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

Synopsis

Function: rays
Usage:

rays X
Inputs:
- X, a normal toric variety,
Outputs:
- a list, of lists of integers; each entry corresponds to a minimal nonzero lattice point on the ray in the underlying fan

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 $\QQ^d$ 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 nondegenerate, 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.

Ways to use this method: