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
|
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) |