 Solution

The images of the three rays of $\sigma$ areNo cone of the complete fan of ${\mathbb{P}}^2$ contains all three rays, so $\hat{\phi }(\sigma )$ is not contained in one target cone. Thus $\hat{\phi }$ is not a morphism of fans and does not define a regular toric morphism on all of $U_{\sigma }$.

On the dense algebraic torus, the pullbacks of the two standard target characters areUnder the standard torus chart $[u:v:1]\in {\mathbb{P}}^2$, the resulting rational map of toric varieties from a lattice homomorphism isIts homogeneous coordinates vanish simultaneously exactly at the origin. Hence its indeterminacy locus is $\{0\}$.