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A fan is complete when the union of all its cones is the whole ambient real vector space. A toric variety is proper exactly when its fan is complete.

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.