] . {\displaystyle k[X_{1},\ldots ,X_{n}].} An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by...

quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as varieties. Varieties isomorphic to affine algebraic...

irreducible affine algebraic set is also called an affine variety.: 3  (Some authors use the phrase affine variety to refer to any affine algebraic set...

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...

an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space:...

of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the...

Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety;...

It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse...

algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety...

I\right\}.} Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation...

}}K{\text{-algebras}}),} and, restricting to affine varieties (i.e. irreducible affine algebraic sets), Γ : ( affine algebraic varieties ) o p p → ( integral finitely...

algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull...

of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles...

Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of abstract...

mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general...

containing p elements. Affine n-space over a field F is denoted by An(F). Results and definitions stated below, for affine varieties, can be translated to...

asserts, for an algebraic variety X, the existence of a surjection with affine fibers from an affine variety W to X. The variety W is therefore homotopy-equivalent...

algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective...

morphism of affine varieties X → A k d {\displaystyle X\to \mathbb {A} _{k}^{d}} : that is, any affine variety is a branched covering of affine space. When...

The affine cipher is a type of monoalphabetic substitution cipher, where each letter in an alphabet is mapped to its numeric equivalent, encrypted using...

be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at...

or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective...

algebraic variety. Various complications arise. For example, if X is an affine variety, then one can try to construct X/G as Spec of the ring of invariants...

properties from Euclidean space are no longer relevant. For example, in affine space, there is no concept of distance, so there are no reflections or motions...

homogeneous, one can consider Y r either as an affine variety in mn-dimensional affine space, or as a projective variety in (mn − 1)-dimensional projective space...

polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space. André Weil saw that geometric reasoning...

of affine varieties X → A k d {\displaystyle X\to \mathbb {A} _{k}^{d}} . The conclusion is that any affine variety is a branched covering of affine space...

allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Suppose X {\displaystyle X} is...

"regular ring". An affine algebraic variety corresponds to a prime ideal in a polynomial ring, and the points of such an affine variety correspond to the...

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance...