Projective d-space is a smooth complete normal toric variety. The rays are generated by the standard basis e1, e2, …,ed of ℤd together with vector -e1-e2-…-ed. The maximal cones in the fan correspond to the d-element subsets of {0,1, …,d}.
The examples illustrate the projective line and projective 3-space.
i1 : PP1 = projectiveSpace 1; |
i2 : rays PP1 o2 = {{-1}, {1}} o2 : List |
i3 : max PP1 o3 = {{0}, {1}} o3 : List |
i4 : dim PP1 o4 = 1 |
i5 : ring PP1 o5 = QQ[x , x ] 0 1 o5 : PolynomialRing |
i6 : ideal PP1 o6 = ideal (x , x ) 1 0 o6 : Ideal of QQ[x , x ] 0 1 |
i7 : assert(isSmooth PP1 and isComplete PP1) |
i8 : PP3 = projectiveSpace(3, CoefficientRing => ZZ/32003, Variable => y); |
i9 : rays PP3 o9 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}} o9 : List |
i10 : max PP3 o10 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}} o10 : List |
i11 : dim PP3 o11 = 3 |
i12 : ring PP3 ZZ o12 = -----[y , y , y , y ] 32003 0 1 2 3 o12 : PolynomialRing |
i13 : ideal PP3 o13 = ideal (y , y , y , y ) 3 2 1 0 ZZ o13 : Ideal of -----[y , y , y , y ] 32003 0 1 2 3 |
i14 : assert(isSmooth PP3 and isComplete PP3) |