The purpose of this paper is to give basic tools for the classification of
nonsingular toric Fano varieties by means of the notions of primitive
collections and primitive relations due to Batyrev. By using them we can easily
deal with equivariant blow-ups and blow-downs, and get an easy criterion to
determine whether a given nonsingular toric variety is a Fano variety or not.
As applications of these results, we get a toric version of a theorem of Mori,
and can classify, in principle, all nonsingular toric Fano varieties obtained
from a given nonsingular toric Fano variety by finite successions of
equivariant blow-ups and blow-downs through nonsingular toric Fano varieties.
Especially, we get a new method for the classification of nonsingular toric
Fano varieties of dimension at most four. These methods are extended to the
case of Gorenstein toric Fano varieties endowed with natural resolutions of
singularities. Especially, we easily get a new method for the classification of
Gorenstein toric Fano surfaces.Comment: 36 pages, Latex2e, to appear in Tohoku Math.
