# Picard group of toric varieties

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?

• You might want to have a look at Section VI.2 in G. Ewald, Combinatorial convexity and algebraic geometry, Springer GTM 168 (1996). – Fred Rohrer Nov 9 '17 at 7:01
• @FredRohrer Thanks for the helpful comment. On page 273 of this book, I found 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? – Mtheorist Nov 9 '17 at 15:50
• Are you prevented from taking $Q_i^{(r)} = Q_i^{(s)}$ or other redundancies in the equivalence relation? – AHusain Nov 9 '17 at 18:34
• Dear @Mtheorist, I have not thought about these things for quite some time now, and hence, unfortunately, I do not know the answer to your question(s). – Fred Rohrer Nov 9 '17 at 19:08
• By quickly looking at the paper you linked, it seems to me that "the physicists" definition of toric varieties they are working with includes the assumtion that the fan is simplicial. They specifically mention "fan and triangulation". Simplicial toric varieties are exactly those for which $\lambda=0$, but are also characterized as those for which the variety is a geometric quotient from the equivalence relation (as opposed to a categorical quotient in general). – Gjergji Zaimi Nov 12 '17 at 6:26