David Mumford - Classification of Surfaces

Classification of Surfaces

In three papers written between 1969 and 1976 (the last two in collaboration with E. Bombieri), Mumford extended the Enriques–Kodaira classification of smooth projective surfaces from the case of the complex groundfield to the case of an algebraically closed groundfield of characteristic p. The final answer turns out to be essentially as the answer in the complex case (though the methods employed are sometimes quite different), once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about when p-torsion in the Picard scheme degenerates to a non-reduced group scheme. The second is the possibility of obtaining quasi-elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one with a cusp.

Once these adjustments are made, the surfaces are divided into four classes by their Kodaira dimension, as in the complex case. The four classes are: a) Kodaira dimension minus infinity. These are the ruled surfaces. b) Kodaira dimension 0. These are the K3 surfaces, abelian surfaces, hyperelliptic and quasi-hyperelliptic surfaces, and Enriques surfaces. There are classical and non-classical examples in the last two Kodaira dimension zero cases. c) Kodaira dimension 1. These are the elliptic and quasi-elliptic surfaces not contained in the last two groups. d) Kodaira dimension 2. These are the surfaces of general type.