Stable Varieties with a Twist

1.1. Moduli of stable varieties: the case of surfaces. In the paper [KSB88], Kollár and Shepherd-Barron introduced stable surfaces as a generalization of stable curves. This class is natural from the point of view of the minimal model program, which shows that any one-parameter family of surfaces of general type admits a unique stable limit. Indeed, the stable reduction process of Deligne and Mumford can be interpreted using the language of minimal models of surfaces. Stable surfaces admit semilog canonical singularities. Codimension-one semilog canonical singularities are nodes, but in codimension two more complicated singularities arise. However, these singularities are all reduced, satisfy Serre’s condition S2 (hence are Cohen–Macaulay in codimension 2), and admit a well-defined Q-Cartier canonical divisor KS. Stable varieties in any dimension are similarly defined as proper varieties with semilog canonical singularities and ample canonical divisor. Kollár and Shepherd-Barron proposed the moduli space of stable surfaces as the natural compactification of the moduli space of surfaces of general type. Much of this was established in [KSB88], which also offered a detailed classification of the singularities that arise. Boundedness of the class of stable surfaces with fixed invariant (KS) 2 was shown by Alexeev [Ale94].