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 QQ –factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of ZZ –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 QQ –Fano, QQ –Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied.