= Solution
Insert the primitive ray
$$
v_3=(0,-1)=\frac12(v_0+v_1)
$$
inside the singular cone. The corresponding <star subdivision> replaces $\langle v_0,v_1\rangle$ by $\langle v_0,v_3\rangle$ and $\langle v_3,v_1\rangle$. Both new determinants have absolute value one, as do the two unchanged cones, so the subdivided fan is smooth. The induced proper birational <toric morphism> $\widetilde X\to X$ is therefore a <toric resolution of singularities>. Its exceptional invariant curve has self-intersection $-2$, and $\widetilde X$ is the <Hirzebruch surface> $\mathbb F_2$.
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