Homogeneous coordinate ring information

In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is...

In algebraic geometry, a Cox ring is a sort of universal homogeneous coordinate ring for a projective variety, and is (roughly speaking) a direct sum...

multiplication may not be commutative. For the general ring A, a projective line over A can be defined with homogeneous factors acting on the left and the projective...

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

] / 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...

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

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

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...

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...

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...

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...

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...

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

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...

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...

}=\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 )...

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...

vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which...

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...

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}...

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...

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...

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...

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