Projectivization

Given a vector space V, its projectivization P(V), sometimes written P(V-0), is the set of equivalence classes x∼lambdax for any lambda!=0 in V-0. For example, complex projective space has homogeneous coordinates [x_0,...,x_n], with not all x_i=0.

The projectivization is a manifold with one less dimension than V. In fact, it is covered by the n+1 affine coordinate charts,

U_0={[1,x_1,...,x_n]},...,U_n={[x_0,...,x_(n-1),1]}.