Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous coordinate $z_i$. These are the coordinates in $\mathbb{C}^n$. Let us try to work with $\mathbb{P}^2$, namely the complex projective plane which is itself a toric variety. We have the matrix $v = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ -1 & -1 \ \end{array} \right)$. This comes from the fan of $\mathbb{P}^2$ which has three one dimensional cones $v_1=(1,0), v_2=(0,1), v_3=(-1,-1)$ corresponding to $z_1,z_2,z_3$ of the $\mathbb{C}^3$ out of which we construct $\mathbb{P}^2$. Then there exists the induced map $\phi: \mathbb{C}^3 \to \mathbb{C}^2$ for which maps $$(z_1,z_2,z_3) \mapsto (\prod_{i=1}^nz_i^{v_i^1},\prod_{i=1}^nz_i^{v_i^2} )=(z_1z_3^{-1}, z_2z_3^{-1}).$$ We define a group $\tilde{G} = (\mathbb{C}^*)^{3-2=1} = \text{Ker}(\phi)$. What exactly are the elements of this kernel? I see that $x_1$ is not mapped under $\phi$. Is that it? Then, we see that $\tilde{G}$ acts on $\mathbb{C}^n$ as $$ \tilde{G} \supset (\mathbb{C}^*)_a: (z_1, z_2, z_3) \mapsto (\lambda^{Q_1^a}z_1, \lambda^{Q_2^a}z_2, \lambda^{Q_3^a}z_3 ) $$ for each $a$, where the charge vectors are in the kernel of $\phi$.
My confusions:
- I cannot really understand what are the elements of the kernel. I see that the element $x_1x_2$ is not mapped under $\phi$.
- I do not see how these charges $Q_i^a$ are related to the kernel of $\phi$.
- Finally nor I see why it is true being in the kernel implies that $$ \sum_i (v_i^k)Q_i^a = 0$$