A $\mathbb{Q}$--factorial complete toric variety with Picard number 2 is projective

TL;DR

This paper proves that all Q-factorial complete toric varieties with Picard number 2 are projective, and explores the minimal Picard number for non-projective examples using Gale duality.

Contribution

It establishes that Q-factorial complete toric varieties with Picard number 2 are necessarily projective and identifies the minimal Picard number for non-projective cases.

Findings

All Q-factorial complete toric varieties with Picard number 2 are projective.

The minimal Picard number for non-projective examples is 3.

A 3-dimensional example shows the Picard number threshold for nef cone vanishing.

Abstract

This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a Q-factorial complete toric variety. The first problem is about the existence of ample divisors when the Picard number is 2: we give a positive answer to this question, by studying the secondary fan by means of Z-linear Gale duality. The second problem is about the minimum value of the Picard number allowing the vanishing of the Nef cone: we present a 3-dimensional example showing that this value cannot be greater then 3, which, under the previous result, is also the minimum value guaranteeing the existence of non-projective examples.