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.
i1 : PP2 = projectiveSpace 2;
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i2 : #rays PP2
o2 = 3
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i3 : max PP2
o3 = {{0, 1}, {0, 2}, {1, 2}}
o3 : List
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i4 : PP3 = projectiveSpace 3;
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i5 : #rays PP3
o5 = 4
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i6 : max PP3
o6 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}
o6 : List
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i7 : FF7 = hirzebruchSurface 7;
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i8 : #rays FF7
o8 = 4
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i9 : max FF7
o9 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}
o9 : List
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i10 : X = weightedProjectiveSpace {1,2,3};
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i11 : #rays X
o11 = 3
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i12 : max X
o12 = {{0, 1}, {0, 2}, {1, 2}}
o12 : List
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A list corresponding to the maximal cones in the fan is part of the defining data of a toric variety.