TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, 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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
, 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, a projective space of dimension is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of dimension . Equivalently, it is the quotient set of by the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
"being on the same vector line". As a vector line intersects the unit sphere of in two
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
, 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s, which can be distinguished only by their intersections with the line at infinity: two intersection points for
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s; one for the
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
, which is tangent to the line at infinity; and no real intersection point of
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s. In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, and more specifically in manifold theory, projective spaces play a fundamental role, being typical examples of non-orientable manifolds.

# Motivation

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 Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
." Such statements are suggested by the study of perspective, which may be considered as a
central projection In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projec ...
of the three dimensional space onto a plane (see
Pinhole camera model The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a p ...
). 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 of the space (the intersection of the axes in the figure); the projection plane (, in blue on the figure) is a plane not passing through , which is often chosen to be the plane of equation , when
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
are considered. Then, the central projection maps a point to the intersection of the line with the projection plane. Such an intersection exists if and only if the point does not belong to the plane (, in green on the figure) that passes through and is parallel to . It follows that the lines passing through split in two disjoint subsets: the lines that are not contained in , which are in one to one correspondence with the points of , and those contained in , which are in one to one correspondence with the directions of parallel lines in . This suggests to define the ''points'' (called here ''projective points'' for clarity) of the projective plane as the lines passing through . A ''projective line'' in this plane consists of all projective points (which are lines) contained in a plane passing through . As the intersection of two planes passing through is a line passing through , the intersection of two distinct projective lines consists of a single projective point. The plane defines a projective line which is called the ''line at infinity'' of . By identifying each point of with the corresponding projective point, one can thus say that the projective plane is the disjoint union of and the (projective) line at infinity. As an affine space with a distinguished point may be identified with its associated
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(see ), 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 can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension . 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 class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es 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 as the set of pairs of
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
in a sphere of dimension (in a space of dimension ).

# Definition

Given a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field , the ''projective space'' is the set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of under the equivalence relation defined by if there is a nonzero element of such that . If is a topological vector space, the quotient space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, endowed with the quotient topology of the subspace topology of . This is the case when is the field $\mathbb R$ of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or the field $\mathbb C$ of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. If is finite dimensional, the ''dimension'' of is the dimension of minus one. In the common case where , the projective space is denoted (as well as or , although this notation may be confused with exponentiation). The space is often called ''the'' projective space of dimension over , or ''the projective -space'', since all projective spaces of dimension are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to it (because every vector space of dimension is isomorphic to ). The elements of a projective space are commonly called ''
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
''. If a basis of has been chosen, and, in particular if , the
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinate system, Cartesian coordinates are u ...
of a point ''P'' are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted , 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 are projective coordinates of a point, then are also projective coordinates of the same point, for any nonzero in . Also, the above definition implies that are projective coordinates of a point if and only if at least one of the coordinates is nonzero. If is the field of real or complex numbers, a projective space is called a real projective space or a complex projective space, respectively. If is one or two, a projective space of dimension is called a projective line or a projective plane, respectively. The complex projective line is also called the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. All these definitions extend naturally to the case where is a division ring; see, for example,
Quaternionic projective space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''n'' ...
. The notation is sometimes used for . If is a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with elements, is often denoted (see
PG(3,2) In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. It also has the following properties: * Each point is contained in ...
).

# Related concepts

## Subspace

Let be a projective space, where is a vector space over a field , and $p:V\to \mathbf P(V)$ be the ''canonical map'' that maps a nonzero vector to its equivalence class, which is the vector line containing with the zero vector removed. Every
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of is a union of lines. It follows that is a projective space, which can be identified with . A ''projective subspace'' is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines . If and are two different points of , the vectors and are linearly independent. It follows that: * ''There is exactly one projective line that passes through two different points of'' , and * ''A subset of'' ''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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of projective subspaces is a projective subspace. It follows that for every subset of a projective space, there is a smallest projective subspace containing , the intersection of all projective subspaces containing . This projective subspace is called the ''projective span'' of , and is a spanning set for it. A set of points is ''projectively independent'' if its span is not the span of any proper subset of . If is a spanning set of a projective space , then there is a subset of that spans and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of is , such an independent spanning set has elements. Contrarily to the cases of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s and affine spaces, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.

## Frame

A ''projective frame'' is an ordered set of points in a projective space that allows defining coordinates. More precisely, in a -dimensional projective space, a projective frame is a tuple of points such that any of them are independent—that is are not contained in a hyperplane. If is a -dimensional vector space, and is the canonical projection from to , then $\left(p\left(e_0\right),\dots, p\left(e_\right)\right)$ is a projective frame if and only if $\left(e_0, \dots, e_n\right)$ is a basis of , and the coefficients of $e_$ on this basis are all nonzero. By rescaling the first vectors, any frame can be rewritten as $\left(p\left(e\text{'}_0\right),\dots, p\left(e\text{'}_\right)\right)$ such that $e\text{'}_ = e\text{'}_0 + \dots + e\text{'}_n;$ this representation is unique up to the multiplication of all $e\text{'}_i$ with a common nonzero factor. The ''projective coordinates'' or ''homogeneous coordinates'' of a point on a frame $\left(p\left(e_0\right),\dots, p\left(e_\right)\right)$ with $e_=e_0+\dots+ e_n$ are the coordinates of on the basis $\left(e_0, \dots, e_n\right).$ They are again only defined up to scaling with a common nonzero factor. The ''canonical frame'' of the projective space consists of images by of the elements of the canonical basis of (the tuples with only one nonzero entry, equal to 1), and the image by of their sum.

# Topology

A projective space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, as endowed with the quotient topology of the topology of a finite dimensional real vector space. Let be the unit sphere in a normed vector space , and consider the function $\pi: S \to \mathbf P(V)$ that maps a point of to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of consist of two antipodal points. As spheres are
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s, it follows that: For every point of , the restriction of to a neighborhood of is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
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 An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
can be provided, as follows. As soon as a basis has been chosen for , any vector can be identified with its coordinates on the basis, and any point of may be identified with its
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
. For , the set $U_i = \$ is an open subset of , and $\mathbf P(V) = \bigcup_^n U_i$ since every point of has at least one nonzero coordinate. To each is associated a
chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent ...
, which is the
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
s such that where hats means that the corresponding term is missing. These charts form an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
, and, as the transition maps are
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s, it results that projective spaces are analytic manifolds. For example, in the case of , that is of a projective line, there are only two , which can each be identified to a copy of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is $x\mapsto \frac 1 x$ 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 A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
structure, as can be obtained from by attaching an -cell with the quotient projection as the attaching map.

# Algebraic geometry

Originally,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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 has exactly 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 and consists of exactly points if one consider complex points in the projective plane, and if one counts the points with their multiplicity. 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 In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
are common zeros of a set of
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s. 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 In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ...
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
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s can be built by gluing together open sets of $\R^n.$ 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.

# Synthetic geometry

In synthetic geometry, a projective space ''S'' can be defined axiomatically as a set ''P'' (the set of points), together with a set ''L'' of subsets of ''P'' (the set of lines), satisfying these axioms: * Each two distinct points ''p'' and ''q'' are in exactly one line. * Veblen's axiom: If ''a'', ''b'', ''c'', ''d'' are distinct points and the lines through ''ab'' and ''cd'' meet, then so do the lines through ''ac'' and ''bd''. * 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 consisting of a set ''P'' of points, a set ''L'' of lines, and an incidence relation ''I'' 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 In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...
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. It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. gives such an extension due to Bachmann. 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; * 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''; * The three diagonal points of a complete quadrangle are never collinear. A subspace of the projective space is a subset ''X'', such that any line containing two points of ''X'' is a subset of ''X'' (that is, completely contained in ''X''). The full space and the empty space are always subspaces. The geometric dimension of the space is said to be ''n'' if that is the largest number for which there is a strictly ascending chain of subspaces of this form: $\varnothing = X_\subset X_\subset \cdots X_=P.$ A subspace $X_i$ in such a chain is said to have (geometric) dimension $i$. Subspaces of dimension 0 are called ''points'', those of dimension 1 are called ''lines'' and so on. If the full space has dimension $n$ then any subspace of dimension $n-1$ is called a
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
. 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 lattice In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition, ;Modular law: implies where are arbitrary elements in the lattice,  ≤  is the partial order, and & ...
s.

## 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 is equivalent to a projective plane. These are much harder to classify, as not all of them are isomorphic with a . The Desarguesian planes (those that are isomorphic with a 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. proved the Veblen–Young theorem, to the effect that every projective space of dimension is isomorphic with a , the ''n''-dimensional projective space over some division ring ''K''.

## Finite projective spaces and planes

A ''finite projective space'' is a projective space where ''P'' 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 In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, GF(''q''), whose order (that is, number of elements) is ''q'' (a prime power). A finite projective space defined over such a finite field has points on a line, so the two concepts of order coincide. Notationally, is usually written as . 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 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, with 7 points and 7 lines. The smallest 3-dimensional projective spaces is
PG(3,2) In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. It also has the following properties: * Each point is contained in ...
, with 15 points, 35 lines and 15 planes.

# Morphisms

Injective
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s between two vector spaces ''V'' and ''W'' over the same field ''k'' induce mappings of the corresponding projective spaces via: where ''v'' is a non-zero element of ''V'' 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 ''T'' is not injective, it has a
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
larger than ; in this case the meaning of the class of ''T''(''v'') is problematic if ''v'' 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 ''S'' and ''T'' in induce the same map between P(''V'') and P(''W'')
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
they differ by a scalar multiple, that is if for some . Thus if one identifies the scalar multiples of the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
with the underlying field ''K'', the set of ''K''-linear morphisms from P(''V'') to P(''W'') is simply . The
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s can be described more concretely. (We deal only with automorphisms preserving the base field ''K''). 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 ''V''. The latter form the group GL(''V''). By identifying maps that differ by a scalar, one concludes that the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
of GL(''V'') modulo the matrices that are scalar multiples of the identity. (These matrices form the center of Aut(''V'').) The groups PGL are called projective linear groups. The automorphisms of the complex projective line P1(C) are called Möbius transformations.

# Dual projective space

When the construction above is applied to the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
''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 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.

# 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 space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s parametrize objects such as
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s 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 ''k'', which become isomorphic to projective spaces after an extension of the base field ''k''. Another generalization of projective spaces are
weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''. Prope ...
s; these are themselves special cases of
toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
.

## Generalizations

* Grassmannian manifold * Projective line over a ring *
Space (mathematics) In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, ...

## Projective geometry

* projective transformation *
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where ...

## Related

*
Geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...

# References

* * * * * * * Greenberg, M.J.; ''Euclidean and non-Euclidean geometries'', 2nd ed. Freeman (1980). * , 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). * * (Reprint of 1910 edition)