# toricProjectiveSpace(ZZ) -- make a projective space

## Synopsis

• Usage:
toricProjectiveSpace d
• Function: toricProjectiveSpace
• Inputs:
• Optional inputs:
• CoefficientRing => a ring, default value QQ, that specifies the coefficient ring of the total coordinate ring
• Variable => , default value x, the base symbol for the indexed variables in the total coordinate ring
• Outputs:
• , projective d-space

## Description

Projective d-space is a smooth complete normal toric variety. The rays are generated by the standard basis e1, e2, …,ed of d 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 = toricProjectiveSpace 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 = toricProjectiveSpace (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)`