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NormalToricVarieties :: affineSpace(ZZ)

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

Synopsis

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)

See also