Projective space
In
This definition of a projective space has the disadvantage of not being
Using
Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in projective geometry, they intersect in exactly one point. Also, there is only one class of conic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points for hyperbolas; one for the parabola, which is tangent to the line at infinity; and no real intersection point of ellipses.
In
Motivation
As outlined above, projective spaces were introduced for formalizing statements like "two
Mathematically, the center of projection is a point O 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 through O, which is often chosen to be the plane of equation z = 1, when
It follows that the lines passing through O split in two disjoint subsets: the lines that are not contained in P1, which are in one to one correspondence with the points of P2, and those contained in P1, which are in one to one correspondence with the directions of parallel lines in P2. This suggests to define the points (called here projective points for clarity) of the projective plane as the lines passing through O. A projective line in this plane consists of all projective points (which are lines) contained in a plane passing through O. As the intersection of two planes passing through O is a line passing through O, the intersection of two distinct projective lines consists of a single projective point. The plane P1 defines a projective line which is called the line at infinity of P2. By identifying each point of P2 with the corresponding projective point, one can thus say that the projective plane is the disjoint union of P2 and the (projective) line at infinity.
As an affine space with a distinguished point O may be identified with its associated vector space (see Affine space § Vector spaces as affine spaces), 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 n can be defined as the set of
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 n as the set of pairs of
Definition
Given a
In the common case where V = Kn+1, the projective space P(V) is denoted Pn(K) (as well as KPn or Pn(K), although this notation may be confused with exponentiation). The space Pn(K) is often called the projective space of dimension n over K, or the projective n-space, since all projective spaces of dimension n are isomorphic to it (because every K vector space of dimension n + 1 is isomorphic to Kn+1).
The elements of a projective space P(V) are commonly called
If K is the field of real or complex numbers, a projective space is called a real projective space or a complex projective space, respectively. If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The complex projective line is also called the Riemann sphere.
All these definitions extend naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).[1] If K is a finite field with q elements, Pn(K) is often denoted PG(n, q) (see PG(3,2)).[a]
Related concepts
Subspace
Let P(V) be a projective space, where V is a vector space over a field K, and be the canonical map that maps a nonzero vector v to its equivalence class, which is the
Every linear subspace W of V is a union of lines. It follows that p(W) is a projective space, which can be identified with P(W).
A projective subspace is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines P(V).
If p(v) and p(w) are two different points of P(V), the vectors v and w are
- There is exactly one projective line that passes through two different points of P(V), and
- A subset of P(V) 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
A set S of points is projectively independent if its span is not the span of any proper subset of S. If S is a spanning set of a projective space P, then there is a subset of S that spans P and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of P is n, such an independent spanning set has n + 1 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
A projective frame or projective basis is an ordered set of points in a projective space that allows defining coordinates.[2] More precisely, in an n-dimensional projective space, a projective frame is a tuple of n + 2 points such that any n + 1 of them are independent; that is, they are not contained in a hyperplane.
If V is an (n + 1)-dimensional vector space, and p is the canonical projection from V to P(V), then (p(e0), ..., p(en+1)) is a projective frame if and only if (e0, ..., en) is a basis of V and the coefficients of en+1 on this basis are all nonzero. By rescaling the first n vectors, any frame can be rewritten as (p(e′0), ..., p(e′n+1)) such that e′n+1 = e′0 + ... + e′n; this representation is unique up to the multiplication of all e′i with a common nonzero factor.
The projective coordinates or homogeneous coordinates of a point p(v) on a frame (p(e0), ..., p(en+1)) with en+1 = e0 + ... + en are the coordinates of v on the basis (e0, ..., en). They are only defined up to scaling with a common nonzero factor.
The canonical frame of the projective space Pn(K) consists of images by p of the elements of the canonical basis of Kn+1 (that is, the
Projective geometry
In
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 a
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of
Projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.[3] It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental 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 study
Topology
A projective space is a
Let S be the unit sphere in a normed vector space V, and consider the function that maps a point of S to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of P(V) consist of two antipodal points. As spheres are compact spaces, it follows that:
For every point P of S, the restriction of π to a neighborhood of P 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 V, any vector can be identified with its coordinates on the basis, and any point of P(V) may be identified with its homogeneous coordinates. For i = 0, ..., n, the set is an open subset of P(V), and since every point of P(V) has at least one nonzero coordinate.
To each Ui is associated a
These charts form an
For example, in the case of n = 1, that is of a projective line, there are only two Ui, which can each be identified to a copy of the
CW complex structure
Real projective spaces have a simple CW complex structure, as Pn(R) can be obtained from Pn−1(R) by attaching an n-cell with the quotient projection Sn−1 → Pn−1(R) as the attaching map.
Algebraic geometry
Originally,
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
An important property of projective spaces and projective varieties is that the image of a projective variety under a
A projective space is itself a projective variety, being the set of zeros of the zero polynomial.
Scheme theory
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:[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
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
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
- 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:
A subspace Xi 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
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 n = 2 is equivalent to a 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
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
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 spaces is PG(3, 2), with 15 points, 35 lines and 15 planes.
Morphisms
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:
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
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
the
Dual projective space
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 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 manifoldis 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.
Another generalization of projective spaces are weighted projective spaces; these are themselves special cases of toric varieties.[8]
See also
- Generalizations
- Grassmannian manifold
- Projective line over a ring
- Space (mathematics)
- Projective geometry
- projective transformation
- projective representation
Notes
- ^ The absence of space after the comma is common for this notation.
- ^ The correct definition of the multiplicity if 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
- 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
- ISBN 978-0-13-022269-5, p. 109.
- ^ Mukai 2003, example 3.72
References
- 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, MR 1629468
- ISBN 0-471-18283-4
- OCLC 977732
- Dembowski, P. (1968), Finite geometries, MR 0233275
- Greenberg, M.J.; Euclidean and non-Euclidean geometries, 2nd ed. Freeman (1980).
- 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, ISBN 978-0-521-80906-1
- MR 0179666(Reprint of 1910 edition)