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 $Nef(X)cappartialoverline{Eff}(X) eq{0}$. 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 $Nef(X)cappartialoverline{Eff}(X)={0}$. 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.