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

- 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.

## See also

- making normal toric varieties -- information about the basic constructors
- finding attributes and properties -- information about accessing features of a normal toric variety
- max(NormalToricVariety) -- get the maximal cones in the associated fan
- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/ToricVarietiesDocumentation.m2:1444:0.