Projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
Using linear algebra, a projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently, it is the quotient set of V \ {0} by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of V in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in projective geometry, they intersect in exactly one point. Also, there is only one class of conic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points for hyperbolas; one for the parabola, which is tangent to the line at infinity; and no real intersection point of ellipses.
In topology, and more specifically in manifold theory, projective spaces play a fundamental role, being typical examples of non-orientable manifolds.
Definition[edit]
Given a vector space V over a field K, the projective space P(V) is the set of equivalence classes of V \ {0} under the equivalence relation ~ defined by x ~ y if there is a nonzero element λ of K such that x = λy. If V is a topological vector space, the quotient space P(V) is a topological space, endowed with the quotient topology of the subspace topology of V \ {0}. This is the case when K is the field R of the real numbers or the field C of the complex numbers. If V is finite dimensional, the dimension of P(V) is the dimension of V minus one.
In the common case where V = Kn+1, the projective space P(V) is denoted Pn(K) (as well as KPn or Pn(K), although this notation may be confused with exponentiation). The space Pn(K) is often called the projective space of dimension n over K, or the projective n-space, since all projective spaces of dimension n are isomorphic to it (because every K vector space of dimension n + 1 is isomorphic to Kn+1).
The elements of a projective space P(V) are commonly called points. If a basis of V has been chosen, and, in particular if V = Kn+1, the projective coordinates of a point P are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted [x0 : ... : xn], the colons and the brackets being used for distinguishing from usual coordinates, and emphasizing that this is an equivalence class, which is defined up to the multiplication by a non zero constant. That is, if [x0 : ... : xn] are projective coordinates of a point, then [λx0 : ... : λxn] are also projective coordinates of the same point, for any nonzero λ in K. Also, the above definition implies that [x0 : ... : xn] are projective coordinates of a point if and only if at least one of the coordinates is nonzero.
If K is the field of real or complex numbers, a projective space is called a real projective space or a complex projective space, respectively. If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The complex projective line is also called the Riemann sphere.
All these definitions extend naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).[1] If K is a finite field with q elements, Pn(K) is often denoted PG(n, q) (see PG(3,2)).[a]
Dual projective space[edit]
When the construction above is applied to the dual space V∗ rather than V, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of V. That is, if V is n-dimensional, then P(V∗) is the Grassmannian of n − 1 planes in V.
In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles. One would like to be able to associate a projective space to every quasi-coherent sheaf E over a scheme Y, not just the locally free ones. See EGAII, Chap. II, par. 4 for more details.