Toric Variety - The Toric Variety of A Convex Polytope

The Toric Variety of A Convex Polytope

The fan of a rational convex polytope in N consists of the cones over its proper faces. The toric variety of the polytope is the toric variety of its fan. A variation of this construction is to take a rational polytope in the dual of N and take the toric variety of its polar set in N.

The toric variety has a map to the polytope in the dual of N whose fibers are topological tori. For example, the complex projective plane CP2 may be represented by three complex coordinates satisfying

where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following U(1) action:

The approach of toric geometry is to write

The coordinates are non-negative, and they parameterize a triangle because

that is,

The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of ; the phase of can be chosen real and positive by the symmetry.

However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at or or because the phase of becomes inconsequential, respectively.

The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case).