I cannot understand which is the kernel of the following map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$ with $$ (t_1,t_2,t_3) \mapsto \left(\frac{t_2}{t_1}, \frac{t_3}{t_1}\right) $$ In other words I do not see which elements of $ \mathbb{C^*}^3$ map to zero in $ \mathbb{C^*}^2$ under $\phi$. Of course $t_1$ cannot be because we divide by it. Therefore which ones belong to the kernel?
Note that this is the standard map induced when one tries to construct toric projective varieties.
First of all, the map is clearly not a map from $\mathbb C^3$ to $\mathbb C^2$, but a map from $C^3\setminus\{(0,t_2,t_3)|t_2,t_3\in\mathbb C\}$ to $\mathbb C^2$. The map is clearly undefined on $(0,1,1)$ for example.
Now, if $\phi(t_1,t_2,t_3)$ is defined, then you can find the kernel using this hint: $$\frac{t_2}{t_1} = 0 \iff t_2 = 0$$