Fibration and classification of smooth projective toric varieties of low Picard number

In this paper, we show that a smooth toric variety X of Picard number r ≤ 3 always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone Nef(X) of numerically effective divisors and cutting a facet of the pseudo-effective cone Eff(X), that is Nef(X) Eff¯(X){0}. In particular, this means that X admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of X, so giving rise to a classification of smooth and complete toric varieties with r ≤ 3. Moreover, we revise and improve results of Oda-Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension n = 3 and Picard number r = 4, allowing us to classifying all these threefolds. We then improve results of Fujino-Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension n ≥ 4 and Picard number r = 4 whose non-trivial nef divisors are big, that is Nef(X) Eff¯(X) = {0}. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for n = 4, the given example turns out to be a weak Fano toric fourfold of Picard number 4.