# projectiveSpace -- constructs projective space as a GKM variety

## Synopsis

• Usage:
projectiveSpace n
projectiveSpace(n,R)
• Inputs:
• Outputs:
• ,

## Description

Given an integer $n$ this method constructs the n-dimensional projective space, $\mathbb P^n$, as a GKM variety. The action of $(\mathbb C^*)^{n+1}$ on $\mathbb P^n$ is defined by $(t_0, \ldots, t_n) \cdot (x_0, \ldots, x_n) = (t_0^{-1}x_0, \ldots, t_n^{-1}x_n)$.

 i1 : PP4 = projectiveSpace 4; i2 : peek PP4 o2 = GKMVariety{cache => CacheTable{...1...} } characterRing => ZZ[T ..T ] 0 4 charts => HashTable{set {0} => {{-1, 1, 0, 0, 0}, {-1, 0, 1, 0, 0}, {-1, 0, 0, 1, 0}, {-1, 0, 0, 0, 1}}} set {1} => {{1, -1, 0, 0, 0}, {0, -1, 1, 0, 0}, {0, -1, 0, 1, 0}, {0, -1, 0, 0, 1}} set {2} => {{1, 0, -1, 0, 0}, {0, 1, -1, 0, 0}, {0, 0, -1, 1, 0}, {0, 0, -1, 0, 1}} set {3} => {{1, 0, 0, -1, 0}, {0, 1, 0, -1, 0}, {0, 0, 1, -1, 0}, {0, 0, 0, -1, 1}} set {4} => {{1, 0, 0, 0, -1}, {0, 1, 0, 0, -1}, {0, 0, 1, 0, -1}, {0, 0, 0, 1, -1}} momentGraph => a "moment graph" on 5 vertices with 10 edges points => {set {0}, set {1}, set {2}, set {3}, set {4}}