A $$\mathbb {Q}$$ Q -factorial complete toric variety is a quotient of a poly weighted space

Abstract

We prove that every \(\mathbb {Q}\)-factorial complete toric variety is a finite abelian quotient of a poly weighted space (PWS), as defined in our previous work (Rossi and Terracini in Linear Algebra Appl 495:256–288, 2016. doi:10.1016/j.laa.2016.01.039). This generalizes the Batyrev–Cox and Conrads description of a \(\mathbb {Q}\)-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) (Duke Math J 75:293–338, 1994, Lemma 2.11) and (Manuscr Math 107:215–227, 2002, Prop. 4.7), to every possible Picard number, by replacing the covering WPS with a PWS. By Buczynska’s results (2008), we get a universal picture of coverings in codimension 1 for every \(\mathbb {Q}\)-factorial complete toric variety, as topological counterpart of the \(\mathbb {Z}\)-linear universal property of the double Gale dual of a fan matrix. 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 \(\mathbb {Q}\)-factorial complete toric variety the description given in Rossi and Terracini (2016, Thm. 2.9) for a PWS.