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In algebraic geometry, the homogeneous coordinate ring is a certain commutative ring assigned to any projective variety. If V is an algebraic variety
Homogeneous_coordinate_ring
Universal homogenous coordinate ring of a projective variety
In algebraic geometry, a Cox ring (or total coordinate ring) is a sort of universal homogeneous coordinate ring for a projective variety, and is (roughly
Cox_ring
Algebraic variety in a projective space
] / I {\displaystyle k[x_{0},\ldots ,x_{n}]/I} is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension
Projective_variety
Type of algebraic structure
is the close relationship between homogeneous polynomials and projective varieties (cf. Homogeneous coordinate ring). Another example of a graded algebra
Graded_ring
Coordinate system used in projective geometry
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are
Homogeneous_coordinates
Method for specifying point positions
or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems
Coordinate_system
Branch of algebra
For a projective variety, there is an analogous ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they
Ring_theory
transform. The goal of the theory is to prove results on the homogeneous coordinate ring of the embedded abelian variety A, that is, set in a projective
Equations defining abelian varieties
Equations_defining_abelian_varieties
{P} ^{n}} be a projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If H = { f = 0 } ⊂ P n {\displaystyle H=\{f=0\}\subset
Scheme-theoretic_intersection
Branch of mathematics
set, whose homogeneous coordinate ring is an integral domain, the projective coordinates ring being defined as the quotient of the graded ring or the polynomials
Algebraic_geometry
example Nagata's compactification theorem. Cox ring A generalization of a homogeneous coordinate ring. See Cox ring. crepant A crepant morphism f : X → Y {\displaystyle
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
such embedding of C and the minimal free resolution for its homogeneous coordinate ring, for the minimum index i for which βi, i + 1 is zero, then the
Gonality of an algebraic curve
Gonality_of_an_algebraic_curve
Concept in mathematics
function on X is of the form g/h for some homogeneous elements g, h of the same degree in the homogeneous coordinate ring k [ X ¯ ] {\displaystyle k[{\overline
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Concept in algebraic geometry
curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated
Canonical_bundle
Combinatorial object in representation theory
for invariant theory, starting from the work of Hodge on the homogeneous coordinate ring of the Grassmannian and further explored by Gian-Carlo Rota with
Young_tableau
Tool in mathematical dimension theory
as the Hilbert polynomial of the homogeneous coordinate ring of V. Polynomial rings and their quotients by homogeneous ideals are typical graded algebras
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
invariant a(C) in terms of the minimal free resolution of the homogeneous coordinate ring of C in its canonical embedding, as the largest index i for which
Clifford's theorem on special divisors
Clifford's_theorem_on_special_divisors
Mathematical element
integral over R. The integral closure of the homogeneous coordinate ring of a normal projective variety X is the ring of sections ⨁ n ≥ 0 H 0 ( X , O X ( n
Integral_element
Algebraic structure
mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by
Ring_of_polynomial_functions
Branch of mathematics
commutative ring. This construction builds a projective algebraic variety together with a very ample line bundle whose homogeneous coordinate ring is the original
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Space formed by the ''n''-tuples of real numbers
In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or R n {\displaystyle \mathbb {R} ^{n}} , is the set
Real_coordinate_space
Generalization of a vector bundle
(whence the terminology). If X = Spec k is a point and R is a homogeneous coordinate ring, then the affine cone of R is the (usual) affine cone over the
Cone_(algebraic_geometry)
vector space m/m2 over the residue field. Cox ring A Cox ring is a sort of universal homogeneous coordinate ring for a projective variety. decomposable A module
Glossary of commutative algebra
Glossary_of_commutative_algebra
Italian mathematician (1865–1952)
Castelnuovo–Richmond–Igusa quartic Noether–Castelnouvo theorem Homogeneous coordinate ring Riemann–Roch theorem for surfaces Italian school of algebraic
Guido_Castelnuovo
Mathematical object studied in the field of algebraic geometry
all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this
Algebraic_variety
Generalization of algebraic variety
fundamental idea that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and
Scheme_(mathematics)
Dwarf planet with a ring and two moons
further studies of the visible and near infrared spectra suggest a homogeneous surface covered by an intimate 1:1 mixture of amorphous and crystalline
Haumea
Topics referred to by the same term
(AKA 1-normal), a property in algebraic geometry related to homogeneous coordinate rings Linearly unique polytope (AKA linearly stable polytope), a centrally
Linear_(disambiguation)
Typically linear operator defined in terms of differentiation of functions
}=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}.} The highest homogeneous component of the symbol, namely, σ ( x , ξ ) = ∑ | α | = m a α ( x )
Differential_operator
Topology on prime ideals and algebraic varieties
Equivalently, it can be checked that: The elements of the affine coordinate ring A ( X ) = k [ x 1 , … , x n ] / I ( X ) {\displaystyle A(X)=k[x_{1}
Zariski_topology
Mathematical space
W ] {\displaystyle [W]} . A coordinate atlas ensures that for any n × k {\displaystyle n\times k} homogeneous coordinate matrix W {\displaystyle W} ,
Grassmannian
Short exact sequence of sheaves on projective space
uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescaling, or "independent of the radial coordinate". A function
Euler_sequence
For a ring R and ideal I, multiplication in gr I R {\displaystyle \operatorname {gr} _{I}R} is defined as follows: First, consider homogeneous elements
Associated_graded_ring
"Smallest" commutative algebra that contains a vector space
polynomial ring K[B], where the elements of B are considered as indeterminates. Therefore, the symmetric algebra over V can be viewed as a "coordinate free"
Symmetric_algebra
Euclidean space without distance and angles
surjective. Examples of n-coordinate system in an (n−1)-dimensional space are barycentric coordinates and affine "homogeneous" coordinates (1, x1, … , xn−1)
Affine_space
Algebraic structure in linear algebra
vector spaces, whose study belongs to functional analysis. Systems of homogeneous linear equations are closely tied to vector spaces. For example, the
Vector_space
products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex Grassmannians. Seshadri (1978) initiated a program
Standard_monomial_theory
Class of mathematical expression
defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using
Division_by_zero
Affine space over the complex numbers
affine functions on A defines a ring of functions, called the affine coordinate ring in algebraic geometry. This ring carries a filtration, by degree
Complex_affine_space
(2011-12-23). "Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings". Journal of Algebra. 322 (7): 2508–2527. arXiv:0812.0609. doi:10
Sklyanin_algebra
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Property of a space in which the local dimensionality is the same everywhere
not equidimensional. An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.[clarification needed] A differential
Equidimensionality
Number used in algebraic geometry
the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may
Degree of an algebraic variety
Degree_of_an_algebraic_variety
Construction in projective geometry
geometries; in such contexts, the coordinate components always obey the structure of a PTR. By contrast, the homogeneous coordinates typically used in projective
Planar_ternary_ring
Type of integral domain
Q[x, y]/(x2 + y2 − 1) is not a UFD, but the ring Q(i)[x, y]/(x2 + y2 − 1) is. Similarly the coordinate ring R[X, Y, Z]/(X2 + Y2 + Z2 − 1) of the 2-dimensional
Unique_factorization_domain
Overview of and topical guide to geometry
Tropical geometry Chirality Handedness Relative direction Mirror image Coordinate-free treatment Four-dimensional space Infinitesimal transformation Geometric
Outline_of_geometry
Graded vector space with applications to theoretical physics
first p {\displaystyle p} coordinate basis vectors and the odd space is spanned by the last q {\displaystyle q} . A homogeneous subspace of a super vector
Super_vector_space
Result of commutative algebra
geometric interpretation. Suppose A is the coordinate ring of an affine variety X, and consider S as the coordinate ring of a d-dimensional affine space A k
Noether_normalization_lemma
Mathematical study of invariants under symmetries
follows. The ring R is a polynomial ring so is graded by degrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of
Invariant_theory
Mathematical concept
the homogeneous polynomials of positive degree: ⨁ n > 0 S n . {\displaystyle \bigoplus _{n>0}S_{n}.} Define Proj S to be the set of all homogeneous prime
Complex_projective_space
In mathematics, invariant of square matrices
(finite) rank n {\displaystyle n} over a commutative ring R {\displaystyle R} can be formulated in a coordinate-free manner by considering the n {\displaystyle
Determinant
Chemical compound
organoiridium compound with the formula [C8H12IrP(C6H11)3C5H5N]PF6. It is a homogeneous catalyst for hydrogenation and hydrogen-transfer reactions, developed
Crabtree's_catalyst
Organic reaction involving the breakup and reassembly of alkene double bonds
investigated for small-scale reactions or in academic research. The homogeneous catalysts are often classified as Schrock catalysts and Grubbs catalysts
Olefin_metathesis
Type of topological space
CW complex with 1 cell in every dimension. In homogeneous coordinates (x1 ... xn+1) on Sn, the coordinate neighborhood U1 = {(x1 ... xn+1) | x1 ≠ 0} can
Real_projective_space
Measure of a mathematical object studied in the field of algebraic geometry
algebraic set defined as the set of the common zeros of a homogeneous ideal I in a polynomial ring R = K [ x 0 , x 1 , … , x n ] {\displaystyle R=K[x_{0}
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Number of intersection points of algebraic curves and hypersurfaces
two point have the same Cartesian x-coordinate. The resultant R(x ,t) of P and Q with respect to y is a homogeneous polynomial in x and t that has the
Bézout's_theorem
Concept in projective geometry
underlying vector space (with a companion antiautomorphism) and conversely. Homogeneous coordinates may be used to give an algebraic description of dualities
Duality_(projective_geometry)
scalar multiples. The non-zero scalar multiples, written as coordinate triples, are the homogeneous coordinates of the given point, called point coordinates
Incidence_(geometry)
Geometric concept of a 2D space with "points at infinity" adjoined
construct a coordinate "ring"—a so-called planar ternary ring (not a genuine ring)—corresponding to any projective plane. A planar ternary ring need not
Projective_plane
Family of linear transformations
transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity
Lorentz_transformation
Fourteenth letter in the Greek alphabet
A parameter denoted as warped time used to derive the equations for homogeneous azeotropic distillation State Price Density in mathematical finance The
Xi_(letter)
Type of geometry
dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems. In a foundational sense, projective geometry and ordered
Projective_geometry
Mn(R) over another von Neumann regular ring R. Here a complemented modular lattice has order n if it has a homogeneous basis of n elements, where a basis
Continuous_geometry
Concept in physics and mathematics
below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean
Galilean_transformation
Condition in which spacetime itself breaks down
A.K. (1992). "The oscillatory mode of approach to a singularity in homogeneous cosmological models with rotating axes". Perspectives in Theoretical
Gravitational_singularity
Matrix representing a Euclidean rotation
counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates
Rotation_matrix
inclusions in a split-ring resonator designed as an anisotropic metamaterial. The configuration can be viewed as alternating layers of homogeneous isotropic dielectric
Seismic_metamaterial
Algebraic object with geometric applications
often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional
Tensor
varieties in arbitrary characteristic. Brion, Michel (2007). "The total coordinate ring of a wonderful variety". Journal of Algebra. 313 (1): 61–99. arXiv:math/0603157
Spherical_variety
Generalization of the tangent space to a manifold to the case of certain spaces
polynomial f, let in ( f ) {\displaystyle \operatorname {in} (f)} be the homogeneous component of f of the lowest degree, the initial term of f, and let in
Tangent_cone
Motion of a certain space that preserves at least one point
more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis)
Rotation_(mathematics)
Metric based on the exact solution of Einstein's field equations of general relativity
metric (FLRW; /ˈfriːdmən ləˈmɛtrə ... /) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting, oscillating or constant)
Friedmann–Lemaître–Robertson–Walker metric
Friedmann–Lemaître–Robertson–Walker_metric
kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all G a {\displaystyle \mathbb {G} _{a}}
Locally_nilpotent_derivation
Geometry theorem
the parallelity A c ∥ C a {\displaystyle \;Ac\parallel Ca\;} . Choose homogeneous coordinates with C = ( 1 , 0 , 0 ) , c = ( 0 , 1 , 0 ) , X = ( 0 , 0
Pappus's_hexagon_theorem
Vector-valued function of multiple vectors, linear in each argument
of its arguments is zero. Algebraic form Multilinear form Homogeneous polynomial Homogeneous function Tensors Lang, Serge (2005) [2002]. "XIII. Matrices
Multilinear_map
Solution of Einstein field equations
tensor contains two terms: the first representing the matter density of a homogeneous distribution of swirling dust particles (see Dust solution), and the
Gödel_metric
Transformations induced by a mathematical group
group, which is ( n − 1 ) {\displaystyle (n-1)} -homogeneous (since it is transitive, and thus 1-homogeneous) without being ( n − 1 ) {\displaystyle (n-1)}
Group_action
Curve defined as zeros of polynomials
projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be
Algebraic_curve
Polynomial with all terms of degree two
polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y − 3 y 2 {\displaystyle 4x^{2}+2xy-3y^{2}}
Quadratic_form
Four-dimensional number system
quaternions, called rotors, can be very useful for applications involving homogeneous coordinates. But it is only in 3D that the number of basis bivectors
Quaternion
General relativity model near spacetime singularities
equations describing the asymptotics come from a class of spatially homogeneous solutions which constitute the Mixmaster dynamics: a complicated oscillatory
BKL_singularity
Operation in differential geometry
where ( z k + 1 ) {\displaystyle (z^{k+1})} is the ideal generated by homogeneous polynomials of order ≥ k + 1. We now move to the composition of jets
Jet_(mathematics)
Curve from a cone intersecting a plane
atrix}x\\y\\1\end{pmatrix}}=0.} This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below)
Conic_section
Vector spaces associated to a matrix
of real numbers. The row and column spaces are subspaces of the real coordinate spaces R n {\displaystyle \mathbb {R} ^{n}} and R m {\displaystyle \mathbb
Row_and_column_spaces
Ion or molecule bound to a metal atom
used in homogeneous catalysis, such as asymmetric hydrogenation. Hemilabile ligands contain at least two electronically different coordinating groups and
Ligand
Simple curve of Euclidean geometry
inspire the development of geometry, astronomy and calculus. Annulus: a ring-shaped object, the region bounded by two concentric circles. Arc: any connected
Circle
Branch of mathematics
is an element of the preimage of v by T. Let (S′) be the associated homogeneous system, where the right-hand sides of the equations are put to zero:
Linear_algebra
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform, P ( t ) = [ T ( t ) ] p = { P 1 } = [ A ( t ) d ( t ) 0 1
Screw_theory
Type of mathematical model
more than one stationary homogeneous solution, a typical solution is given by travelling fronts connecting the homogeneous states. These solutions move
Reaction–diffusion_system
Ligand with unclear oxidation state
redox non-innocent ligands are also used as controlling factors to steer homogeneous catalysis. : Porphyrin ligands can be innocent (−2) or noninnocent (−1)
Non-innocent_ligand
Hypothetical phenomenon
(OSD) model illustrates the collapse of a spherical cloud composed of homogeneous dust (pressureless matter). In this scenario, all the matter converges
Naked_singularity
Mathematical construct in computer algebra
basis is a particular kind of generating set of an ideal in a polynomial ring K [ x 1 , … , x n ] {\displaystyle K[x_{1},\ldots ,x_{n}]} over a field K
Gröbner_basis
About polynomials in several variables
the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form F = ( X 1 + H 1 , … , X n + H n ) {\displaystyle
Jacobian_conjecture
Supergeometric generalization of a manifold
superalgebras and supergroups. Alongside the standard locally ringed-space formulation, more concrete coordinate-based formalisms are also used, especially in the
Supermanifold
Algebra associated to any vector space
themselves blades, but linear combinations of blades; a sum of blades of homogeneous degree k {\displaystyle k} is called a k-vector, while a more general
Exterior_algebra
Group in group theory and physics
commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting
Heisenberg_group
bond. Iminophosphoranes have found diverse applications as ligands for homogeneous catalysis (i.e. cross coupling, polymerization, etc.), superbasic or
Iminophosphorane
Family of chemical compounds
"Steric effects of phosphorus ligands in organometallic chemistry and homogeneous catalysis". Chemical Reviews. 77 (3): 313–348. doi:10.1021/cr60307a002
Cyclic_alkyl_amino_carbenes
result in different coordination geometries and catalytic behavior in homogeneous catalysts. Many widely used diphosphine ligands have the general formula
Diphosphine_ligands
Class of compact connected topological spaces
three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of an abelian compact
Solenoid_(mathematics)
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HOMOGENEOUS COORDINATE-RING
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