Enriques–Kodaira classification

In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.

Max Noether began the systematic study of algebraic surfaces, and Guido Castelnuovo proved important parts of the classification. Federigo Enriques (1914, 1949) described the classification of complex projective surfaces. Kunihiko Kodaira (1964, 1966, 1968a, 1968b) later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun by David Mumford (1969) and completed by Enrico Bombieri and David Mumford (1976, 1977); it is similar to the characteristic 0 projective case, except that one also gets singular and supersingular Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3.