A normal toric variety corresponds to a strongly convex rational polyhedral fan in affine space. In this package, the fan associated to a normal
d-dimensional toric variety lies in the rational vector space
ℚd with underlying lattice
N = ℤd. The fan is encoded by the minimal nonzero lattice points on its rays and the set of rays defining the maximal cones (i.e. a maximal cone is not properly contained in another cone in the fan). The rays are ordered and indexed by nonnegative integers:
0,…, n. Using this indexing, a maximal cone in the fan corresponds to a sublist of
{0,…,n}; the entries index the rays that generate the cone.
The examples show the maximal cones for the projective plane, projective 3-space, a Hirzebruch surface, and a weighted projective space.
PP2 = projectiveSpace 2; |
#rays PP2 |
max PP2 |
PP3 = projectiveSpace 3; |
#rays PP3 |
max PP3 |
FF7 = hirzebruchSurface 7; |
#rays FF7 |
max FF7 |
X = weightedProjectiveSpace {1,2,3}; |
#rays X |
max X |
A list corresponding to the maximal cones in the fan is part of the defining data of a toric variety.