Cox rings of toric varieties Let $X$ be a projective toric variety over a field $k$.  Is it true that the Cox ring of $X$ is the polynomial ring over the Picard group of $X$?  If not, what is the significance of the Picard group of a toric variety in terms of its Cox ring?
 A: I'm not sure what you mean exactly, since the Cox ring of a toric variety is finitely-generated, while the Picard group is infinite. My guess is that you mean to ask if the Cox ring is a polynomial ring with generators given by a basis of the Picard group. If so, that is not true.
The Cox ring (of any variety) is (more or less) the ring of sections of all effective line bundles. So first, line bundles that don't have sections don't contribute generators to the Cox ring. Second, a single line bundle (hence a single element of the Picard group) may have lots of linearly independent section, so may contribute many generators.
Already $\mathbf{P}^n$ gives a counterexample. Here the Picard group is $\mathbf{Z}$, generated by the line bundle $O(1)$. Negative powers of $O(1)$ don't have any sections, whereas sections of positive powers come from sections of $O(1)$ by tensoring. So the Cox ring is a polynomial ring generated by a basis of the sections of $O(1)$; that is, isomorphic to $k[x_0,\ldots,x_n]$.
It is, however, still the case that the Cox ring of a toric variety is always a polynomial ring, with generators given by certain bases of sections of all effective line bundles. Cox's original paper (http://arxiv.org/pdf/alg-geom/9210008v2.pdf) gives the details. 
Finally, a possible answer for your last question: the significance of the Picard group is that (for any variety with first Betti number zero, in particular toric varieties), the Cox ring is multigraded by the monoid of effective line bundles in the Picard group. It can happen, for example, that this monoid isn't finitely generated, which tells you that the Cox ring isn't either. 
