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First make a toric fan simplicial by compatible rational subdivisions. In a nonregular simplicial cone, choose a nonregular face minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, with every $0<{\alpha }_i<1$: a zero coefficient would put it in a regular proper face and force all coefficients to be integers. A star subdivision at the primitive point on this ray replaces a generator by $w$ and multiplies the affected lattice determinant by a number strictly between zero and one. The maximum multiplicity and the number of maximal cones realizing it decrease lexicographically, giving a finite toric resolution of singularities.

Insert the ray generated by $w=(0,1)$ between $v_1=(1,0)$ and $v_2=(-1,3)$. Sincethe subdivided fan is smooth and gives a toric resolution of singularities. The unique exceptional divisor is $D_w$. Asthe self-intersection formula for a toric surface divisor gives $D_w^2=-3$.

The primitive ray generators of $\sigma$ areand $|\det (v_0,v_3)|=3$. They do not form a lattice basis, so the smoothness criterion for a toric variety shows that $U_{\sigma }$ is singular. The presentation from part (i) gives the same conclusion by the Jacobian criterion: all derivatives of $B^3-AC$ vanish at $A=B=C=0$.

Insert the primitive raysLet $\Sigma$ consist of the conesand all their faces. Sinceevery cone of $\Sigma$ is smooth. Its support is $\sigma$, so this fan subdivision induces a proper birational toric morphismThe source is smooth, and $p$ is an isomorphism over the dense algebraic torus; it is therefore a toric resolution of singularities.

Insert the primitive rayinside the singular cone. The corresponding star subdivision replaces $⟨v_0,v_1⟩$ by $⟨v_0,v_3⟩$ and $⟨v_3,v_1⟩$. 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 $\tilde{X}\to X$ is therefore a toric resolution of singularities. Its exceptional invariant curve has self-intersection $-2$, and $\tilde{X}$ is the Hirzebruch surface ${\mathbb{F}}_2$.