# toricProjectiveSpace(ZZ) -- make a projective space as a normal toric variety

## Synopsis

• Function: toricProjectiveSpace
• Usage:
toricProjectiveSpace 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 base name for the indexed variables in the total coordinate ring
• Outputs:
• , that is projective $d$-space

## Description

Projective $d$-space is a smooth complete normal toric variety. The rays are generated by the standard basis $e_1, e_2, \dots,e_d$ of $\ZZ^d$ together with vector $-e_1-e_2-\dots-e_d$. The maximal cones in the fan correspond to the $d$-element subsets of $\{ 0,1, \dots,d\}$.

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 (isWellDefined PP1 and 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 ] 32003 0 3 o12 : PolynomialRing i13 : ideal PP3 o13 = ideal (y , y , y , y ) 3 2 1 0 ZZ o13 : Ideal of -----[y ..y ] 32003 0 3 i14 : assert (isWellDefined PP3 and isSmooth PP3 and isComplete PP3)