A numerical ampleness criterion via Gale duality

The main object of the present paper is a numerical criterion determining when a Weil divisor of a Q–factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of Z–linear interpretation of Gale duality and secondary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak Q–Fano, Q–Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied.