# affineSpace(ZZ) -- make an affine space as a normal toric variety

## Synopsis

• Function: affineSpace
• Usage:
affineSpace d
• Inputs:
• d, an integer, a positive integer
• Optional inputs:
• CoefficientRing => a ring, default value QQ, that specifies the coefficient ring of the total coordinate ring
• Variable => , default value x, that specifies the base name for the indexed variables in the total coordinate ring
• Outputs:
• , that is affine $d$-space

## Description

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)