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

TL;DR

This paper classifies smooth projective toric varieties with low Picard number, showing their geometric structure and extending results to threefolds and fourfolds, with implications for nef divisors and fiber contractions.

Contribution

It provides a classification of smooth projective toric varieties with Picard number up to 3 and extends the classification to certain fourfolds, improving previous results and constructing explicit examples.

Findings

Smooth toric varieties with Picard number ≤ 3 have nef primitive collections supporting hyperplanes.

Classification of threefolds with Picard number 4 is achieved.

Counterexamples of higher-dimensional toric varieties with nef divisors that are big are constructed.

Abstract

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.