A Q-factorial complete toric variety with Picard number 2 is projective

This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a $\mathbb{Q}$-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 $\mathbb{Z}$-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.