|
In mathematics, a projective space is a fundamental
construction from any vector space. It generalises the projective plane that may be constructed from a three-dimensional vector
space, over any field. While the theory of projective
planes has an aspect that belongs to combinatorics too, that is absent in
the general case. Projective space is basic in algebraic
geometry, through the rich field of projective geometry
developed in the nineteenth century but also in the constructions of the modern theory (based on graded commutative rings). Projective spaces and their
generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and
algebraic groups, and their representation theory.
The basic construction, given a vector space V over a field K, is to form the set of equivalence classes of
non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to
w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions
of perspective. If the field K is the real numbers, and V has dimension n, then the projective space P(V) - which we
can talk about as the space of lines through the zero element 0 of V - carries a natural structure of a compact smooth manifold of
dimension n − 1. It is also highly symmetric, since any linear automorphism of V gives rise to a symmetry
of P(V). These in the classical examples identify with 'perspectivity' and 'projectivity' transformations described
geometrically, and account for the name. The group of these symmetries is the quotient of the general linear group of V by the subgroup of non-zero
scalar multiples of the identity.
The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the
projective space associated to a four-dimensional real vector space. In that way geometrical ideas introduced by Poncelet and others become part of a theory founded on linear algebra. The part of a projective space not 'at infinity' is called
affine space; but the symmetries of P(V) do not respect that
division. Use of a basis of V allows, if required, the introduction of homogeneous co-ordinates for the handling of concrete calculations.
Use of vector spaces over the field of complex numbers gives rise to
different manifolds, also used by geometers. There are good reasons for using them, in order to get a theory about intersections
of algebraic varieties with predictable properties. In the theory of Alexander Grothendieck there are reasons for applying the construction outlined above rather to the
dual vector space V*.
See also
|