Picard group of toric varieties

Question

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?

Answer 1

Edit: I have elaborated on this approach to the Picard group in Section 2 of my preprint.

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.