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 $\QQ^d$ with underlying lattice $N = {\ZZ}^d$. 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. There is a canonical bijection between the rays and torus-invariant Weil divisor on the toric variety.
i1 : PP2 = toricProjectiveSpace 2; |
i2 : rays PP2
o2 = {{-1, -1}, {1, 0}, {0, 1}}
o2 : List
|
i3 : dim PP2 o3 = 2 |
i4 : weilDivisorGroup PP2
3
o4 = ZZ
o4 : ZZ-module, free
|
i5 : PP2_0
o5 = PP2
0
o5 : ToricDivisor on PP2
|
i6 : PP3 = toricProjectiveSpace 3; |
i7 : rays PP3
o7 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
o7 : List
|
i8 : dim PP3 o8 = 3 |
i9 : weilDivisorGroup PP3
4
o9 = ZZ
o9 : ZZ-module, free
|
i10 : FF7 = hirzebruchSurface 7; |
i11 : rays FF7
o11 = {{1, 0}, {0, 1}, {-1, 7}, {0, -1}}
o11 : List
|
i12 : dim FF7 o12 = 2 |
i13 : weilDivisorGroup FF7
4
o13 = ZZ
o13 : ZZ-module, free
|
i14 : X = weightedProjectiveSpace {1,2,3};
|
i15 : rays X
o15 = {{-2, -3}, {1, 0}, {0, 1}}
o15 : List
|
i16 : weilDivisorGroup X
3
o16 = ZZ
o16 : ZZ-module, free
|
When the normal toric variety is nondegerenate, the number of rays equals the number of variables in the total coordinate ring.
i17 : #rays X == numgens ring X o17 = true |
In this package, 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, so this method does no computation.