  In this paper we show that a smooth toric variety $X$ of Picard number $r\leq
3$ 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
$\Nef(X)\cap\partial\overline{\Eff}(X)\neq\{0\}$. 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 $r\leq 3$. 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
$n\geq4$ and Picard number $r=4$ whose non-trivial nef divisors are big, that
is $\Nef(X)\cap\partial\overline{\Eff}(X)=\{0\}$. 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.
