Take the <weighted projective plane> $X=\mathbb P(1,1,2)$. Its <fan in toric geometry>[fan] in $N=\mathbb Z^2$ has rays generated by
$$
v_0=(-1,-2),\qquad v_1=(1,0),\qquad v_2=(0,1),
$$
and all three two-dimensional cones spanned by adjacent rays. Their union is $N_{\mathbb R}$, so the fan is <complete fan>[complete] and the <toric variety> $X$ is <proper toric variety>[proper].

The determinants of the cones $\langle v_1,v_2\rangle$ and $\langle v_2,v_0\rangle$ have absolute value one, while
$$
|\det(v_0,v_1)|=2.
$$
The <smoothness criterion for a toric variety> therefore shows that the affine chart for $\langle v_0,v_1\rangle$ is singular; it is the cyclic quotient singularity of type $\frac12(1,1)$. Thus $X$ is a singular proper toric surface.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 1 / a / Solution

A toric variety is proper exactly when its <fan in toric geometry>[fan] is <complete fan>[complete].

Proper toric variety

{title2=$\Sigma$}
{wiki=Fan_(geometry)}

A fan $\Sigma$ in $N_{\mathbb R}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.


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 Incoming links: Fan in toric geometry