Affine $d$-space is a smooth normal toric variety. The rays are generated by the standard basis $e_1, e_2, \dots, e_d$ of $\ZZ^d$, and the maximal cone in the fan correspond to the $d$-element subsets of $\{ 0, 1, \dots, d-1 \}$.
The examples illustrate the affine line and affine $3$-space.
i1 : AA1 = affineSpace 1; |
i2 : rays AA1
o2 = {{1}}
o2 : List
|
i3 : max AA1
o3 = {{0}}
o3 : List
|
i4 : dim AA1 o4 = 1 |
i5 : assert (isWellDefined AA1 and not isComplete AA1 and isSmooth AA1) |
i6 : AA3 = affineSpace (3, CoefficientRing => ZZ/32003, Variable => y); |
i7 : rays AA3
o7 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
o7 : List
|
i8 : max AA3
o8 = {{0, 1, 2}}
o8 : List
|
i9 : dim AA3 o9 = 3 |
i10 : ring AA3
ZZ
o10 = -----[y ..y ]
32003 0 2
o10 : PolynomialRing
|
i11 : assert (isWellDefined AA3 and not isComplete AA3 and isSmooth AA3) |