A note on the $G$-Sarkisov program

TL;DR

This paper extends the Sarkisov program to the equivariant setting with a connected algebraic group, providing a new characterization of certain subgroups of the birational automorphism group.

Contribution

It proves the $G$-equivariant Sarkisov program for connected algebraic groups, following the approach of Hacon and McKernan, and characterizes connected subgroups of $Bir(Z)$ acting rationally.

Findings

Established the $G$-equivariant Sarkisov program.

Characterized connected subgroups of $Bir(Z)$.

Extended the Sarkisov framework to equivariant contexts.

Abstract

The purpose of this note is to prove the $G$-equivariant Sarkisov program for a connected algebraic group $G$ following the proof of the Sarkisov program by Hacon and McKernan. As a consequence, we obtain a characterisation of connected subgroups of $Bir(Z)$ acting rationally on $Z$.