  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.

Algebraic Geometry