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](https://mathoverflow.net/questions/136888/reference-for-weighted-projective-stacks/164806#164806) - 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?



Picard group of toric varieties
Edit: I have elaborated on this approach to the Picard group in Section 2 of my [preprint](https://arxiv.org/abs/2308.08879).


The question was answered in the comments above, but only for the case of torsion-free Picard group. However, for non-complete toric varieties, the Picard group may have torsion (see example below). Since this showed up in my research, let me describe a method of determining the Picard group of a normal toric variety, where the fan is not necessarily simplicial or complete. I will give a criterion for the Picard group to be torsion-free and describe how to determine its rank.


Let $X$ be a $n$-dimensional normal toric variety associated to a fan $\Sigma$ in a lattice $N\cong\mathbb{Z}^n$. Let $v\_1, \dots, v\_r\in N$ be generators of the rays (one-dimensional cones) of $\Sigma$ and write $F:=\mathbb{Z}^r$. I assume that $X$ has no torus factor, that means $v\_1, \dots, v\_r$ generate $N \otimes \mathbb{Q}$ as a vector space. We have a canonical map of lattices
$$
P \colon F \to N, \qquad e\_i \mapsto v\_i.
$$
Writing $E:=F^\*$ and $M:=N^\*$ for the dual lattices, standard toric geometry gives an exact sequence
$$\require{AMScd}
\begin{CD}
0 @>>> M @>{P^\*}>> E @>>> \mathrm{Cl}(X) @>>> 0,
\end{CD}$$
where $\mathrm{Cl}(X)$ is the divisor class group (Weil divisors modulo principal divisors) of $X$. I view the Picard group $\mathrm{Pic}(X)$ as Cartier divisors modulo principal divisors, hence it is a subgroup of $\mathrm{Cl(X)}$. Note that a Weil Divisor on a toric variety is Cartier iff it is principal on all affine toric charts associated to the cones $\sigma \in \Sigma$.


So let's work with a single affine chart $U\_\sigma$ for a cone $\sigma = \mathrm{poshull}(v\_{i\_1}, \dots, v\_{i\_{r\_\sigma}}) \in \Sigma$. Define
$$
N(\sigma) := \mathrm{lin}\_{N\otimes\mathbb{Q}}(\sigma) \cap N, \qquad F(\sigma) := \mathbb{Z}^{r\_\sigma}.
$$
Note that $\dim(N(\sigma)) \leq r\_\sigma$ and equality holds iff $\sigma$ is simplicial. Let $\alpha\_\sigma\colon N(\sigma) \to N$ be the inclusion and $\beta\_\sigma\colon F(\sigma) \to F, e\_j \mapsto e\_{i\_j}$. Setting $M(\sigma) := N(\sigma)^\*$ and $E(\sigma):= F(\sigma)^\*$, we obtain a commutative diagram with exact rows
$$\require{AMScd}
\begin{CD}
0 @>>> M @>{P^\*}>> E @>>> \mathrm{Cl}(X) @>>> 0 \\
@.@V{\alpha\_\sigma^\*}VV @V{\beta\_\sigma^\*}VV @V{\pi\_\sigma}VV @. \\
0 @>>> M(\sigma) @>>> E(\sigma) @>>> \mathrm{Cl}(U\_\sigma) @>>> 0.
\end{CD}$$
where $\pi\_\sigma$ maps a divisor class $[D]$ to the restriction $[D|\_{U\_\sigma}]$. Being principal on $U\_\sigma$ means being in the kernel of $\pi\_\sigma$. For the Picard group, this means
$$
\mathrm{Pic}(X) = \bigcap\_{\sigma \in \Sigma} \ker{\pi\_\sigma}.
$$
We now draw the above diagram again, but for all cones $\sigma\in\Sigma$ at the same time. That is, we define
$$
\mathbf{F}:=\bigoplus\_{\sigma\in\Sigma} F(\sigma),\quad \mathbf{N}:=\bigoplus\_{\sigma\in\Sigma} N(\sigma),\quad \mathbf{M}:=\mathbf{N}^\*,\quad\mathbf{E}:=\mathbf{F}^\*.
$$
Furthermore, we obtain maps $\alpha\colon \mathbf{N} \to N, \beta\colon \mathbf{F} \to F$ and $\pi\colon \mathrm{Cl}(X) \to \bigoplus\_{\sigma\in\Sigma} \mathrm{Cl}(U\_\sigma)$ and we have $\mathrm{Pic}(X) = \ker{\pi}$. Note that $\beta$ is surjective, hence $\beta^\*$ is injective. Furthermore, since $v\_1, \dots, v\_r$ generate $N \otimes \mathbb{Q}$ as a vector space, the cokernel of $\alpha$ is finite, so $\alpha^\*$ is injective as well. We obtain a diagram with exact rows and columns
$$\require{AMScd}
\begin{CD}
@. 0 @. 0 @. \mathrm{Pic}(X) @. \\
@. @VVV @VVV @VVV @. \\
0 @>>> M @>{P^\*}>> E @>>> \mathrm{Cl}(X) @>>> 0 \\
@.@V{\alpha^\*}VV @V{\beta^\*}VV @V{\pi}VV @. \\
0 @>>> \mathbf{M} @>>> \mathbf{E} @>>> \bigoplus\_{\sigma\in\Sigma} \mathrm{Cl}(U\_\sigma) @>>> 0. \\
@. @VVV @VVV @VVV @. \\
@.\mathrm{coker}(\alpha^\*) @>>> \mathrm{coker}(\beta^\*) @>>> \mathrm{coker}(\pi) @>>> 0 \\
\end{CD}$$
The snake lemma now identifies $\mathrm{Pic}(X)$ with a subgroup of $\mathrm{coker}(\alpha^\*)$. This gives a criterion for torsion-freeness of the Picard group: If $\alpha$ is surjective, $\mathrm{coker}(\alpha^\*)$ is torsion-free, hence so is the Picard group.


Note that as soon as $\Sigma$ contains a cone of maximal dimension (for instance if it is complete), this holds trivially, since then $N(\sigma)=N$. A counterexample is the fan $\Sigma$ in $\mathbb{Z}^2$ having as maximal cones the two rays generated by $(1,0)$ and $(1,2)$. The Picard group of the associated smooth toric surface has torsion, indeed it has $\mathrm{Pic}(X)=\mathrm{Cl}(X)\cong\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.


To determine the rank of the Picard group, look again at the exact sequence obtained by applying the snake lemma to the above diagram. Since the $\pi\_\sigma$ are all surjective, the cokernel of $\pi$ is finite. Since the alternating sum of dimensions in an exact sequence of vector spaces vanishes, we obtain


$$
\begin{align\*}
\mathrm{rank}(\mathrm{Pic}(X)) & = \mathrm{rank}(\mathrm{coker}(\alpha^\*)) - \mathrm{rank}(\mathrm{coker}(\beta^\*)) \\
& = r - n - \left(\sum\_{\sigma\in\Sigma} r\_\sigma - \dim(\sigma)\right)
\end{align\*}
$$


Note that this is exactly the formula found in Ewald's *Combinatorial Complexity and Algebraic Geometry* mentioned in the comments above.


