Editing Projective space (section)

== Generalizations ==
; dimension : The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space {{math|''V''}} is generalized to [[Grassmannian manifold]], which is parametrizing higher-dimensional subspaces (of some fixed dimension) of {{math|''V''}}.
; sequence of subspaces : More generally [[flag manifold]] is the space of flags, i.e., chains of linear subspaces of {{math|''V''}}.
; other subvarieties : Even more generally, [[moduli space]]s parametrize objects such as [[elliptic curve]]s of a given kind.
; other rings : Generalizing to associative [[ring (mathematics)|ring]]s (rather than only fields) yields, for example, the [[projective line over a ring]].
; patching : Patching projective spaces together yields [[projective space bundles]].

[[Severi–Brauer variety|Severi–Brauer varieties]] are [[algebraic varieties]] over a field&nbsp;{{math|''K''}}, which become isomorphic to projective spaces after an extension of the base field&nbsp;{{math|''K''}}.

Another generalization of projective spaces are [[weighted projective space]]s; these are themselves special cases of [[toric variety|toric varieties]].<ref>{{ harvnb | Mukai | 2003 | loc=example 3.72 }}</ref>