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I am looking for an example of projective toric threefold $X = \mathbb P_\Delta$ that comes from a rational polytope $\Delta$, where $\Delta$ is a triangular bipyramid (click the word for its image).

I wish that $X$ has three singularities of type $A_1$ that correspond to the vertices with valence 4 of $\Delta$ and no other singuarities. I am wondering if anyone know any example.

Thanks for your attention.

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I assume you mean dim 3 ODPs? The $A_1$ singularity typically refers to surface ODPs.

I don't think such variety can be constructed. On the fan side, the minimum generators of the rays form something that looks like a triangular prism. The ODP condition means that the sides are flat and are minimum squares. Then the two triangular faces are at integer distance $1$ from each other, and there is no place for the origin inside the polytope.

You can presumably construct a toric DM stack by sticking the origin inside a triangular prism and enlarging the lattice, but I don't know if you want to go this route.

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