ON THE CLASSIFICATION OF SMOOTH PROJECTIVE TORIC VARIETIES

We investigate the problem of the classification of smooth projective toric varieties V of dimension d with a given Picard number-rho over an algebraically closed field. For that purpose we introduce a convenient combinatorial description of such varieties by means of primitive relations among d + rho-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 rho. We prove this conjecture for rho less-than-or-equal-to 3 and give the classification of d-dimensional smooth complete toric varieties with rho = 3.