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

Abstract

In this paper, we show that a smooth toric variety [Formula: see text] of Picard number [Formula: see text] always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone [Formula: see text] of numerically effective divisors and cutting a facet of the pseudo-effective cone [Formula: see text], that is [Formula: see text]. In particular, this means that [Formula: see text] 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 [Formula: see text], so giving rise to a classification of smooth and complete toric varieties with [Formula: see text]. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension [Formula: see text] and Picard number [Formula: see text], 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 [Formula: see text] and Picard number [Formula: see text] whose non-trivial nef divisors are big, that is [Formula: see text]. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for [Formula: see text], the given example turns out to be a weak Fano toric fourfold of Picard number 4.