Relations in the Sarkisov program

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, a birational map f: X-→ Y is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of Z over W with ρ (Z/W)= 3. © 2013 Foundation Compositio Mathematica.