I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .

Here, a toric variety has homogeneous coordinates $H:=\{x_i : i=1,\ldots, I\}$ equipped with a number $R$ of equivalence relations $$ (x_1,\dots,x_I)\sim (\lambda_r^{Q^{(r)}_1} x_1, \ldots, \lambda_r^{Q^{(r)}_I} x_I), \\ $$ for $r=1,\dots, R$ with the weights $Q^{(r)}_i\in \mathbb Z$ and $\lambda_r\in \mathbb{C}^* = \mathbb{C}-\{0\}$.

They go on to say (above equation (7)) that for each divisor $D_i$ of a toric variety, there is a line bundle $$\tag{7} L_i={\cal O}_X \bigl(Q^{(1)}_i, \ldots, Q^{(R)}_i \bigr)\; . $$

It is not clear to me why the weights $Q^{(r)}_i\in \mathbb Z$ govern the classification of line bundles, but this seems to imply that for a particular toric variety, $X$, the Picard group is
$$
Pic(X)=\mathbb{Z}^R.
$$
However, I have also read here -Reference for Weighted Projective Stacks - that for weighted projective spaces, the Picard group is cyclic. (Edit: It seems that this link talks about weighted projective *stacks*, which are not toric varieties.)

What is the rationale for describing line bundles in terms of weights as in equation (7), and how does one find the Picard group of a toric variety in general?

**Edit:** Fred Rohrer suggested in the comments that I look at Ewald's Combinatorial Complexity and Algebraic Geometry. On page 273 we find that for an arbitrary $n$-dimensional toric variety, $Pic(X)=\mathbb{Z}^{k-n-\lambda}$, $k$ being the number of 1-cones of the corresponding fan $\Sigma$, $\lambda$ being the total dimension of the spaces of linear dependencies of generators of all maximal cones which are not simplex cones. This, however, translates to $$Pic(X)=\mathbb{Z}^{R-\lambda},$$ which is different from the expression above. Why is there this discrepancy?

Combinatorial convexity and algebraic geometry,Springer GTM 168 (1996). $\endgroup$ – Fred Rohrer Nov 9 '17 at 7:01geometric quotientfrom the equivalence relation (as opposed to a categorical quotient in general). $\endgroup$ – Gjergji Zaimi Nov 12 '17 at 6:264more comments