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. As a result, each ray in the fan is determined by the minimal nonzero lattice point it contains. Each such lattice point is given as a
list of
d integers.
The examples show the rays for the projective plane, projective
3-space, a Hirzebruch surface, and a weighted projective space. Observe that there is a bijection between the rays and torus-invariant Weil divisor on the toric variety.
PP2 = projectiveSpace 2; |
rays PP2 |
dim PP2 |
wDiv PP2 |
PP3 = projectiveSpace 3; |
rays PP3 |
dim PP3 |
wDiv PP3 |
FF7 = hirzebruchSurface 7; |
rays FF7 |
dim FF7 |
wDiv FF7 |
X = weightedProjectiveSpace {1,2,3}; |
rays X |
dim X |
wDiv X |
When
X is nondegerenate, the number of rays equals the number of variables in the total coordinate ring.
#rays X == numgens ring X |
An ordered list of the minimal nonzero lattice points on the rays in the fan is part of the defining data of a toric variety.