I am trying to solve exercise 3.3.10 from Cox' book on toric varieties. I have no trouble with part (a), but it seems to me that for part (b) I have to wirte down the variety $X_{\Sigma}$ explicitly, and I'm having some trouble doing that. They consider the lattice $N= \mathbb{Z}^{n+1}/\mathbb{Z}(1,1,2)$ for this variety, and I know that this is isomorphic to $\mathbb{Z^2}$ as 1 and 2 are coprime, but I do now know how to explicitly write down the dual lattice, so this is the first problem. In the example the exercise is referring to, they took the generators of the dual cones to be in $N$ also, so I find this a bit confusing also.
Any help would be appreciated!
Quick update: I managed to calculate all the dual cones now, and the next thing I need to do is calculate all the affine toric varieties associated to these cones, and how they are glued together to a new toric variety. The problem is that I still have no clue how to explicitly calculate the map they want in part (b), and also I have no idea which variety that will be isomorphic to at the end.