We usually follow the definitions and notation of Hartshorne (1977) and Kollár and Mori (1998). Those that may be less familiar or are used inconsistently in the literature are recalled in Section 1.1.

The rest of the chapter is more advanced. We suggest skipping it at first reading and then returning to these topics when they are used later.

The classical theory of minimal models is summarized in Section 1.2. Minimal and canonical models of pairs are treated in greater detail in Section 1.3. Our basic reference is Kollár and Mori (1998), but several of the results that we discuss were not yet available when Kollár and Mori (1998) appeared. In Section 1.4 we collect various theorems that can be used to improve the singularities of a variety while changing the global structure only mildly. Random facts about some singularities are collected in Section 1.5.

Assumptions Throughout this book, all schemes are assumed noetherian and separated. Further restrictions are noted at the beginning of every chapter.

All the concepts discussed were originally developed for projective varieties over ℂ. We made a serious effort to develop everything for rather general schemes. This has been fairly successful for the basic results in Chapter 2, but most of the later theorems are known only in characteristic 0.

Notation and conventions

Notation 1.1 The singular locus of a scheme X is denoted by SingX. It is a closed, reduced subscheme if X is excellent.