On the classification of smooth projective toric varieties

Abstract

We investigate the problem of the classification of smooth projective toric varieties V of dimension d with a given Picard number p over an algebraically closed field. For that purpose we introduce a convenient combinatorial description of such varieties by means of primitive relations among d+p integral generators of the associated complete regular fan of convex cones in /-dimensional real space. The main conjecture asserts that the number of the primitive relations is bounded by an absolute constant depending only on p. We prove this conjecture for p < 3 and give the classification of /-dimensional smooth complete toric varieties with p = 3.