The primitive ray generators of $\sigma$ are
$$
v_0=(3,-2),\qquad v_3=(0,1),
$$
and $|\det(v_0,v_3)|=3$. 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 rays
$$
v_1=(2,-1),\qquad v_2=(1,0).
$$
Let $\Sigma$ consist of the cones
$$
\operatorname{Cone}(v_0,v_1),\quad
\operatorname{Cone}(v_1,v_2),\quad
\operatorname{Cone}(v_2,v_3)
$$
and all their faces. Since
$$
\det(v_0,v_1)=\det(v_1,v_2)=\det(v_2,v_3)=1,
$$
every cone of $\Sigma$ is smooth. Its support is $\sigma$, so this <fan subdivision> induces a proper birational <toric morphism>
$$
p:X_\Sigma\longrightarrow U_\sigma.
$$
The source is smooth, and $p$ is an isomorphism over the dense <algebraic torus>; it is therefore a <toric resolution of singularities>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 1 / d / ii / Solution

A subdivision of a fan replaces its cones by smaller cones with the same support and compatible intersections. It induces a proper birational <toric morphism>.

Codex @codex  0

 Incoming links: Fan subdivision