Projective space

A nonzero vector $a=(a_0,\ldots,a_n)$ spans a point written in homogeneous coordinates as $[a_0:\cdots:a_n]$, with

for every $\lambda\in k^\times$. Thus

This formula describes the $k$-valued points; the scheme $\mathbb P_k^n$ contains additional information after extension of the ground field.

For each $i$, the condition $a_i\ne0$ defines a standard open subset $U_i$. Dividing the other coordinates by $a_i$ gives an isomorphism

Consequently, $\mathbb P_k^n=\bigcup_{i=0}^nU_i$ is covered by $n+1$ copies of affine $n$-space, proving directly that it is a scheme.

The assignment $L\mapsto L\subset V$ identifies $\mathbb P(V)$ with the Grassmannian of lines $\operatorname{Gr}_1(V)$. A linear subspace $0\ne W\subseteq V$ determines the projective subspace $\mathbb P(W)\subseteq\mathbb P(V)$, whose projective dimension is $\dim_kW-1$.

There are two standard projectivization conventions. This knowl uses $\operatorname{Proj}\operatorname{Sym}(V^\vee)$, whose points are lines in $V$. Some algebraic-geometry texts instead write $\mathbb P(V)=\operatorname{Proj}\operatorname{Sym}(V)$; that scheme parameterizes one-dimensional quotients of $V$, equivalently lines in $V^\vee$. Formulas involving tautological line bundles and duals must be translated when conventions differ.

Projective space is the ambient incidence space of projective geometry: its projective lines and higher-dimensional projective subspaces come from linear subspaces of $V$.

The projective line $\mathbb P_k^1$ is the affine line together with one point at infinity. For $k=\mathbb R$ or $\mathbb C$, the sets of lines carry their familiar smooth structures as real projective space and complex projective space. Invertible linear maps act through the projective general linear group.

Unlike affine space, projective space of positive dimension is not an affine scheme.

1. Robin Hartshorne, Algebraic Geometry, Springer, 1977. Publisher record. Relevant: Chapter II, §2, especially the Proj construction and projective space.
2. Joe Harris, Algebraic Geometry: A First Course, Springer, 1992. Publisher record. Relevant: Lecture 1, homogeneous coordinates and projective varieties.