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NormalToricVarieties :: rays(NormalToricVariety)

rays(NormalToricVariety) -- get the rays of the associated fan

Synopsis

Description

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 d 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.

See also