orbits(NormalToricVariety,ZZ) -- get a list of the torus orbits (a.k.a. cones in the fan) of a given dimension

Synopsis

Function: orbits
Usage:

orbits (X, i)
Inputs:
- X, a normal toric variety,
- i, an integer, determining the dimension of the orbits
Outputs:
- a list, of lists of integers indexing the torus orbits

Description

A normal toric variety is a disjoint union of its orbits under the action of its algebraic torus. These orbits are in bijection with the cones in the associated fan. Each cone is determined by the rays it contains. In this package, the rays are ordered and indexed by nonnegative integers: $0, 1, \dots, n$. Using this indexing, an orbit or cone corresponds to a sublist of $\{ 0, 1, \dots, n \}$; the entries index the rays that generate the cone.

The projective plane has three fixed points and three fixed curves (under the action of its torus), and projective $3$-space has four fixed points, six fixed curves, and four divisors.

i1 : PP2 = toricProjectiveSpace 2;

i2 : orbits (PP2,0)

o2 = {{0, 1}, {0, 2}, {1, 2}}

o2 : List

i3 : orbits (PP2,1)

o3 = {{0}, {1}, {2}}

o3 : List

i4 : orbits (PP2,2)

o4 = {{}}

o4 : List

i5 : PP3 = toricProjectiveSpace 3;

i6 : orbits (PP3,0)

o6 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}

o6 : List

i7 : orbits (PP3,1)

o7 = {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}}

o7 : List

i8 : orbits (PP3,2)

o8 = {{0}, {1}, {2}, {3}}

o8 : List

i9 : orbits (PP3,3)

o9 = {{}}

o9 : List

Here is a non-simplicial example. Since it is nondegenerate, the fixed points correspond to the maximal cones in the fan. The rays always correspond to the divisors.

i10 : X = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));

i11 : orbits (X,0)

o11 = {{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7},
      -----------------------------------------------------------------------
      {4, 5, 6, 7}}

o11 : List

i12 : assert (orbits (X,0) === max X)

i13 : orbits (X,1)

o13 = {{0, 1}, {0, 2}, {0, 4}, {1, 3}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4,
      -----------------------------------------------------------------------
      5}, {4, 6}, {5, 7}, {6, 7}}

o13 : List

i14 : orbits (X,2)

o14 = {{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}}

o14 : List

i15 : assert (orbits (X,2) === apply (#rays X, i -> {i}))

i16 : orbits (X,3)

o16 = {{}}

o16 : List

i17 : assert (orbits (X,3) === {{}})

The following degenerate example has no fixed points.

i18 : U = normalToricVariety ({{4,-1,0},{0,1,0}},{{0,1}});

i19 : assert isDegenerate U

i20 : orbits (U,0)

o20 = {}

o20 : List

i21 : orbits (U,1)

o21 = {{0, 1}}

o21 : List

i22 : orbits (U,2)

o22 = {{0}, {1}}

o22 : List

i23 : orbits (U,3)

o23 = {{}}

o23 : List

i24 : dim U

o24 = 3

To routine extracts the requested list from hashTable returned by orbits(NormalToricVariety).

Ways to use this method: