Logarithmic Mori Program

Abstract

The purpose of this chapter is twofold. The first is to review what we have learned so far by going over the main ingredients of the Mori program in the framework of the logarithmic category. The reader who feels confident of and comfortable with the material so far should have no trouble understanding this part and should even skip this review chapter. Mastery and understanding of the log minimal model program will be rewarded when we utilize it as an essential part of the Sarkisov program in Chapter 13. The second is to present some of the subtleties that inevitably arise as we go from the usual category to the logarithmic category. There are some open conjectures even in dimension 3, though their statements are the natural generalizations in the logarithmic category, according to Iitaka’s philosophy, of the corresponding ones in the usual category.