Calculating the fan of projective $n$-space I am reading Fulton's book on Toric Geometry, one of the exercises is to calculate the fan of projective $n$-space. I have no idea how to do this, any advice would be welcome.
 A: I will assume for simplicity that action of torus $T=(\mathbb{C}^\times)^n$ is given by the formula $(t_1,\ldots,t_n)\circ [z_0:\ldots :z_n]=[z_0:t_1 z_1:\ldots :t_n z_n].$
Observe that standard affine covering of $\mathbb{C}P^n$ is in fact $T$-invariant. 
And also it is clear that on each copy of $\mathbb{A}^n$ torus $T$ acts with dense orbit (i.e. it is an affine toric variety). We know that all varieties of this type (at least normal) are given by the cones.
First consider $\mathbb{A}_0^n=\mathbb{C}P^n\setminus\{z_0=0\}.$ This toric variety is given by the cone $\mathbb{Z}_{\ge 0}\langle e_1,\ldots, e_n\rangle$ since torus acts on coordinate functions by its basis characters.
Now let us consider an arbitrary $\mathbb{A}_i^n=\mathbb{C}P^n\setminus\{z_i=0\}.$ We can compute the formula for $T$-action on this affine variety:
$$
(t_1,\ldots,t_n)\circ(\frac{z_0}{z_i},\ldots,\frac{z_j}{z_i},\ldots, \widehat{\frac{z_i}{z_i}}, \ldots, \frac{z_n}{z_i})=(\frac{z_0}{z_i}t_i^{-1},\ldots,\frac{z_j}{z_i}t_jt_i^{-1},\ldots, \widehat{\frac{z_i}{z_i}}, \ldots, \frac{z_n}{z_i}t_nt_i^{-1}).
$$
We see now that $T$ acts on coordinate functions by characters of the form $\{-e^*_i, \ldots, e^*_j-e^*_i, \ldots, e^*_n-e^*_i\}$, they define certain cone $\sigma^\vee$ in character lattice. By passing from character lattice to the subgroups lattice we obtain a dual cone $\sigma=\mathbb{Z}_{\ge 0}\langle-e_1-e_2, e_1, e_2, \ldots, \widehat{e_i}, \ldots, e_n\rangle.$ This cone gives us desired toric variety $\mathbb{A}^n_i.$ 
Finally the fan could be glued from this data in an evident manner. 
