  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.
