# nefGenerators(NormalToricVariety) -- compute generators of the nef cone

## Synopsis

• Function: nefGenerators
• Usage:
nefGenerators X
• Inputs:
• X, ,
• Outputs:
• , whose columns generate the nef cone of X as a convex cone in the Picard group

## Description

The nef cone of a variety is the cone generated by classes of nef Cartier divisors in vector space of Cartier divisors modulo numerical equivalence. On a normal toric variety, numerical equivalence and linear equivalence coincide, so the nef cone lies in the Picard group. Assume that the normal toric variety is non-degenerate, its nef cone is a rational polyhedral cone in the Picard group; see Theorem 6.3.20 in Cox-Little-Schenck's Toric Varieties. This function calculates generators for the rays of this cone, and returns a matrix whose columns correspond to these generates (expressed as vectors in the chosen basis for the Picard group).

For some of our favourite normal toric varieties, we choose a basis for the Picard group which makes the nef cone into the positive orthant.

 i1 : nefGenerators toricProjectiveSpace 1 o1 = | 1 | 1 1 o1 : Matrix ZZ <--- ZZ i2 : nefGenerators toricProjectiveSpace 3 o2 = | 1 | 1 1 o2 : Matrix ZZ <--- ZZ i3 : nefGenerators normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3)) o3 = | 1 | 1 1 o3 : Matrix ZZ <--- ZZ i4 : nefGenerators hirzebruchSurface 7 o4 = | 1 0 | | 0 1 | 2 2 o4 : Matrix ZZ <--- ZZ i5 : nefGenerators kleinschmidt (3,{0,1}) o5 = | 1 0 | | 0 1 | 2 2 o5 : Matrix ZZ <--- ZZ i6 : nefGenerators smoothFanoToricVariety (2,3) o6 = | 1 0 0 | | 0 1 0 | | 0 0 1 | 3 3 o6 : Matrix ZZ <--- ZZ i7 : nefGenerators smoothFanoToricVariety (3,12) o7 = | 1 0 0 0 | | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | 4 4 o7 : Matrix ZZ <--- ZZ i8 : nefGenerators smoothFanoToricVariety (4,90) o8 = | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 1 0 | | 0 0 0 0 1 | 5 5 o8 : Matrix ZZ <--- ZZ

In general, the nef cone need not even be simplicial.

 i9 : nefGenerators smoothFanoToricVariety (2,4) o9 = | 1 0 0 1 0 | | 0 1 1 0 0 | | 0 0 1 0 1 | | 0 0 0 1 1 | 4 5 o9 : Matrix ZZ <--- ZZ i10 : nefGenerators smoothFanoToricVariety (3,16) o10 = | 1 0 1 0 0 0 | | 1 0 0 0 0 1 | | 0 1 1 0 0 0 | | 0 0 0 1 0 0 | | 0 0 0 0 1 1 | 5 6 o10 : Matrix ZZ <--- ZZ i11 : nefGenerators smoothFanoToricVariety (4,120) o11 = | 1 0 1 0 0 0 0 0 0 0 | | 0 1 0 1 0 0 0 0 0 0 | | 0 0 1 0 1 0 0 0 0 0 | | 0 0 0 1 1 0 0 0 0 0 | | 0 0 0 0 0 1 0 1 0 0 | | 0 0 0 0 0 0 1 0 1 0 | | 0 0 0 0 0 0 0 1 0 1 | | 0 0 0 0 0 0 0 0 1 1 | 8 10 o11 : Matrix ZZ <--- ZZ

There are smooth complete normal toric varieties with no nontrivial nef divisors.

 i12 : X = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{0,-1,2},{0,0,-1},{-1,1,-1},{-1,0,-1},{-1,-1,0}},{{0,1,2},{0,2,3},{0,3,4},{0,4,5},{0,1,5},{1,2,7},{2,3,7},{3,4,7},{4,5,6},{4,6,7},{5,6,7},{1,5,7}}); i13 : assert (isComplete X and not isProjective X and isSmooth X) i14 : picardGroup X 5 o14 = ZZ o14 : ZZ-module, free i15 : assert (nefGenerators X == 0)