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.
i1 : PP2 = projectiveSpace 2;
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i2 : rays PP2
o2 = {{-1, -1}, {1, 0}, {0, 1}}
o2 : List
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i3 : dim PP2
o3 = 2
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i4 : wDiv PP2
3
o4 = ZZ
o4 : ZZ-module, free
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i5 : PP3 = projectiveSpace 3;
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i6 : rays PP3
o6 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
o6 : List
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i7 : dim PP3
o7 = 3
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i8 : wDiv PP3
4
o8 = ZZ
o8 : ZZ-module, free
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i9 : FF7 = hirzebruchSurface 7;
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i10 : rays FF7
o10 = {{1, 0}, {0, 1}, {-1, 7}, {0, -1}}
o10 : List
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i11 : dim FF7
o11 = 2
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i12 : wDiv FF7
4
o12 = ZZ
o12 : ZZ-module, free
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i13 : X = weightedProjectiveSpace {1,2,3};
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i14 : rays X
o14 = {{-2, -3}, {1, 0}, {0, 1}}
o14 : List
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i15 : dim X
o15 = 2
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i16 : wDiv X
3
o16 = ZZ
o16 : ZZ-module, free
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When
X is nondegerenate, the number of rays equals the number of variables in the total coordinate ring.
i17 : #rays X == numgens ring X
o17 = true
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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.