Explicit example of a toric flip

Question

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes).

Does anyone know where I can find something like this?

Answer 1

Consider the cube with vertices $(\pm1,\pm1,\pm1)$ and subdivide one of the faces by its diagonal. Consider the fan whose cones are spanned by the cells of the resulting polyhedral complex. Now, you can do this in two ways, so we get two fans $F_1$ and $F_2$.

Moreover, if we subdivide that face with the two diagonals, and build the cones on those cells, we get a third fan $F_3$ which refines both $F_1$ and $F_2$.

The swap $F_1\leftarrow F_3\rightarrow F_2$ is a flip, the corresponding construction on the associated toric varieties is a toric flip.

More generally, pick any polyhedron and subdivide one of its faces until you get triangles. Then —unless you picked a face which was a triangle!— you can always pick two triangles sharing a side and flip the cuadrilateral they form by removing that segment and adding back in the other diagonal of the cuadrilateral.