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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 edit real projective spaces have a simple cw complex structure as p n r can be obtained from p n 1 r by attaching an n cell with the quotient projection s n 1 p n 1 r as the attaching map algebraic geometry edit 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 n has exactly n 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 d and e consists of exactly de points if one consider complex points in the projective plane and if one counts the points with their multiplicity b 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 c 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 edit 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 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 see also algebraic geometry of projective spaces synthetic geometry edit 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 4 each two distinct points p and q are in exactly one line veblen s axiom d 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 p l i 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 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 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 there exist four points no three of which are collinear to avoid the non desarguesian planes include pappus s theorem as an axiom e 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 f 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 x 1 x 0 x n p displaystyle varnothing x_ 1 subset x_ 0 subset cdots x_ n 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 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 7 classification edit 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 n 2 is equivalent to a projective plane these are much harder to classify as not all of them are isomorphic with a pg d k the desarguesian planes those that are isomorphic with a pg 2 k 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 veblen young 1965 proved the veblen young theorem to the effect that every projective space of dimension n 3 is isomorphic with a pg n k the n dimensional projective space over some division ring k finite projective spaces and planes edit further information on finite projective planes projective plane finite projective planes the fano plane 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 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 q 1 points on a line so the two concepts of order coincide notationally pg n gf q is usually written as pg 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 1 1 1 1 0 1 1 4 0 sequence a001231 in the oeis 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 pg 2 2 with 7 points and 7 lines the smallest 3 dimensional projective space is pg 3 2 with 15 points 35 lines and 15 planes morphisms edit injective linear maps t l v w between two vector spaces v and w over the same field k induce mappings of the corresponding projective spaces p v p w via v t v 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 larger than 0 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 l v w induce the same map between p v and p w if and only if they differ by a scalar multiple that is if t λs for some λ 0 thus if one identifies the scalar multiples of the identity map with the underlying field k the set of k linear morphisms from p v to p w is simply p l v w the automorphisms p v p v 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 aut p v aut v k gl v k pgl v the quotient group 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 p 1 c are called möbius transformations dual projective space edit when the construction above is applied to the dual space v rather than v one obtains the dual projective space which can be canonically identified with the space of hyperplanes through the origin of v that is if v is n dimensional then p v is the grassmannian of n 1 planes in v in algebraic geometry this construction allows for greater flexibility in the construction of projective bundles one would like to be able to associate a projective space to every quasi coherent sheaf e over a scheme y not just the locally free ones clarification needed see ega ii chap ii par 4 for more details generalizations edit 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 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 k which become isomorphic to projective spaces after an extension of the base field k another generalization of projective spaces are weighted projective spaces these are themselves special cases of toric varieties 8 see also edit geometric algebra generalizations grassmannian manifold projective line over a ring space mathematics projective geometry projective transformation projective representation notes edit the absence of space after the comma is common for this notation the correct definition of the multiplicity is not easy and dates only from the middle of 20th century homogeneous required in order that a zero remains a zero when the homogeneous coordinates are multiplied by a nonzero scalar also referred to as the veblen young axiom and mistakenly as the axiom of pasch beutelspacher rosenbaum 1998 pp 6 7 pasch was concerned with real projective space and was attempting to introduce order which is not a concern of the veblen young axiom as pappus s theorem implies desargues s theorem this eliminates the non desarguesian planes and also implies that the space is defined over a field and not a division ring this restriction allows the real and complex fields to be used zero characteristic but removes the fano plane and other planes that exhibit atypical behavior citations edit mauro biliotti vikram jha norman l johnson 2001 foundations of translation planes p 506 marcel dekker isbn 0 8247 0609 9 berger 2009 chapter 4 4 projective bases berger 2009 chapter 4 beutelspacher rosenbaum 1998 pp 6 7 baer 2005 p 71 bachmann f 1959 aufbau der geometrie aus dem spiegelsbegriff grundlehren der mathematischen wissenschaftern 96 berlin springer pp 76 77 peter crawley and robert p dilworth 1973 algebraic theory of lattices prentice hall isbn 978 0 13 022269 5 p 109 mukai 2003 example 3 72 references edit afanas ev v v 2001 1994 projective space encyclopedia of mathematics ems press baer reinhold 2005 first published 1952 linear algebra and projective geometry dover isbn 978 0 486 44565 6 berger marcel 2009 geometry i springer verlag isbn 978 3 540 11658 5 translated from the 1977 french original by m cole and s levy fourth printing of the 1987 english translation beutelspacher albrecht rosenbaum ute 1998 projective geometry from foundations to applications cambridge university press isbn 978 0 521 48277 6 mr 1629468 coxeter harold scott macdonald 1974 introduction to geometry new york john wiley sons isbn 0 471 18283 4 coxeter harold scott macdonald 1969 projective geometry toronto ont university of toronto press isbn 0 8020 2104 2 mr 0346652 oclc 977732 dembowski p 1968 finite geometries ergebnisse der mathematik und ihrer grenzgebiete band 44 berlin new york springer verlag isbn 3 540 61786 8 mr 0233275 greenberg m j euclidean and non euclidean geometries 2nd ed freeman 1980 hartshorne robin 1977 algebraic geometry berlin new york springer verlag isbn 978 0 387 90244 9 mr 0463157 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 mukai shigeru 2003 an introduction to invariants and moduli cambridge studies in advanced mathematics cambridge university press isbn 978 0 521 80906 1 veblen oswald young john wesley 1965 projective geometry vols 1 2 blaisdell publishing co ginn and co new york toronto london mr 0179666 reprint of 1910 edition external links edit weisstein eric w projective space mathworld projective space at planetmath projective planes of small order v t e dimension dimensional spaces vector space euclidean space affine space projective space free module manifold algebraic variety spacetime other di...