beyond this limitation, thanks to the presence of a singularity on the variety where the maps act. We provide all details of the calculations.