max(NormalToricVariety) -- get the maximal cones in the associated fan

Synopsis

Function: max
Usage:

max X
Inputs:
- X, a normal toric variety,
Outputs:
- a list, of lists of nonnegative integers; each entry indexes the rays that generate a maximal cone in the 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 $N = \ZZ^d$. The fan is encoded by the minimal nonzero lattice points on its rays and the set of rays defining the maximal cones (where a maximal cone is not properly contained in another cone in the fan). The rays are ordered and indexed by nonnegative integers: $0, 1, \dots, n-1$. Using this indexing, a maximal cone in the fan corresponds to a sublist of $\{ 0, 1, \dots, n-1 \}$; the entries index the rays that generate the cone.

The examples show the maximal cones for the projective line, projective $3$-space, a Hirzebruch surface, and a weighted projective space.

i1 : PP1 = toricProjectiveSpace 1;

i2 : # rays PP1

o2 = 2

i3 : max PP1

o3 = {{0}, {1}}

o3 : List

i4 : PP3 = toricProjectiveSpace 3;

i5 : # rays PP3

o5 = 4

i6 : max PP3

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

o6 : List

i7 : FF7 = hirzebruchSurface 7;

i8 : # rays FF7

o8 = 4

i9 : max FF7

o9 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}

o9 : List

i10 : X = weightedProjectiveSpace {1,2,3};

i11 : # rays X

o11 = 3

i12 : max X

o12 = {{0, 1}, {0, 2}, {1, 2}}

o12 : List

In this package, a list corresponding to the maximal cones in the fan is part of the defining data of a normal toric variety, so this method does no computation.

Ways to use this method: