A note on minimization of rational surfaces obtained from birational dynamical systems

TL;DR

This paper introduces a method to obtain relatively minimal rational surfaces from birational dynamical systems using blowing down structures, aiding the analysis of integrable and linearizable mappings.

Contribution

It presents a novel technique for minimalization of rational surfaces derived from birational systems, enhancing understanding of their geometric properties.

Findings

Successfully applied to integrable mappings

Effective in desingularization and minimalization

Provides clearer geometric insights

Abstract

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.