In algebraic geometry, the homogeneous coordinate ring is a certain commutative ring assigned to any projective variety. If V is an algebraic variety...

] / 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...

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...

between homogeneous polynomials and projective varieties (cf. Homogeneous coordinate ring.) The above definitions have been generalized to rings graded...

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are...

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...

homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values. Some other common coordinate...

{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...

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...

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...

For a projective variety, there is an analogous ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they...

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...

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...

as the Hilbert polynomial of the homogeneous coordinate ring of V. Polynomial rings and their quotients by homogeneous ideals are typical graded algebras...

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...

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...

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...

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...

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...

} 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}...

commutative ring. This construction builds a projective algebraic variety together with a very ample line bundle whose homogeneous coordinate ring is the original...

Castelnuovo–Richmond–Igusa quartic Noether–Castelnouvo theorem Homogeneous coordinate ring Riemann–Roch theorem for surfaces Italian school of algebraic...

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...

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...

}=\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 )...

(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...

{\displaystyle [W]} . We now define a coordinate atlas. For any n × k {\displaystyle n\times k} homogeneous coordinate matrix W {\displaystyle W} , we can...

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...

vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast...

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...