Relations in the Sarkisov Program

TL;DR

This paper investigates the structure of relations in the Sarkisov Program, demonstrating that all such relations are generated by elementary relations arising from specific MMP configurations.

Contribution

It proves that relations in the Sarkisov Program are generated by elementary relations, clarifying the foundational structure of these relations.

Findings

Relations are generated by elementary relations.

Elementary relations are related to MMP end products with Picard rank 3.

Provides a structural understanding of Sarkisov relations.

Abstract

The Sarkisov Program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If X and Y are terminal Q-factorial projective varieties endowed with a structure of Mori fibre space, any birational map between them can be decomposed into a finite number of elementary Sarkisov links. This decomposition is not unique in general, and any two distinct decompositions define a relation in the Sarkisov Program. This paper shows that relations in the Sarkisov Program are generated by some elementary relations. Roughly speaking, elementary relations are the relations among the end products of the MMP of Z over W, for suitable Z and W with relative Picard rank 3.