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