Projective space

Summary

The projective space $\mathbb{P}^n$ over a field $k$ is the set of lines through the origin in $k^{n+1}$. A point is written in homogeneous coordinates $[x_0 : \dots : x_n]$, defined up to a common nonzero factor. Projective space adds points at infinity to affine space. Then, for example, any two distinct lines in the projective plane meet in exactly one point. It is the ambient space of projective varieties, the compact objects of algebraic geometry.