= Solution
The images of the three rays of $\sigma$ are
$$
\widehat\varphi(e_1)=f_1,qquad
\widehat\varphi(e_2)=f_2,qquad
\widehat\varphi(e_3)=-f_1-f_2.
$$
No cone of the complete fan of $\mathbb P^2$ contains all three rays, so $\widehat\varphi(\sigma)$ is not contained in one target cone. Thus $\widehat\varphi$ 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 are
$$
u\longmapsto x_1/x_3,
\qquad
v\longmapsto x_2/x_3.
$$
Under the standard torus chart $[u:v:1]\in\mathbb P^2$, the resulting <rational map of toric varieties from a lattice homomorphism> is
$$
f:\mathbb A^3\dashrightarrow\mathbb P^2,
\qquad
(x_1,x_2,x_3)\longmapsto[x_1:x_2:x_3].
$$
Its homogeneous coordinates vanish simultaneously exactly at the origin. Hence its <indeterminacy locus> is $\{0\}$.
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