Inmathematics, the concept of aprojective space originated from the visual effect ofperspective, where parallel lines seem to meetat infinity. A projective space may thus be viewed as the extension of aEuclidean space, or, more generally, anaffine space withpoints at infinity, in such a way that there is one point at infinity of eachdirection ofparallel lines.
This definition of a projective space has the disadvantage of not beingisotropic, 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. Insynthetic geometry,point andline 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 theaxioms 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.
Usinglinear algebra, a projective space of dimensionn is defined as the set of thevector lines (that is, vector subspaces of dimension one) in avector spaceV of dimensionn + 1. Equivalently, it is thequotient set ofV \ {0} by theequivalence relation "being on the same vector line". As a vector line intersects theunit sphere ofV in twoantipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is aprojective line, and a projective space of dimension 2 is aprojective plane.
Projective spaces are widely used ingeometry, allowing for simpler statements and simpler proofs. For example, inaffine geometry, two distinct lines in a plane intersect in at most one point, while, inprojective geometry, they intersect in exactly one point. Also, there is only one class ofconic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points forhyperbolas; one for theparabola, which is tangent to the line at infinity; and no real intersection point ofellipses.
Intopology, and more specifically inmanifold theory, projective spaces play a fundamental role, being typical examples ofnon-orientable manifolds.
As outlined above, projective spaces were introduced for formalizing statements like "twocoplanar lines intersect in exactly one point, and this point is at infinity if the lines areparallel". Such statements are suggested by the study ofperspective, which may be considered as acentral projection of thethree dimensional space onto aplane (seePinhole camera model). More precisely, the entrance pupil of a camera or of the eye of an observer is thecenter of projection, and the image is formed on theprojection plane.
Mathematically, the center of projection is a pointO of the space (the intersection of the axes in the figure); the projection plane (P2, in blue on the figure) is a plane not passing throughO, which is often chosen to be the plane of equationz = 1, whenCartesian coordinates are considered. Then, the central projection maps a pointP to the intersection of the lineOP with the projection plane. Such an intersection exists if and only if the pointP does not belong to the plane (P1, in green on the figure) that passes throughO and is parallel toP2.
It follows that the lines passing throughO split in two disjoint subsets: the lines that are not contained inP1, which are in one to one correspondence with the points ofP2, and those contained inP1, which are in one to one correspondence with the directions of parallel lines inP2. This suggests to define thepoints (called hereprojective points for clarity) of the projective plane as the lines passing throughO. Aprojective line in this plane consists of all projective points (which are lines) contained in a plane passing throughO. As the intersection of two planes passing throughO is a line passing throughO, the intersection of two distinct projective lines consists of a single projective point. The planeP1defines a projective line which is called theline at infinity ofP2. By identifying each point ofP2 with the corresponding projective point, one can thus say that the projective plane is thedisjoint union ofP2 and the (projective) line at infinity.
As anaffine space with a distinguished pointO may be identified with its associatedvector space (seeAffine space § Vector spaces as affine spaces), the preceding construction is generally done by starting from a vector space and is calledprojectivization. Also, the construction can be done by starting with a vector space of any positive dimension.
So, a projective space of dimensionn can be defined as the set ofvector lines (vector subspaces of dimension one) in a vector space of dimensionn + 1. 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 ofequivalence 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 dimensionn as the set of pairs ofantipodal points in a sphere of dimensionn (in a space of dimensionn + 1).
Given avector spaceV over afieldK, theprojective spaceP(V) is the set ofequivalence classes ofV \ {0} under the equivalence relation~ defined byx ~y if there is a nonzero elementλ ofK such thatx =λy. IfV is atopological vector space, the quotient spaceP(V) is atopological space, endowed with thequotient topology of thesubspace topology ofV \ {0}. This is the case whenK is the fieldR of thereal numbers or the fieldC of thecomplex numbers. IfV is finite dimensional, thedimension ofP(V) is the dimension ofV minus one.
In the common case whereV =Kn+1, the projective spaceP(V) is denotedPn(K) (as well asKPn orPn(K), although this notation may be confused with exponentiation). The spacePn(K) is often calledthe projective space of dimensionn overK, orthe projectiven-space, since all projective spaces of dimensionn areisomorphic to it (because everyK vector space of dimensionn + 1 is isomorphic toKn+1).
The elements of a projective spaceP(V) are commonly calledpoints. If abasis ofV has been chosen, and, in particular ifV =Kn+1, theprojective coordinates of a pointP 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 definedup 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 nonzeroλ inK. 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.
IfK is the field of real or complex numbers, a projective space is called areal projective space or acomplex projective space, respectively. Ifn is one or two, a projective space of dimensionn is called aprojective line or aprojective plane, respectively. The complex projective line is also called theRiemann sphere.
All these definitions extend naturally to the case whereK is adivision ring; see, for example,Quaternionic projective space. The notationPG(n,K) is sometimes used forPn(K).[1] IfK is afinite field withq elements,Pn(K) is often denotedPG(n,q) (seePG(3,2)).[a]
LetP(V) be a projective space, whereV is a vector space over a fieldK, andbe thecanonical map that maps a nonzero vectorv to its equivalence class, which is thevector line containingv with the zero vector removed.
Everylinear subspaceW ofV is a union of lines. It follows thatp(W) is a projective space, which can be identified withP(W).
Aprojective subspace is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that definesP(V).
Ifp(v) andp(w) are two different points ofP(V), the vectorsv andw arelinearly independent. It follows that:
Insynthetic geometry, where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.
Everyintersection of projective subspaces is a projective subspace. It follows that for every subsetS of a projective space, there is a smallest projective subspace containingS, the intersection of all projective subspaces containingS. This projective subspace is called theprojective span ofS, andS is a spanning set for it.
A setS of points isprojectively independent if its span is not the span of any proper subset ofS. IfS is a spanning set of a projective spaceP, then there is a subset ofS that spansP and is projectively independent (this results from the similar theorem for vector spaces). If the dimension ofP isn, such an independent spanning set hasn + 1 elements.
Contrarily to the cases ofvector spaces andaffine spaces, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.
Aprojective frame orprojective basis is an ordered set of points in a projective space that allows defining coordinates.[2] More precisely, in ann-dimensional projective space, a projective frame is a tuple ofn + 2 points such that anyn + 1 of them are independent; that is, they are not contained in ahyperplane.
IfV is an(n + 1)-dimensional vector space, andp is the canonical projection fromV toP(V), then(p(e0), ...,p(en+1)) is a projective frame if and only if(e0, ...,en) is a basis ofV and the coefficients ofen+1 on this basis are all nonzero. By rescaling the firstn vectors, any frame can be rewritten as(p(e′0), ..., p(e′n+1)) such thate′n+1 =e′0 + ... +e′n; this representation is unique up to the multiplication of alle′i with a common nonzero factor.
Theprojective coordinates orhomogeneous coordinates of a pointp(v) on a frame(p(e0), ...,p(en+1)) withen+1 =e0 + ... +en are the coordinates ofv on the basis(e0, ...,en). They are only defined up to scaling with a common nonzero factor.
Thecanonical frame of the projective spacePn(K) consists of images byp of the elements of the canonical basis ofKn+1 (that is, thetuples with only one nonzero entry, equal to 1), and the image byp of their sum.
Inmathematics,projective geometry is the study of geometric properties that are invariant with respect toprojective transformations. This means that, compared to elementaryEuclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points thanEuclidean space, for a given dimension, and thatgeometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa.
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by atransformation matrix andtranslations (theaffine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike inEuclidean geometry, the concept of anangle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen inperspective drawing from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in whichparallel lines can be said to meet in apoint at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. SeeProjective plane for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory ofcomplex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (includinginvariant theory, theItalian school of algebraic geometry, andFelix Klein'sErlangen programme resulting in the study of theclassical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, assynthetic geometry. Another topic that developed from axiomatic studies of projective geometry isfinite geometry.
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study ofprojective varieties) andprojective differential geometry (the study ofdifferential invariants of the projective transformations).
Inprojective geometry, ahomography is anisomorphism of projective spaces, induced by an isomorphism of thevector spaces from which the projective spaces derive.[3] It is abijection that mapslines to lines, and thus acollineation. In general, some collineations are not homographies, but thefundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
Historically, homographies (and projective spaces) have been introduced to studyperspective andprojections inEuclidean geometry, and the termhomography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extendedEuclidean andaffine spaces by the addition of new points calledpoints at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of avector space over a givenfield (the above definition is based on this version); this construction facilitates the definition ofprojective coordinates and allows using the tools oflinear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see alsosynthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative)field. EquivalentlyPappus's hexagon theorem andDesargues's theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.
A projective space is atopological space, as endowed with thequotient topology of the topology of a finite dimensional real vector space.
LetS be theunit sphere in a normed vector spaceV, and consider the functionthat maps a point ofS to the vector line passing through it. This function is continuous and surjective. The inverse image of every point ofP(V) consist of twoantipodal points. As spheres arecompact spaces, it follows that:
For every pointP ofS, the restriction ofπ to a neighborhood ofP is ahomeomorphism 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 simpleatlas can be provided, as follows.
As soon as a basis has been chosen forV, any vector can be identified with its coordinates on the basis, and any point ofP(V) may be identified with itshomogeneous coordinates. Fori = 0, ...,n, the setis an open subset ofP(V), andsince every point ofP(V) has at least one nonzero coordinate.
To eachUi is associated achart, which is thehomeomorphismssuch thatwhere hats means that the corresponding term is missing.
These charts form anatlas, and, as thetransition maps areanalytic functions, it results that projective spaces areanalytic manifolds.
For example, in the case ofn = 1, that is of a projective line, there are only twoUi, which can each be identified to a copy of thereal line. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map isin 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.
Real projective spaces have a simpleCW complex structure, asPn(R) can be obtained fromPn−1(R) by attaching ann-cell with the quotient projectionSn−1 →Pn−1(R) as the attaching map.
Originally,algebraic geometry was the study of common zeros of sets ofmultivariate polynomials. These common zeros, calledalgebraic varieties belong to anaffine space. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, thefundamental theorem of algebra asserts that a univariatesquare-free polynomial of degreen has exactlyn complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also considerzeros at infinity. For example,Bézout's theorem asserts that the intersection of two planealgebraic curves of respective degreesd ande consists of exactlyde points if one consider complex points in the projective plane, and if one counts the points with their multiplicity.[b] Another example is thegenus–degree formula that allows computing the genus of a planealgebraic curve from itssingularities in thecomplex projective plane.
So aprojective variety is the set of points in a projective space, whosehomogeneous coordinates are common zeros of a set ofhomogeneous polynomials.[c]
Any affine variety can becompleted, in a unique way, into a projective variety by adding itspoints at infinity, which consists ofhomogenizing the defining polynomials, and removing the components that are contained in the hyperplane at infinity, bysaturating with respect to the homogenizing variable.
An important property of projective spaces and projective varieties is that the image of a projective variety under amorphism of algebraic varieties is closed forZariski topology (that is, it is analgebraic 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, introduced byAlexander Grothendieck during the second half of 20th century, allows defining a generalization of algebraic varieties, calledschemes, by gluing together smaller pieces calledaffine schemes, similarly asmanifolds can be built by gluing together open sets ofRn. TheProj 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.
Insynthetic geometry, aprojective spaceS can be defined axiomatically as a setP (the set of points), together with a setL of subsets ofP (the set of lines), satisfying these axioms:[4]
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 anincidence structure(P,L,I) consisting of a setP of points, a setL of lines, and anincidence relationI 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 theVeblen–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 thetheorem of Desargues and are known asnon-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.[5]
It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space.Coxeter (1969, p. 231) gives such an extension due to Bachmann.[6] To ensure that the dimension is at least two, replace the three point per line axiom above by:
To avoid the non-Desarguesian planes, includePappus's theorem as an axiom;[e]
And, to ensure that the vector space is defined over a field that does not have evencharacteristic includeFano's axiom;[f]
Asubspace of the projective space is a subsetX, such that any line containing two points ofX is a subset ofX (that is, completely contained inX). The full space and the empty space are always subspaces.
The geometric dimension of the space is said to ben if that is the largest number for which there is a strictly ascending chain of subspaces of this form:
A subspaceXi in such a chain is said to have (geometric) dimensioni. Subspaces of dimension 0 are calledpoints, those of dimension 1 are calledlines and so on. If the full space has dimensionn then any subspace of dimensionn − 1 is called ahyperplane.
Projective spaces admit an equivalent formulation in terms oflattice theory. There is a bijective correspondence between projective spaces and geomodular lattices, namely,subdirectly irreducible,compactly generated,complemented,modular lattices.[7]
Afinite projective space is a projective space whereP is a finite set of points. In any finite projective space, each line contains the same number of points and theorder 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 afinite field,GF(q), whose order (that is, number of elements) isq (a prime power). A finite projective space defined over such a finite field hasq + 1 points on a line, so the two concepts of order coincide. Notationally,PG(n, GF(q)) is usually written asPG(n,q).
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 theBruck–Ryser theorem.
The smallest projective plane is theFano plane,PG(2, 2) with 7 points and 7 lines. The smallest 3-dimensional projective space isPG(3, 2), with 15 points, 35 lines and 15 planes.
Injectivelinear mapsT ∈L(V,W) between two vector spacesV andW over the same field K induce mappings of the corresponding projective spacesP(V) →P(W) via:
wherev is a non-zero element ofV 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 iswell-defined. (IfT is not injective, it has anull space larger than{0}; in this case the meaning of the class ofT(v) is problematic ifv is non-zero and in the null space. In this case one obtains a so-calledrational map, see alsoBirational geometry.)
Two linear mapsS andT inL(V,W) induce the same map betweenP(V) andP(W)if and only if they differ by a scalar multiple, that is ifT =λS for someλ ≠ 0. Thus if one identifies the scalar multiples of theidentity map with the underlying field K, the set ofK-linearmorphisms fromP(V) toP(W) is simplyP(L(V,W)).
TheautomorphismsP(V) →P(V) can be described more concretely. (We deal only with automorphisms preserving the base field K). Using the notion ofsheaves 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 spaceV. The latter form thegroupGL(V). By identifying maps that differ by a scalar, one concludes that
thequotient group ofGL(V) modulo the matrices that are scalar multiples of the identity. (These matrices form thecenter ofAut(V).) The groupsPGL are calledprojective linear groups. The automorphisms of the complex projective lineP1(C) are calledMöbius transformations.
When the construction above is applied to thedual spaceV∗ rather thanV, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin ofV. That is, ifV isn-dimensional, thenP(V∗) is theGrassmannian ofn − 1 planes inV.
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 toevery quasi-coherent sheafE over a schemeY, not just the locally free ones.[clarification needed] SeeEGAII, Chap. II, par. 4 for more details.
Severi–Brauer varieties arealgebraic varieties over a field K, which become isomorphic to projective spaces after an extension of the base field K.
Another generalization of projective spaces areweighted projective spaces; these are themselves special cases oftoric varieties.[8]
