The Enriques Kodaira Classification

In this chapter we deal with the Enriques-Kodaira classification. In the first edition of this book the case (2,1) of Iitaka’s conjecture (see the Introduction) was used in the proof of Theorem 1.1. In the meantime it has become customary to give a slightly different proof for the classification theorem. The new proof, a form of which is presented below, rests on a systematic use of nef divisors (IV. Sect. 7). A central role is played by the Rationality Theorem which we prove first. We then show how the full classification of surfaces with K_x not nef follows in an astonishingly simple way, and deduce Castelnuovo’s criterion as a corollary. After all this the classification of minimal algebraic surfaces becomes quite easy. The classification of non-algebraic surfaces which follows next, is the same as in the first edition of the book. In Sect. 8 we prove Iitaka’s results concerning the stability of the ten classes under deformations.