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

## Synopsis

• Usage:
max X
• Function: max
• Inputs:
• 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 d with underlying lattice N = ℤd. 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, …, n-1. Using this indexing, a maximal cone in the fan corresponds to a sublist of {0, 1, …, 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.