Projective space ($\mathbf{P}(V)$)

For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]

Projective space is a fundamental concept in both mathematics and geometry, particularly in the fields of projective geometry and algebraic geometry. It can be intuitively thought of as an extension of the concept of Euclidean space. Here are some key points to understand projective space: ### Definition 1.