A Q-factorial complete toric variety is a quotient of a poly weighted space

Abstract: We prove that every Q-factorial complete toric variety is a finite quotient of a poly weighted space (PWS), as defined in our previous work arXiv:1501.05244. This generalizes the Batyrev-Cox and Conrads description of a Q-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) \cite[Lemma~2.11]{BC} and \cite[Prop.~4.7]{Conrads}, to every possible Picard number, by replacing the covering WPS with a PWS. As a consequence we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every Q-factorial complete toric variety the description given in arXiv:1501.05244, Thm. 2.9, for a PWS.