LOG SARKISOV PROGRAM

Andrea Bruno Kenji Matsuki

Abstract. The purpose of this paper is two-fold. The first is to give a tutorial introduction to the so called Sarkisov program, a three-dimensional generalization of Castelnuovo-Noether Theorem “untwisting” birational maps between Mori tiber spaces, after Sarkisov, Reid, and Corti. The second is an attempt to give a logarithmic generalization of the program by introducing the notion of Sarkisov relation.

We also simplify and visualize the flowchart for the program, make explicit the necessary ingredients toward higher dimension, and clarify the mechanism of termination connected to the boundedness conjecture of (log) Q-Fano n-folds by Borisov.

Typeset by AMS-TEX