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

## Synopsis

• Function: max
• Usage:
max X
• Inputs:
• X, ,
• 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.