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

TL;DR

This paper proves that all Q-factorial complete toric varieties can be represented as finite quotients of poly weighted spaces, extending previous results from Picard number 1 to arbitrary Picard numbers.

Contribution

It generalizes the description of Q-factorial complete toric varieties as quotients of weighted projective spaces to the broader class of poly weighted spaces for all Picard numbers.

Findings

Every Q-factorial complete toric variety is a finite quotient of a poly weighted space.

Provides explicit descriptions of divisor subgroup bases within the class group.

Extends known results from Picard number 1 to arbitrary Picard numbers.

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.