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Completion of the usual space with "points at infinity"
point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be
Projective_space
Mathematical concept
complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label
Complex_projective_space
Geometric concept of a 2D space with "points at infinity" adjoined
the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can
Projective_plane
Algebraic variety in a projective space
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in
Projective_variety
Type of topological space
In mathematics, real projective space, denoted R P n {\displaystyle \mathbb {RP} ^{n}} or P n ( R ) , {\displaystyle \mathbb {P} _{n}(\mathbb {R}
Real_projective_space
Type of geometry
compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts
Projective_geometry
Generalized Euclidean space in mathematics
space. In quantum mechanics, the equivalence classes [ v ] {\displaystyle [v]} are also referred to as rays or projective rays. Each such projective ray
Projective_Hilbert_space
Compact non-orientable two-dimensional manifold
real projective plane, denoted R P 2 {\displaystyle \mathbf {RP} ^{2}} or P 2 {\displaystyle \mathbb {P} _{2}} , is a two-dimensional projective space
Real_projective_plane
Mathematical set with some added structure
subsets of projective space. Projective varieties were subsets defined by a set of homogeneous polynomials. At each point of the projective variety, all
Space_(mathematics)
Concept in mathematics
mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates
Quaternionic_projective_space
Concept in projective geometry
duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may be defined axiomatically
Duality_(projective_geometry)
Construction in group theory
especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action
Projective_linear_group
Isomorphism of projective spaces in geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces
Homography
In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn]
Weighted_projective_space
Manifold or algebraic variety of dimension n in a space of dimension n+1
which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces
Hypersurface
Geometric system with a finite number of points
line, called a projective line. Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for n = 2 is a projective plane. These
Finite_geometry
Complex algebraic variety
isomorphic to it. There are exactly 50 fake projective planes. Prasad & Yeung (2006) found four examples of fake projective 4-folds, and showed that no arithmetic
Fake_projective_space
Geometric space whose points represent algebro-geometric objects of some fixed kind
{\displaystyle \theta } . Real projective space P n ( R ) {\displaystyle \mathbb {P} ^{n}(\mathbb {R} )} is the moduli space of lines through the origin
Moduli_space
Fiber bundle whose fibers are projective spaces
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle
Projective_bundle
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety
Hilbert_scheme
Plane tiling corresponding to a polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations
Projective_polyhedron
Projective analogue of the spectrum of a ring
schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental
Proj_construction
Fundamental space of geometry
defining a projective space as the set of the vector lines in a vector space of dimension one more. As for affine spaces, projective spaces are defined
Euclidean_space
Mathematical object studied in the field of algebraic geometry
by definition quasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of a projective space. For example, in Chapter
Algebraic_variety
Line with a point at infinity added
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a point
Projective_line
Quotient of special unitary group by its center
isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices,
Projective_unitary_group
Concept in geometry
In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set
Polar_space
Euclidean group Poincaré group Galilean group Projective space Projective transformation Projective geometry Projective linear group Quadric and conic section
Outline_of_linear_algebra
quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of
Quasi-projective_variety
The concept of a projective space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Topological space in group theory
as hyperbolic space. A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional
Homogeneous_space
Vector bundle of rank 1
tautological line bundle on projective space. The projectivization P ( V ) {\displaystyle \mathbf {P} (V)} of a vector space V {\displaystyle V} over a
Line_bundle
2-dimensional complex projective space
Pezzo surface Toric geometry Fake projective plane C. E. Springer (1964) Geometry and Analysis of Projective Spaces, pages 140–3, W. H. Freeman and Company
Complex_projective_plane
Locus of the zeros of a polynomial of degree two
points of the projective completion are the points of the projective space whose projective coordinates are zeros of P. So, a projective quadric is the
Quadric
Rational surface in 5-dimensional projective space
projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective
Veronese_surface
Smooth manifold with an inner product on each tangent space
and real projective spaces with their standard metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley
Riemannian_manifold
Space with one dimension
itself. The projective line over K , {\displaystyle K,} denoted P 1 ( K ) , {\displaystyle \mathbf {P} ^{1}(K),} is a one-dimensional space. In particular
One-dimensional_space
Space in mathematics and theoretical physics
metric between the two are considered. It turns out that complex projective 3-space C P 3 {\displaystyle \mathbb {CP} ^{3}} parametrizes such isomorphisms
Twistor_space
commutative ring assigned to any projective variety. If V is an algebraic variety given as a subvariety of projective space of a given dimension N, its homogeneous
Homogeneous_coordinate_ring
Euclidean space without distance and angles
group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are
Affine_space
Subspace of n-space whose dimension is (n-1)
the solution of a single linear equation. Projective hyperplanes are used in projective geometry. A projective subspace is a set of points with the property
Hyperplane
Group of rotations in 3 dimensions
is used to identify SO(3) topologically with three-dimensional real projective space. Consider the solid ball in R 3 {\displaystyle \mathbb {R} ^{3}} of
3D_rotation_group
conventional projective space to a point. More concretely, in a real projective space, complex projective space or quaternionic projective space K P n {\displaystyle
Stunted_projective_space
Embedding a topological space into a compact space as a dense subset
that the projective plane RP2 is not the one-point compactification of the plane R2 since more than one point is added. Complex projective space CPn is
Compactification (mathematics)
Compactification_(mathematics)
Coordinate system used in projective geometry
homogeneous coordinates are required to specify a point in the projective plane. The real projective plane can be thought of as the Euclidean plane with additional
Homogeneous_coordinates
Projective line over the real numbers
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically
Real_projective_line
Model of the extended complex plane plus a point at infinity
manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line P 1 ( C
Riemann_sphere
Type of transport in differential geometry
having the same unparametrized geodesics. Projective connections are modeled on the geometry of projective space. In modern terms, they may be described
Projective_connection
Concept in algebraic geometry
projective variety X {\displaystyle X} embedded in P r {\displaystyle \mathbb {P} ^{r}} has a natural linear system determining a map to projective space
Linear_system_of_divisors
Sphere-like surface
projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space
Ovoid_(projective_geometry)
Branch of mathematics
an intersection of spaces: Let Y 1 , Y 2 ⊂ X {\displaystyle Y_{1},Y_{2}\subset X} be projective subvarieties of a smooth projective variety. Then, we can
K-theory
Measure of a mathematical object studied in the field of algebraic geometry
as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. Let K be a field, and
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Algebraic structure in linear algebra
one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting
Vector_space
finite projective plane of even order. A k-arc which can not be extended to a larger arc is called a complete arc. In the Desarguesian projective planes
Arc_(projective_geometry)
Geometric transformation that preserves lines but not angles nor the origin
Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations
Affine_transformation
Concept in geometry
all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity
Point_at_infinity
Surface containing a line through every point
being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ruled surfaces
Ruled_surface
Branch of mathematics
the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather
Algebraic_geometry
Polynomial characterizing lines in projective 3-space
lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent
Klein_quadric
Mathematical space with two coordinates
coordinates. A two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has
Two-dimensional_space
2D surface which extends indefinitely
the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can
Plane_(mathematics)
NASA program from 1984 to 1990
The Teacher in Space Project (TISP) was a NASA program announced by U.S. President Ronald Reagan in 1984 designed to inspire students, honor teachers
Teacher_in_Space_Project
Circle-like pointset in a geometric plane
Indeed, there exist abstract ovals which can not lie in any projective plane. In a projective plane a set Ω of points is called an oval, if: Any line l
Oval_(projective_plane)
fourfold is a degree 3 hypersurface of dimension 4 in 5-dimensional projective space. Although some cubic fourfolds are known to be rational, it is known
Cubic_fourfold
Analogs of homology groups for algebraic varieties
equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same
Chow_group
Geometry
Oriented projective geometry is an oriented version of real projective geometry. Whereas the real projective plane describes the set of all unoriented
Oriented_projective_geometry
In projective geometry, a bijection between projective spaces that preserves collinearity
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to
Collineation
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V
Projective_orthogonal_group
Short exact sequence of sheaves on projective space
sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably
Euler_sequence
In tropical geometry, a tropical projective space is the tropical analog of the classic projective space. Given a module M over the tropical semiring
Tropical_projective_space
Curve defined as zeros of polynomials
zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three
Algebraic_curve
Optimal stable 2-systolic inequality
{CP} ^{n})} , valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study
Gromov's inequality for complex projective space
Gromov's_inequality_for_complex_projective_space
precisely the space of solutions to massless field equations, to sheaf cohomology groups on complex projective space. The projective space in question is
Penrose_transform
Embedding of a Grassmannian into projective space
three-dimensional space (which, as projective lines in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). The image
Plücker_embedding
Concept in algebraic geometry
to a projective scheme with O ( 1 ) = L {\displaystyle {\mathcal {O}}(1)=L} (not just to a proper scheme). Algebraic geometry of projective spaces Fano
Ample_line_bundle
Riemannian manifold with SU(n) holonomy
variety embedded in a projective space is a Kähler manifold, because there is a natural Fubini–Study metric on a projective space which one can restrict
Calabi–Yau_manifold
Characterises non-singular projective varieties amongst compact Kähler manifolds
so Kodaira's results states that Hodge manifolds are projective. The converse that projective manifolds are Hodge manifolds is more elementary and was
Kodaira_embedding_theorem
Concept in algebraic geometry
leads to many numerical invariants for projective varieties. For example, if X {\displaystyle X} is a smooth projective curve over an algebraically closed
Coherent_sheaf_cohomology
Shape
and no three points are collinear (on a common line). An ovoid in a projective space is a set Ω of points such that: Any line intersects Ω in at most 2
Oval
not just a copy of the projective plane, but a copy of the projective plane together with an embedding into projective 5-space. Varieties were often considered
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Subspace defined by a polynomial of degree 2 over a field
N-dimensional space defined by a polynomial equation of degree 2 over a field. The theory is simplified by working in projective space rather than affine space. An
Quadric_(algebraic_geometry)
Generalization of covers
construct a degree 2 branched covering of the corresponding projective elliptic curve to the projective line. The previous example may be generalized to any
Branched_covering
In projective geometry, points that define coordinates
and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining
Projective_frame
Type of mathematical space
generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action
Generalized_flag_variety
Theorem in projective geometry
Poncelet. This results in a projective plane. Desargues's theorem is true for the real projective plane and for any projective space defined arithmetically
Desargues's_theorem
Non-Euclidean geometry
as projective geometry. The points of n-dimensional projective space can be identified with lines through the origin in (n + 1)-dimensional space, and
Elliptic_geometry
Generalizations of codimension-1 subvarieties of algebraic varieties
system of D. A projective linear subspace of this projective space is called a linear system of divisors. One reason to study the space of global sections
Divisor_(algebraic_geometry)
Geometric model of the physical space
Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions
Three-dimensional_space
Algebraic variety containing an algebraic torus
of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. A precise definition is that
Toric_variety
Geometric space with five dimensions
five-dimensional (5D) space is a mathematical or physical space that has five independent dimensions. In physics and geometry, such a space extends the familiar
Five-dimensional_space
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric,
Simple_Lie_group
Lines not in the same plane
parallel nor intersect. In affine d-space, two flats of any dimension may be parallel. However, in projective space, parallelism does not exist; two flats
Skew_lines
Points of small height in projective space lie in a finite number of hyperplanes
mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang
Subspace_theorem
Quotient of a weakly contractible space by a free action
infinite-dimensional projective space R P ∞ {\displaystyle \mathbb {RP} ^{\infty }} (the direct limit of finite-dimensional projective spaces) is a classifying space for
Classifying_space
Concept in projective geometry
In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension
Correlation (projective geometry)
Correlation_(projective_geometry)
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
known as Hopf fibrations. First, one can replace the projective line by an n-dimensional projective space. Second, one can replace the complex numbers by any
Hopf_fibration
complex vector space C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . The projective model of the complex hyperbolic space is the projectivized space of all negative
Complex_hyperbolic_space
Tool to track locally defined data attached to the open sets of a topological space
{\displaystyle {\mathcal {F}}\to {\mathcal {G}}} is a sheaf, since projective limits commute with projective limits. On the other hand, the cokernel is not always
Sheaf_(mathematics)
Field of algebraic geometry
birational to a projective variety (Chow's lemma). So, for the purposes of birational classification, it is enough to work only with projective varieties,
Birational_geometry
Characteristic classes of vector bundles
1-dimensional projective spaces over many other fields. See Chern–Simons theory for more discussion. Given a complex vector bundle E over a topological space X,
Chern_class
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