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Projective space

In , projective space \mathbb{P}^n(F) over a F is defined as the set of one-dimensional subspaces (lines through the ) of the F^{n+1}, or equivalently, the quotient space of F^{n+1} \setminus \{0\} by the of by nonzero elements of F. This construction generalizes by incorporating points at , allowing to intersect and eliminating the distinction between and affine transformations in favor of projective ones. The concept of projective space emerged from early efforts in perspective drawing during the , where architect developed linear perspective around 1425 to represent three-dimensional scenes on two-dimensional surfaces, leading to the observation that parallel lines converge at a . In the , Gérard Desargues formalized these ideas in his 1639 work Brouillon projet d'une atteinte aux événements des rencontres du cônne avec un plan, introducing projective methods for conic sections, while contributed theorems on hexagons inscribed in conics. The saw projective mature as an independent field, with key developments by , who axiomatized it in 1822, and , emphasizing invariants like the under projections. In , projective space serves as a foundational ambient space for studying varieties, constructed as the Proj of a in n+1 variables, enabling the use of to compactify affine varieties and handle phenomena at . Points in \mathbb{P}^n are represented by [x_0 : x_1 : \dots : x_n], where does not change the point, and it admits a natural structure that supports morphisms and embeddings of algebraic sets. This framework is essential for theorems like Bézout's on intersections and for classifying curves and surfaces up to projective equivalence. Beyond pure mathematics, projective space finds applications in , where it models projections from three-dimensional scenes onto planes, facilitating tasks like camera , estimation, and structure-from-motion . In this context, the projective plane \mathbb{P}^2 represents coordinates, allowing to handle distortions and multiple . It also appears in for pose estimation and in for rendering transformations, bridging geometric theory with practical computation.

Motivation

Geometric Intuition

In the , denoted \mathbb{RP}^2, from are understood to intersect at points at , forming a line at that completes the space and resolves the exception where parallels do not meet. This intuition arises from considering the plane as the set of directions from a viewpoint, where all lines in a parallel class converge to a single ideal point on the horizon, eliminating the need to treat parallelism as a special case. For visualization, imagine a diagram of \mathbb{RP}^2 as a disk with opposite boundary points identified: finite points lie inside, while the boundary represents the line at , and lines appear as great circles crossing the boundary at their infinite endpoints, showing how Euclidean parallels meet there. This geometric idea has historical roots in , where artists like and employed techniques to create illusions of depth, converging —such as architectural edges or receding floors—to vanishing points on the horizon line. In works like Masaccio's The Holy Trinity (c. 1427), these vanishing points simulate the meeting of parallels at infinity, prefiguring the mathematical formalization of by figures like in the 17th century, who recognized such projections as invariant properties of . This artistic thus provided an intuitive bridge to projective concepts, treating the canvas as a where visual convergence mirrors geometric completion. A further intuition comes from conic sections, where slicing a with planes parallel to its side yields a parabola tangent to the line at , while hyperbolas cross it at two points and ellipses avoid it entirely; in projective space, these distinctions dissolve as the space completes the by adding the infinite line, allowing all non-degenerate conics to be projectively equivalent without degeneration exceptions at . Projective space thus serves as a natural compactification of , incorporating these infinite behaviors seamlessly. Overall, projective space unifies points, lines, and planes by treating infinite elements as ordinary points, ensuring that any two lines intersect at exactly one point and any two planes intersect in a line, without special rules for directions at infinity.

Algebraic Perspective

In , projective space arises naturally as a construction from linear . Specifically, the real projective space \mathbb{RP}^n is the of the \mathbb{R}^{n+1} minus the origin by the of by nonzero reals, where two nonzero vectors v, w \in \mathbb{R}^{n+1} are identified if w = \lambda v for some \lambda \in \mathbb{R} \setminus \{0\}. Each represents a one-dimensional , or line through the origin, of \mathbb{R}^{n+1}. For instance, \mathbb{RP}^2 is obtained as \mathbb{R}^3 \setminus \{0\} / \sim, parameterizing lines through the origin in three-dimensional . This provides significant benefits for studying by enabling homogenization. An affine f(x_1, \dots, x_n) of d is homogenized to a projective F(X_0, \dots, X_n) by introducing a new variable X_0 and multiplying each term of less than d by appropriate powers of X_0 to make all terms homogeneous of d. For example, the affine x^2 + y^2 = 1 homogenizes to X^2 + Y^2 - Z^2 = 0 in \mathbb{P}^2, where the original affine plane corresponds to the patch Z \neq 0 via dehomogenization x = X/Z, y = Y/Z. This process defines the projective closure of the affine , incorporating points at and ensuring the is invariant under scaling. Projective space thus resolves coordinate singularities inherent in affine descriptions by covering the space with multiple affine patches without global inconsistencies. In the circle example, the affine is compact but lacks a uniform treatment at "" in affine coordinates; its projective as the conic X^2 + Y^2 - Z^2 = 0 embeds it smoothly into \mathbb{P}^2, adding ideal points at [1 : \pm i : 0] over the complexes and providing a closed, proper free from affine boundary artifacts.

Definitions

Homogeneous Coordinates

The n-dimensional projective space over a field K, denoted \mathbb{P}^n(K), is defined as the set of equivalence classes of (n+1)-tuples (x_0, x_1, \dots, x_n) \in K^{n+1} \setminus \{0\}, where two tuples (x_0, \dots, x_n) and (y_0, \dots, y_n) are equivalent if there exists a nonzero \lambda \in K such that y_i = \lambda x_i for all i = 0, \dots, n. This construction identifies points that differ by , capturing lines through the origin in the K^{n+1}. Points in \mathbb{P}^n(K) are typically denoted by square brackets [x_0 : x_1 : \dots : x_n], with the understanding that [x_0 : \dots : x_n] = [\lambda x_0 : \dots : \lambda x_n] for any \lambda \in K \setminus \{0\}. To recover affine coordinates, standard affine charts cover \mathbb{P}^n(K): for each i = 0, \dots, n, define the U_i = \{ [x_0 : \dots : x_n] \mid x_i \neq 0 \}, which is isomorphic to the K^n via dehomogenization, mapping [x_0 : \dots : x_n] to (x_0 / x_i, \dots, \hat{x}_i / x_i, \dots, x_n / x_i), where the hat indicates omission of the i-th term. These charts provide a coordinate atlas, allowing local affine descriptions of . A concrete example is the real projective line \mathbb{RP}^1, which consists of points [x : y] with (x, y) \in \mathbb{R}^2 \setminus \{(0,0)\} up to scaling by nonzero \lambda \in \mathbb{R}. Topologically, \mathbb{RP}^1 is homeomorphic to the circle S^1, obtained by identifying antipodal points on the unit circle \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}, such as [0:1] representing the origin and [1:0] representing the point at infinity. Homogeneous coordinates facilitate the embedding of affine n-space K^n into \mathbb{P}^n(K) as the chart U_n = \{ [x_0 : \dots : x_{n-1} : 1] \mid x_0, \dots, x_{n-1} \in K \}, with the complementary hyperplane at infinity \{ [x_0 : \dots : x_{n-1} : 0] \mid (x_0, \dots, x_{n-1}) \neq (0, \dots, 0) \} consisting of directions of parallel lines in the affine space. This structure resolves issues like parallel lines intersecting at infinity, providing a unified framework for affine and projective geometries.

Quotient Construction

The projective space associated to a vector space V over a field K, denoted P(V), is formally defined as the quotient set (V \setminus \{0\}) / K^\times, where K^\times = K \setminus \{0\} acts on the nonzero vectors by scalar multiplication. Two nonzero vectors v, w \in V are equivalent, written v \sim w, if there exists a nonzero scalar \lambda \in K^\times such that w = \lambda v; each equivalence class $$ thus corresponds to a one-dimensional subspace (or ray) through the origin in V. This quotient construction satisfies a : any f: V \setminus \{0\} \to X to another set X that is constant on equivalence classes (i.e., f(\lambda v) = f(v) for all \lambda \in K^\times) factors uniquely through the projection map \pi: V \setminus \{0\} \to P(V), yielding a well-defined map \overline{f}: P(V) \to X such that f = \overline{f} \circ \pi. When V is an (n+1)-dimensional over K, the resulting projective space P^n(K) = P(V) has n, since each point in P^n(K) represents a one-dimensional of V. A concrete realization of this construction uses homogeneous coordinates, where points are represented by equivalence classes of (n+1)-tuples in K^{n+1} \setminus \{0\}. For example, over the complex numbers, the one-dimensional projective space \mathbb{CP}^1 = P(\mathbb{C}^2) is isomorphic to the \mathbb{C} \cup \{\infty\}, with the isomorphism given by from the unit sphere in \mathbb{R}^3.

Basic Structures

Subspaces and Dimensions

In , a of a projective space \mathbb{P}(V), where V is a over a K of n+1, is defined as the image under the quotient map \pi: V \setminus \{0\} \to \mathbb{P}(V) of a W \subseteq V with \dim W \geq 1. When \dim W = 1, \mathbb{P}(W) is a point, the 0-dimensional . Such a projective , denoted \mathbb{P}(W), consists of all lines through the origin in W. The dimension of a projective subspace \mathbb{P}(W) is given by \dim \mathbb{P}(W) = \dim W - 1. Equivalently, a k-dimensional projective subspace corresponds to a (k+1)-dimensional subspace of V. This relation establishes the linear algebraic foundation for the geometry of projective spaces, linking their subspaces directly to those of the underlying . For instance, in the real projective space \mathbb{RP}^3, which arises from a 4-dimensional vector space, a projective line is \mathbb{P}(L) where L \subseteq \mathbb{R}^4 is a 2-dimensional subspace, and a projective plane is \mathbb{P}(M) where M \subseteq \mathbb{R}^4 is a 3-dimensional subspace. The entire projective space \mathbb{P}^n has dimension n, while its points, which are the 1-dimensional subspaces of V, form the 0-dimensional subspaces.

Lines and Incidence

In projective space \mathbb{P}^n(K) over a K, a is defined as a one-dimensional projective subspace, corresponding to a two-dimensional vector subspace of the underlying V^{n+1}. Such a line can be parametrized by any two distinct points on it, which generate the line as their span. Incidence between a point and a line in \mathbb{P}^n(K) is determined by the corresponding vector subspaces: a point P, represented by a one-dimensional subspace U \subseteq V, lies on a line L, represented by a two-dimensional subspace W \subseteq V, if and only if U \cap W \neq \{0\}. This relation satisfies the axioms of projective geometry, ensuring that any two distinct points determine a unique line. In higher dimensions, two lines intersect at a point if and only if they are contained in a common plane. The join of a set of points is the smallest projective containing them, obtained as the projectivization of the of their representing vectors. Dually, the meet of a set of subspaces is their , provided it is nonempty and of the appropriate dimension; for two lines, if they intersect, their meet is the unique point of . In the real projective plane \mathbb{RP}^2, any two distinct points determine a unique line, and there are no —all lines intersect at exactly one point, reflecting the absence of parallelism in . This incidence structure implies theorems such as Desargues' theorem, which concerns the of intersection points of corresponding sides of two perspective triangles.

Topology

Topological Properties

The real projective space \mathbb{RP}^n is equipped with the quotient induced from the projection map p: \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{RP}^n, where points are identified under by nonzero reals, or equivalently from the double covering S^n \to \mathbb{RP}^n identifying antipodal points. This topology renders \mathbb{RP}^n compact, as it is the continuous image of the compact S^n, and Hausdorff, since the is closed and the projection is open. As a topological space, \mathbb{RP}^n admits the structure of a smooth n-dimensional manifold, with an atlas derived from affine charts corresponding to hyperplanes not containing the origin in \mathbb{R}^{n+1}. Regarding orientability, \mathbb{RP}^n is orientable if and only if n is odd, since the antipodal map on S^n preserves orientation precisely when n+1 is even; for even n, it is non-orientable. For low dimensions, \mathbb{RP}^1 is homeomorphic to the circle S^1, which is a compact orientable 1-manifold, while \mathbb{RP}^2 can be realized as the disk D^2 with antipodal points on the boundary \partial D^2 identified, yielding a compact non-orientable surface diffeomorphic to the real projective plane. Key and invariants further characterize \mathbb{RP}^n. The is \pi_1(\mathbb{RP}^n) \cong \mathbb{Z}/2\mathbb{Z} for n \geq 2, arising from the two-sheeted S^n \to \mathbb{RP}^n and the action of the antipodal map. With \mathbb{Z}/2\mathbb{Z}-coefficients, the groups are H_k(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} for $0 \leq k \leq n and zero otherwise, computed via of the CW structure with one cell per dimension up to n. The \mathbb{CP}^n inherits its topology from the quotient map \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^n under multiplication by nonzero scalars, or equivalently from the S^{2n+1} \to \mathbb{CP}^n with S^1-fibers. This endows \mathbb{CP}^n with the structure of a compact and a $2n-dimensional real manifold (or n-dimensional ), which is always orientable due to its compatible structure. Unlike \mathbb{RP}^n, \mathbb{CP}^n is simply connected, with trivial .

CW Complex Decomposition

The real projective space \mathbb{RP}^n admits a structure consisting of one open cell e^k in each k from 0 to n. This construction arises from viewing \mathbb{RP}^n as the of the n- S^n by the antipodal map, which identifies each point x with -x. Equivalently, \mathbb{RP}^n can be built inductively by taking the (k-1)-skeleton to be \mathbb{RP}^{k-1} and attaching a k-cell via the quotient map q: S^{k-1} \to \mathbb{RP}^{k-1}, which is the projection induced by the antipodal identification and serves as a degree-2 covering map. In this attaching process, the of the k-disk D^k is mapped to \mathbb{RP}^{k-1} by collapsing antipodal points on S^{k-1}, ensuring the CW structure aligns with the topological . The resulting cellular has \mathbb{Z} in each from 0 to n, with maps d_k: C_k(\mathbb{RP}^n) \to C_{k-1}(\mathbb{RP}^n) given by multiplication by 2 if k is even and by 0 if k is odd. For the example of \mathbb{RP}^2, the CW structure includes one 0-cell e^0 (a point), one 1-cell e^1 (forming a after attachment), and one 2-cell e^2 attached via the degree-2 S^1 \to \mathbb{RP}^1 \cong S^1. The cellular is then $0 \to \mathbb{Z} \xrightarrow{d_2 = \cdot 2} \mathbb{Z} \xrightarrow{d_1 = 0} \mathbb{Z} \to 0, which yields the groups H_0(\mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}, H_1(\mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}, and H_2(\mathbb{RP}^2; \mathbb{Z}) = 0. This decomposition facilitates computations, such as those involving via cellular chains.

Algebraic Aspects

Over Finite Fields

Projective spaces over finite fields \mathbb{F}_q, where q is a , provide analogs of classical with finite point sets and well-defined combinatorial structures. The n-dimensional projective space \mathbb{P}^n(\mathbb{F}_q), often denoted PG(n, q), consists of the 1-dimensional subspaces (lines through the origin) of the (n+1)-dimensional \mathbb{F}_q^{n+1}. Each point in this space corresponds to an of nonzero vectors under by nonzero elements of \mathbb{F}_q. The total number of points is given by the \frac{q^{n+1} - 1}{q - 1}, which counts the distinct directions in the vector space. A key combinatorial feature is the enumeration of subspaces. The number of k-dimensional projective subspaces in PG(n, q) equals the \dbinom{n+1}{k+1}_q, which generalizes the and arises from counting the (k+1)-dimensional subspaces of \mathbb{F}_q^{n+1}. This coefficient is defined as \dbinom{m}{r}_q = \prod_{i=0}^{r-1} \frac{q^{m-i} - 1}{q^{r-i} - 1}, for m = n+1 and r = k+1, and it satisfies recursive properties analogous to those of . These counts highlight the symmetric and balanced nature of the , where subspaces intersect in controlled ways. For the specific case of the projective plane PG(2, q), the formula simplifies to q^2 + q + 1 points, with each line also containing exactly q + 1 points and the same number of lines in total. Every pair of points determines a unique line, and every pair of lines intersects in a unique point, embodying the fundamental incidence axioms. This structure exemplifies the order q of the plane, where lines have q + 1 points. Finite projective spaces exhibit rich combinatorial properties, including their interpretation as partial geometries. In particular, the point-line incidence structure of PG(2, q) forms a partial geometry pg(q, q, 1), where every two non-collinear points have exactly one common neighbor, satisfying the defining axioms of regularity and intersection control. Higher-dimensional spaces extend these properties, serving as frameworks for partial linear spaces with uniform line sizes. These geometries find applications in , notably in the construction of projective Reed-Muller codes, which evaluate polynomials on the points of PG(n, q) to yield error-correcting codes with parameters tied to the space's and size. For instance, these codes generalize classical Reed-Muller codes and achieve minimum distances determined by the geometry's intersection properties.

Projective Modules

In the context of , the notion of projective space can be generalized beyond vector spaces over fields to the projectivization of projective modules over a R. Specifically, for a projective R-module P of constant rank n+1, the projectivization \mathbb{P}(P) is the \operatorname{Proj}_R(\operatorname{Sym}(P^\vee)), where \operatorname{Sym}(P^\vee) is the on the dual module P^\vee, equipped with a morphism to \operatorname{Spec} R. This construction parameterizes the rank-1 locally quotients of P (or dually, line submodules), assuming P is locally of constant rank, which holds when R is such that projective modules of constant rank are locally , as over domains or local rings. When P is , \mathbb{P}(P) recovers the classical projective space \mathbb{P}^n_R. A key property enabling this geometric interpretation is the projectivity of P, which implies that P is locally free over R. That is, there exists a covering of \operatorname{Spec} R by basic open sets D(f_i) such that the localization P_{f_i} is a free R_{f_i}-module of rank n+1. This local freeness ensures that \mathbb{P}(P) behaves geometrically like a projective space locally on \operatorname{Spec} R, bridging the algebraic structure of modules with geometric intuition. Over polynomial rings, such as R = k[x_1, \dots, x_m] for a k, projective modules play a central role in relating this construction to homogeneous ideals in graded rings. By the Quillen-Suslin , every finitely generated projective module over a polynomial ring is , so \mathbb{P}(P) aligns with the standard projective space; more broadly, it connects to the homogeneous spectrum of the on P, providing a foundation for over such rings. This module-theoretic projective space also relates to vector bundles: the classical projective space \mathbb{P}^n_R can be viewed as the projectivization of the trivial of rank n+1 over \operatorname{Spec} R, where points correspond to line subbundles ( projective submodules). In general, for a P, \mathbb{P}(P) parameterizes the line subbundles of the associated , emphasizing the correspondence between projective modules and locally free sheaves.

Synthetic Geometry

Axiomatic Foundations

can be developed synthetically through axiomatic systems that emphasize incidence relations between points and lines, eschewing coordinates to define spaces purely in terms of these primitives. influenced subsequent foundations for , where the parallel postulate (Group III) is replaced by the axiom that any two lines in a intersect at exactly one point. This is achieved by incorporating points at infinity, ensuring all lines meet. (Group I) include: two distinct points determine a unique line (Axiom I, 1), any three non-collinear points determine a unique (Axiom I, 3), and every line contains at least two points while every contains at least three non-collinear points (Axiom I, 7). The order axioms (Group II) define betweenness on lines, the congruence axioms (Group IV) handle segment and angle equivalences, and continuity axioms (Group V) ensure completeness via Dedekind cuts. A more direct synthetic framework for projective spaces appears in the work of and Young, who define a projective space as an satisfying specific on points and lines. Their incidence include: any two distinct points lie on exactly one line (Axiom A1), and the Veblen axiom (A2 or V): given a ABC, if a line intersects sides AB and AC at points not A, B, or C, then it intersects BC (Axiom A2). Extension assumptions guarantee non-degeneracy: every line has at least three points (E0), there exists at least one line (E1), not all points are collinear (E2), and not all points are coplanar (E3). For higher dimensions, additional require that any two intersect in a unique line, and that the space has dimension three or higher, with points existing outside any given . These distinguish projective spaces from affine ones by eliminating parallels. In such axiomatic projective planes, classical theorems emerge as consequences of the incidence structure. For instance, Pappus' theorem states that if two lines are each intersected by three parallel lines (or in projective terms, if a hexagon is inscribed in two lines with alternate vertices on each), then the intersection points of opposite sides are collinear; this holds in any Desarguesian projective plane satisfying the basic incidence axioms (P1–P4) and Desargues' axiom (P7), where P1 asserts any two points determine a unique line, P2 any two lines intersect in a unique point, P3 ensures non-collinearity of all points, and P4 guarantees a line through a point not on a given line. A pivotal result in these systems is the coordinatization theorem, which demonstrates that the axioms independently imply the existence of a without presupposing one. Specifically, for projective spaces of at least three satisfying the Veblen-Young axioms, the is isomorphic to the projective over a defined by a , allowing to be constructed algebraically from the incidence relations alone; this independence underscores the synthetic approach's power in deriving analytic models from pure incidence.

Finite Projective Spaces

Finite projective spaces are incidence structures satisfying the axioms of , restricted to finite point and line sets. For dimensions greater than or equal to 3, the Veblen-Young theorem establishes that all such spaces are Desarguesian, meaning they are isomorphic to the projective geometry PG(n, q) constructed from a over a \mathbb{F}_q, where q is a and n ≥ 3. This classification follows from the fact that any projective space satisfying Desargues' axiom in dimension at least 3 can be coordinatized by a , and by Wedderburn's little theorem, all finite division rings are commutative fields. Thus, no non-Desarguesian examples exist in these higher dimensions. In dimension 2, finite projective planes exhibit greater . Desarguesian planes of q (a prime ) are precisely the PG(2, q) over \mathbb{F}_q, but non-Desarguesian planes also exist, constructed via alternative coordinatizations such as rings that fail to yield rings. Notable examples include the Hughes planes of order p^{2k} for odd prime p and integer k ≥ 1, which are obtained by replacing the field multiplication in the Desarguesian plane with a non-associative derived from a near-field. These planes satisfy the projective plane axioms but violate Desargues' theorem, distinguishing them from their field-based counterparts. The smallest finite projective plane is the Fano plane, denoted PG(2, 2), which has order 2 and consists of 7 points and 7 lines, each line containing 3 points. It serves as the foundational example of a Desarguesian plane over the field with 2 elements and illustrates the basic of . All known finite projective planes have order q, where q is a , and it is conjectured that this holds for all such planes, though this remains an . Existence conditions for finite projective planes are constrained by the Bruck-Ryser-Chowla theorem, which provides a necessary criterion: if a projective plane of order n exists and n ≡ 1 or 2 (mod 4), then n must be expressible as the sum of two integer squares. This theorem rules out planes of certain orders, such as n=6, and has been instrumental in computational searches for non-existent planes, like the unresolved case of order 12.

Transformations

Projective Linear Groups

The projective linear group associated to a V over a K, denoted \mathrm{PGL}(V), is defined as the \mathrm{GL}(V)/Z, where \mathrm{GL}(V) is the general linear group of invertible linear endomorphisms of V and Z is the center consisting of scalar multiples of the map. This identifies linear transformations that differ by a scalar multiple, reflecting the projective nature of the space. The group \mathrm{PGL}(V) acts faithfully on the projective space \mathrm{P}(V) by sending equivalence classes $$ (lines through the origin spanned by x \in V \setminus \{0\}) to [f(x)], where f \in \mathrm{GL}(V). In matrix terms, if V = K^{n+1}, elements of \mathrm{PGL}(n+1, K) can be represented by invertible (n+1) \times (n+1) matrices modulo scalar multiples, acting via [A]() = [A x], where denotes the projective point corresponding to the column vector $x \neq 0$, and the action preserves collinearity: if and $$ lie on a projective line (i.e., x, y are linearly dependent), then so do [A x] and [A y]. This preservation extends to higher-dimensional subspaces, mapping projective subspaces to projective subspaces of the same dimension. A concrete example arises in the real \mathbb{RP}^2, where \mathrm{PGL}(3, \mathbb{R}) is the group of all projective transformations, including perspectivities—central projections from a point outside a line or —and more general maps that generate divisions on lines, such as the inversion of two points in a set of four collinear points. These transformations maintain the and thus preserve properties, which are fundamental invariants in . The fundamental theorem of projective geometry characterizes these groups as the full groups of projective spaces in many cases: any bijective map between projective spaces of at least 2 over a that preserves incidence of points and lines (i.e., a collineation) is induced by a semilinear on the underlying . Over fields, this implies that automorphisms of \mathrm{P}^n(K) (for n \geq 2) arise from elements of the projective semilinear group \mathrm{P\Gamma L}(n+1, K), with \mathrm{PGL}(n+1, K) as the linear when the field automorphism is trivial. This result, tracing back to foundational work by von Staudt and later formalized by Veblen and Young, underscores the rigid structure imposed by projective incidence.

Morphisms Between Spaces

In , a between projective spaces f: \mathbb{P}^m \to \mathbb{P}^n over an k is defined by n+1 homogeneous polynomials F_0, \dots, F_n \in k[x_0, \dots, x_m] of the same degree d > 0 such that F_0, \dots, F_n have no common zeros except the in k^{m+1}, ensuring the map is well-defined and regular everywhere. This construction respects the projective equivalence, mapping [x_0 : \dots : x_m] to [F_0(x) : \dots : F_n(x)], and extends the notion of rational maps by avoiding indeterminacy loci. A special case arises when d=1, where the polynomials are linear forms, inducing f via a linear map \phi: k^{m+1} \to k^{n+1} between the ambient vector spaces, such that f() = [\phi(v)]. If \phi is invertible, f is an ; more generally, such maps induce embeddings of linear subspaces if \phi is injective. Collineations, which are these degree-1 isomorphisms between projective spaces of the same dimension, are precisely those induced by semilinear isomorphisms of the underlying vector spaces, as established by the fundamental theorem of projective geometry. The Veronese embedding provides a example of a non-linear , mapping \mathbb{P}^n into \mathbb{P}^N with N = \binom{n+d}{d} - 1 via all monomials of degree d in the coordinates: \nu_d([x_0 : \dots : x_n]) = [ \dots : x_0^{i_0} \cdots x_n^{i_n} : \dots ], where the indices satisfy i_0 + \dots + i_n = d. This embedding is an isomorphism onto its image, a known as the Veronese variety, and highlights how higher-degree morphisms embed lower-dimensional spaces into higher ones while preserving projective structure. An important birational example is the projection from a point O \in \mathbb{P}^n (not on a target hyperplane H \cong \mathbb{P}^{n-1}) to H, defined rationally by sending a point P to the intersection of the line OP with H; this map is birational, with an inverse given by lines through O from points on H, and is undefined only at O. Such projections illustrate how rational maps between projective spaces can be resolved to morphisms via blow-ups, maintaining birational equivalence. Since projective spaces are complete varieties over k, any morphism f: \mathbb{P}^m \to \mathbb{P}^n is proper, meaning it is separated, of finite type, and universally closed, a property inherited from the properness of the structure morphism to \operatorname{Spec} k. This ensures that images of closed sets remain closed, facilitating compactness-like behavior in algebraic settings.

Dualities and Generalizations

Dual Projective Space

In projective geometry, the dual projective space (\mathbb{P}^n)^*, also denoted \mathbb{P}^n_*, of a projective space \mathbb{P}^n over a field K is constructed such that its points correspond to the hyperplanes of \mathbb{P}^n, and its k-dimensional subspaces correspond to the sets of hyperplanes in \mathbb{P}^n that contain a fixed subspace of codimension k+1. This duality arises from the vector space perspective: if \mathbb{P}^n = \mathbb{P}(V) for a vector space V of dimension n+1, then (\mathbb{P}^n)^* = \mathbb{P}(V^*), where V^* is the dual vector space of linear functionals on V, and each point \in \mathbb{P}(V^*) represents the hyperplane \ker f \subset \mathbb{P}(V). Lines in the dual space consist of pencils of hyperplanes, namely the hyperplanes containing a fixed codimension-2 subspace of \mathbb{P}^n. The duality principle reverses incidence relations: a point p \in \mathbb{P}^n lies on a hyperplane H \subset \mathbb{P}^n the hyperplane corresponding to p in the dual contains the point corresponding to H in (\mathbb{P}^n)^*. This reversal enables the dualization of geometric theorems by interchanging points with hyperplanes and incidence with containment, preserving the logical structure. For instance, Desargues' theorem, which asserts that two triangles in perspective from a point are in perspective from a line, has as its dual the converse statement that two triangles in perspective from a line are in perspective from a point. In the real projective plane \mathbb{RP}^2, the interchanges points and lines, transforming configurations accordingly. A notable example is conic duality: a conic section, viewed as a of points in \mathbb{RP}^2, dualizes to the of its lines, which forms another conic in the plane. Over a K, the projective space \mathbb{P}^n(K) is naturally isomorphic to its (\mathbb{P}^n(K))^* via a nondegenerate on the underlying V, which induces an V \to V^* by v \mapsto \langle \cdot, v \rangle, preserving the projective structure. This identifies points and hyperplanes symmetrically through the pairing.

Abstract Generalizations

In infinite-dimensional settings, the notion of projective space extends beyond finite-dimensional vector spaces to spaces like Hilbert and Banach spaces, where it plays a crucial role in functional analysis and quantum mechanics. For a complex Hilbert space H, the projective space \mathbb{P}(H) consists of one-dimensional subspaces (rays) of H, equipped with a natural Kähler structure induced by the inner product on H. This structure arises from the quotient of the unit sphere in H by the action of the unit circle, providing a manifold model for the space of pure quantum states, where each ray corresponds to an equivalence class of state vectors differing by a phase factor. In quantum mechanics, \mathbb{P}(H) serves as the state space, with observables represented as functions on this space and dynamics governed by a symplectic form derived from the Fubini-Study metric, enabling a classical Hamiltonian formulation of quantum evolution. A further generalization appears in the context of s, where the projective space \mathbb{P}(V) for an infinite-dimensional complex V is defined similarly as the set of one-dimensional subspaces, metrized via an analog of the Fubini-Study adapted to the on V. This ensures \mathbb{P}(V) is paracompact and metrizable, facilitating the study of infinitesimal neighborhoods and structures in infinite-dimensional . Such constructions are essential in for examining properties like injectivity and extensions of operators, though the absence of an inner product complicates the Kähler aspects compared to the Hilbert case. In categorical terms, projective spaces generalize to projective objects within abelian categories, which abstract the notion of subspaces that "project" onto quotients via lifts of homomorphisms. An object P in an abelian category \mathcal{A} is projective if, for every epimorphism A \twoheadrightarrow B and morphism P \to B, there exists a lift P \to A making the diagram commute; this property ensures that \operatorname{Hom}(P, -) preserves epimorphisms. Projective objects thus generalize direct summands of free objects, providing a framework for resolutions and derived categories that extends the subspace structure of classical projective spaces to arbitrary abelian settings, such as modules over rings or sheaves on schemes. The \operatorname{Gr}(k, V), parametrizing k-dimensional subspaces of V, serves as a higher-dimensional analog of projective space, embedding as via the into \mathbb{P}(\bigwedge^k V). For finite k and \dim V = n, \operatorname{Gr}(k, n) generalizes the lines in \mathbb{P}^{n-1} (where k=1) to k-planes, inheriting a rich geometry including Schubert cycles and cohomology rings that mirror aspects of projective space but in higher rank. This structure is foundational in algebraic geometry and representation theory, where Grassmannians classify partial flags and support generalizations of projective duality to multi-dimensional subspaces.