Projective space

File:Railroad-Tracks-Perspective.jpg

In graphical perspective, parallel (horizontal) lines in the plane intersect at a vanishing point (on the horizon).

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 Template:Mvar is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space Template:Mvar of dimension Template:Math. Equivalently, it is the quotient set of Template:Math by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of Template:Mvar 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, allowing for 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.

Motivation

File:Affine space R3.png

Projective plane and central projection

As outlined above, projective spaces were introduced for formalizing statements like "two coplanar lines intersect in exactly one point, and this point is at infinity if the lines are parallel". Such statements are suggested by the study of perspective, which may be considered as a central projection of the three dimensional space onto a plane (see Pinhole camera model). More precisely, the entrance pupil of a camera or of the eye of an observer is the center of projection, and the image is formed on the projection plane.

Mathematically, the center of projection is a point Template:Mvar of the space (the intersection of the axes in the figure); the projection plane (Template:Math, in blue on the figure) is a plane not passing through Template:Mvar, which is often chosen to be the plane of equation Template:Math, when Cartesian coordinates are considered. Then, the central projection maps a point Template:Mvar to the intersection of the line Template:Mvar with the projection plane. Such an intersection exists if and only if the point Template:Mvar does not belong to the plane (Template:Math, in green on the figure) that passes through Template:Mvar and is parallel to Template:Math.

It follows that the lines passing through Template:Mvar split in two disjoint subsets: the lines that are not contained in Template:Math, which are in one to one correspondence with the points of Template:Math, and those contained in Template:Math, which are in one to one correspondence with the directions of parallel lines in Template:Math. This suggests to define the points (called here projective points for clarity) of the projective plane as the lines passing through Template:Mvar. A projective line in this plane consists of all projective points (which are lines) contained in a plane passing through Template:Mvar. As the intersection of two planes passing through Template:Mvar is a line passing through Template:Mvar, the intersection of two distinct projective lines consists of a single projective point. The plane Template:Math defines a projective line which is called the line at infinity of Template:Math. By identifying each point of Template:Math with the corresponding projective point, one can thus say that the projective plane is the disjoint union of Template:Math and the (projective) line at infinity.

As an affine space with a distinguished point Template:Mvar may be identified with its associated vector space (see Template:Slink), the preceding construction is generally done by starting from a vector space and is called projectivization. Also, the construction can be done by starting with a vector space of any positive dimension.

So, a projective space of dimension Template:Mvar can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension Template:Math. A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.

This set can be the set of equivalence classes under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed.

A third equivalent definition is to define a projective space of dimension Template:Mvar as the set of pairs of antipodal points in a sphere of dimension Template:Mvar (in a space of dimension Template:Math).

Definition

Given a vector space Template:Mvar over a field Template:Mvar, the projective space Template:Math is the set of equivalence classes of Template:Math under the equivalence relation Template:Math defined by Template:Math if there is a nonzero element Template:Mvar of Template:Mvar such that Template:Math. If Template:Mvar is a topological vector space, the quotient space Template:Math is a topological space, endowed with the quotient topology of the subspace topology of Template:Math. This is the case when Template:Mvar is the field Template:Math of the real numbers or the field Template:Math of the complex numbers. If Template:Mvar is finite dimensional, the dimension of Template:Math is the dimension of Template:Mvar minus one.

In the common case where Template:Math, the projective space Template:Math is denoted Template:Math (as well as Template:Math or Template:Math, although this notation may be confused with exponentiation). The space Template:Math is often called the projective space of dimension Template:Mvar over Template:Mvar, or the projective Template:Mvar-space, since all projective spaces of dimension Template:Mvar are isomorphic to it (because every Template:Mvar vector space of dimension Template:Math is isomorphic to Template:Math).

The elements of a projective space Template:Math are commonly called points. If a basis of Template:Mvar has been chosen, and, in particular if Template:Math, 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 Template:Math, 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 Template:Math are projective coordinates of a point, then Template:Math are also projective coordinates of the same point, for any nonzero Template:Mvar in Template:Mvar. Also, the above definition implies that Template:Math are projective coordinates of a point if and only if at least one of the coordinates is nonzero.

If Template:Mvar is the field of real or complex numbers, a projective space is called a real projective space or a complex projective space, respectively. If Template:Math is one or two, a projective space of dimension Template:Math 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 Template:Mvar is a division ring; see, for example, Quaternionic projective space. The notation Template:Math is sometimes used for Template:Math.<ref>Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, p. 506, Marcel Dekker Template:Isbn</ref> If Template:Mvar is a finite field with Template:Mvar elements, Template:Math is often denoted Template:Math (see PG(3,2)).Template:Efn

Related concepts

Subspace

Let Template:Math be a projective space, where Template:Mvar is a vector space over a field Template:Mvar, and <math display="block">p:V\to \mathbf P(V)</math> be the canonical map that maps a nonzero vector Template:Mvar to its equivalence class, which is the vector line containing Template:Mvar with the zero vector removed.

Every linear subspace Template:Mvar of Template:Mvar is a union of lines. It follows that Template:Math is a projective space, which can be identified with Template:Math.

A projective subspace is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines Template:Math.

If Template:Math and Template:Math are two different points of Template:Math, the vectors Template:Mvar and Template:Mvar are linearly independent. It follows that:

• There is exactly one projective line that passes through two different points of Template:Math, and
• A subset of Template:Math is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.

In synthetic geometry, where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.

Span

Every intersection of projective subspaces is a projective subspace. It follows that for every subset Template:Mvar of a projective space, there is a smallest projective subspace containing Template:Mvar, the intersection of all projective subspaces containing Template:Mvar. This projective subspace is called the projective span of Template:Mvar, and Template:Mvar is a spanning set for it.

A set Template:Mvar of points is projectively independent if its span is not the span of any proper subset of Template:Mvar. If Template:Mvar is a spanning set of a projective space Template:Mvar, then there is a subset of Template:Mvar that spans Template:Mvar and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of Template:Mvar is Template:Mvar, such an independent spanning set has Template:Math elements.

Contrarily to the cases of vector spaces and affine spaces, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.

Frame

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A projective frame or projective basis is an ordered set of points in a projective space that allows defining coordinates.Template:Sfn More precisely, in an Template:Mvar-dimensional projective space, a projective frame is a tuple of Template:Math points such that any Template:Math of them are independent; that is, they are not contained in a hyperplane.

If Template:Mvar is an Template:Math-dimensional vector space, and Template:Mvar is the canonical projection from Template:Mvar to Template:Math, then Template:Math is a projective frame if and only if Template:Math is a basis of Template:Mvar and the coefficients of Template:Math on this basis are all nonzero. By rescaling the first Template:Mvar vectors, any frame can be rewritten as Template:Math such that Template:Math; this representation is unique up to the multiplication of all Template:Math with a common nonzero factor.

The projective coordinates or homogeneous coordinates of a point Template:Math on a frame Template:Math with Template:Math are the coordinates of Template:Mvar on the basis Template:Math. They are only defined up to scaling with a common nonzero factor.

The canonical frame of the projective space Template:Math consists of images by Template:Mvar of the elements of the canonical basis of Template:Math (that is, the tuples with only one nonzero entry, equal to 1), and the image by Template:Mvar of their sum.

Topology

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A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space.

Let Template:Mvar be the unit sphere in a normed vector space Template:Mvar, and consider the function <math display="block">\pi: S \to \mathbf P(V)</math> that maps a point of Template:Mvar to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of Template:Math consist of two antipodal points. As spheres are compact spaces, it follows that: Template:Block indent

For every point Template:Mvar of Template:Mvar, the restriction of Template:Pi to a neighborhood of Template:Mvar is a homeomorphism onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple atlas can be provided, as follows.

As soon as a basis has been chosen for Template:Mvar, any vector can be identified with its coordinates on the basis, and any point of Template:Math may be identified with its homogeneous coordinates. For Template:Math, the set <math display="block">U_i = \{[x_0:\cdots: x_n], x_i \neq 0\}</math> is an open subset of Template:Math, and <math display="block">\mathbf P(V) = \bigcup_{i=0}^n U_i</math> since every point of Template:Math has at least one nonzero coordinate.

To each Template:Math is associated a chart, which is the homeomorphisms <math display="block">\begin{align} \mathbb \varphi_i: R^n &\to U_i\\ (y_0,\dots,\widehat{y_i},\dots, y_n)&\mapsto [y_0:\cdots:y_{i-1}:1:y_{i+1}:\cdots:y_n], \end{align}</math> such that <math display="block">\varphi_i^{-1}\left([x_0:\cdots:x_n]\right) =\left (\frac{x_0}{x_i}, \dots, \widehat{\frac{x_i}{x_i}}, \dots, \frac{x_n}{x_i} \right ),</math> where hats means that the corresponding term is missing.

File:P1 ako varieta.png

Manifold structure of the real projective line

These charts form an atlas, and, as the transition maps are analytic functions, it results that projective spaces are analytic manifolds.

For example, in the case of Template:Math, that is of a projective line, there are only two Template:Math, which can each be identified to a copy of the real line. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is <math display="block">x\mapsto \frac 1 x</math> in both directions. The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be identified with the projective line with a single point removed.

CW complex structure

Real projective spaces have a simple CW complex structure, as Template:Math can be obtained from Template:Math by attaching an Template:Math-cell with the quotient projection Template:Math as the attaching map.

Algebraic geometry

Originally, algebraic geometry was the study of common zeros of sets of multivariate polynomials. These common zeros, called algebraic varieties belong to an affine space. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the fundamental theorem of algebra asserts that a univariate square-free polynomial of degree Template:Mvar has exactly Template:Mvar complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider zeros at infinity. For example, Bézout's theorem asserts that the intersection of two plane algebraic curves of respective degrees Template:Mvar and Template:Mvar consists of exactly Template:Mvar points if one consider complex points in the projective plane, and if one counts the points with their multiplicity.Template:Efn Another example is the genus–degree formula that allows computing the genus of a plane algebraic curve from its singularities in the complex projective plane.

So a projective variety is the set of points in a projective space, whose homogeneous coordinates are common zeros of a set of homogeneous polynomials.Template:Efn

Any affine variety can be completed, in a unique way, into a projective variety by adding its points at infinity, which consists of homogenizing the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by saturating with respect to the homogenizing variable.

An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization to every ground field of the compactness of the real and complex projective space.

A projective space is itself a projective variety, being the set of zeros of the zero polynomial.

Scheme theory

Scheme theory, introduced by Alexander Grothendieck during the second half of 20th century, allows defining a generalization of algebraic varieties, called schemes, by gluing together smaller pieces called affine schemes, similarly as manifolds can be built by gluing together open sets of Template:Math. The Proj construction is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold. Template:See also

Synthetic geometry

In synthetic geometry, a projective space Template:Math can be defined axiomatically as a set Template:Math (the set of points), together with a set Template:Math of subsets of Template:Math (the set of lines), satisfying these axioms:<ref>Template:Harvnb</ref>

• Each two distinct points Template:Math and Template:Math are in exactly one line.
• Veblen's axiom:Template:Efn If Template:Math, Template:Math, Template:Math, Template:Math are distinct points and the lines through Template:Math and Template:Math meet, then so do the lines through Template:Math and Template:Math.
• Any line has at least 3 points on it.

The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an incidence structure Template:Math consisting of a set Template:Math of points, a set Template:Math of lines, and an incidence relation Template:Math that states which points lie on which lines.

The structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the Veblen–Young theorem, there is no difference. However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the theorem of Desargues and are known as non-Desarguesian planes. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically.<ref>Template:Harvnb</ref>

It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. Template:Harvtxt gives such an extension due to Bachmann.<ref>Template:Citation</ref> To ensure that the dimension is at least two, replace the three point per line axiom above by:

• There exist four points, no three of which are collinear.

To avoid the non-Desarguesian planes, include Pappus's theorem as an axiom;Template:Efn

• If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear.

And, to ensure that the vector space is defined over a field that does not have even characteristic include Fano's axiom;Template:Efn

• The three diagonal points of a complete quadrangle are never collinear.

Template:AnchorA subspace of the projective space is a subset Template:Math, such that any line containing two points of Template:Math is a subset of Template:Math (that is, completely contained in Template:Math). The full space and the empty space are always subspaces.

The geometric dimension of the space is said to be Template:Math if that is the largest number for which there is a strictly ascending chain of subspaces of this form: <math display="block">\varnothing = X_{-1}\subset X_{0}\subset \cdots X_{n}=P.</math>

A subspace Template:Math in such a chain is said to have (geometric) dimension Template:Math. Subspaces of dimension 0 are called points, those of dimension 1 are called lines and so on. If the full space has dimension Template:Math then any subspace of dimension Template:Nowrap is called a hyperplane.

Projective spaces admit an equivalent formulation in terms of lattice theory. There is a bijective correspondence between projective spaces and geomodular lattices, namely, subdirectly irreducible, compactly generated, complemented, modular lattices.<ref>Peter Crawley and Robert P. Dilworth, 1973. Algebraic Theory of Lattices. Prentice-Hall. Template:Isbn, p. 109.</ref>

Classification

• Dimension 0 (no lines): The space is a single point.
• Dimension 1 (exactly one line): All points lie on the unique line.
• Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for Template:Math is equivalent to a projective plane. These are much harder to classify, as not all of them are isomorphic with a Template:Math. The Desarguesian planes (those that are isomorphic with a Template:Math satisfy Desargues's theorem and are projective planes over division rings, but there are many non-Desarguesian planes.
• Dimension at least 3: Two non-intersecting lines exist. Template:Harvtxt proved the Veblen–Young theorem, to the effect that every projective space of dimension Template:Math is isomorphic with a Template:Math, the Template:Math-dimensional projective space over some division ring Template:Math.

Finite projective spaces and planes

File:Fano plane.svg

The Fano plane

A finite projective space is a projective space where Template:Math is a finite set of points. In any finite projective space, each line contains the same number of points and the order of the space is defined as one less than this common number. For finite projective spaces of dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, Template:Math, whose order (that is, number of elements) is Template:Math (a prime power). A finite projective space defined over such a finite field has Template:Math points on a line, so the two concepts of order coincide. Notationally, Template:Math is usually written as Template:Math.

All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are Template:Block indent finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the Bruck–Ryser theorem.

The smallest projective plane is the Fano plane, Template:Math with 7 points and 7 lines. The smallest 3-dimensional projective space is [[PG(3,2)|Template:Math]], with 15 points, 35 lines and 15 planes.

Morphisms

Injective linear maps Template:Math between two vector spaces Template:Math and Template:Math over the same field Template:Math induce mappings of the corresponding projective spaces Template:Math via: Template:Block indent where Template:Math is a non-zero element of Template:Math and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If Template:Math is not injective, it has a null space larger than Template:Math; in this case the meaning of the class of Template:Math is problematic if Template:Math is non-zero and in the null space. In this case one obtains a so-called rational map, see also Birational geometry.)

Two linear maps Template:Math and Template:Math in Template:Math induce the same map between Template:Math and Template:Math if and only if they differ by a scalar multiple, that is if Template:Math for some Template:Math. Thus if one identifies the scalar multiples of the identity map with the underlying field Template:Math, the set of Template:Math-linear morphisms from Template:Math to Template:Math is simply Template:Math.

The automorphisms Template:Math can be described more concretely. (We deal only with automorphisms preserving the base field Template:Math). Using the notion of sheaves generated by global sections, it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e., coming from a (linear) automorphism of the vector space Template:Math. The latter form the group [[general linear group|Template:Math]]. By identifying maps that differ by a scalar, one concludes that

Template:Block indent the quotient group of Template:Math modulo the matrices that are scalar multiples of the identity. (These matrices form the center of Template:Math.) The groups Template:Math are called projective linear groups. The automorphisms of the complex projective line Template:Math are called Möbius transformations.

Dual projective space

When the construction above is applied to the dual space Template:Math rather than Template:Math, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of Template:Math. That is, if Template:Math is Template:Math-dimensional, then Template:Math is the Grassmannian of Template:Math planes in Template:Math.

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 Template:Math over a scheme Template:Math, not just the locally free ones.Template:Clarify See EGA_II, Chap. II, par. 4 for more details.

Generalizations

dimension

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

sequence of subspaces

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

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 only fields) yields, for example, the projective line over a ring.

patching

Patching projective spaces together yields projective space bundles.

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

Another generalization of projective spaces are weighted projective spaces; these are themselves special cases of toric varieties.<ref>Template:Harvnb</ref>

See also

• Geometric algebra

Generalizations

• Grassmannian manifold
• Projective line over a ring
• Space (mathematics)

Projective geometry

• projective transformation
• projective representation

References

• Template:Eom
• Template:Citation
• Template:Citation, translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Greenberg, M.J.; Euclidean and non-Euclidean geometries, 2nd ed. Freeman (1980).
• Template:Citation, esp. chapters I.2, I.7, II.5, and II.7
• Hilbert, D. and Cohn-Vossen, S.; Geometry and the imagination, 2nd ed. Chelsea (1999).
• Template:Citation
• Template:Citation (Reprint of 1910 edition)

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:ProjectiveSpace%7CProjectiveSpace.html}} |title = Projective Space |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Template:PlanetMath
• Projective Planes of Small Order