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

The determinants of the cones $⟨v_1,v_2⟩$ and $⟨v_2,v_0⟩$ have absolute value one, whileThe smoothness criterion for a toric variety therefore shows that the affine chart for $⟨v_0,v_1⟩$ is singular; it is the cyclic quotient singularity of type $\frac{1}{2}(1,1)$. Thus $X$ is a singular proper toric surface.

Solved by gpt-5.6-sol high.

Let $q:N_{\mathbb{R}}\to N(\tau {)}_{\mathbb{R}}=N_{\mathbb{R}}/\mathrm{span}(\tau )$ be the quotient map. The images $q(\sigma )$ for $\tau ⪯\sigma$ satisfy the face and intersection axioms, so $\mathrm{Star}(\tau )$ is a fan.

To prove completeness, take $\bar{v}\in N(\tau {)}_{\mathbb{R}}$, choose a lift $v$, and choose $w$ in the relative interior of $\tau$. Since $\Sigma$ is complete, each $v+mw$ lies in some cone of $\Sigma$. There are finitely many cones, so one cone $\sigma$ contains $v+m_{j}w$ for an unbounded sequence $m_j$. Closedness gives $w\in \sigma$. Because $w$ lies in the relative interior of the cone $\tau$ and $\sigma \cap \tau$ is a face of $\tau$, this forces $\tau ⪯\sigma$. Finally,Thus every point of the quotient lies in the support of $\mathrm{Star}(\tau )$, proving that it is a complete fan.

Solved by gpt-5.6-sol high.

A toric variety is proper exactly when its fan is complete.