] . {\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...

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...

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

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:...

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

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...

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...