Canonical and log canonical singularities

Abstract

In this chapter we introduce the singularities – terminal, canonical, log terminal and log canonical – that appeared in connection with the minimal model program. Section 2.1 contains the basic definitions and their fundamental properties. Whenever possible, we work with schemes, rather than just varieties over a field of characteristic 0. Section 2.2 starts a detailed study of log canonical surface singularities. Here we focus on their general properties; a complete classification is postponed to Section 3.3. Ramified covers are studied in Section 2.3. This method gives a good theoretical framework to study log canonical singularities and their deformations in all dimensions, as long as the degree of the cover is not divisible by the characteristic. Ramified covers have been especially useful in dimensions 2 and 3, leading to a classification of 3-dimensional terminal, canonical and log terminal singularities in characteristic 0. A short summary of this approach is given in Section 2.4. These topics are excellently treated in Reid (1980, 1987). Divisorial log terminal singularities and their rationality properties are investigated in Section 2.5. Roughly speaking, these form the largest well-behaved subclass of log canonical singularities. We prove that they are rational and many important sheaves on them are Cohen-Macaulay (2.88), at least in characteristic 0. This section also shows the two, rather pervasive, problems that arise when one tries to generalize results of birational geometry to positive and mixed characteristics.