Projective Space - Generalizations

Generalizations

dimension

The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space V is generalized to Grassmannian manifold, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of V.

sequence of subspaces

More generally flag manifold is the space of flags, i.e. chains of linear subspaces of V.

other subvarieties

Even more generally, moduli spaces parametrize objects such as elliptic curves of a given kind.

other rings

Generalizing to associative rings (rather than fields) yields inversive ring geometry

patching

Patching projective spaces together yields projective space bundles.

Severi–Brauer varieties are algebraic varieties over a field k which become isomorphic to projective spaces after an extension of the base field k.

Another generalization of projective spaces are weighted projective spaces; these are themselves special cases of toric varieties.