affine variety

Let G be a unipotent group acting on an affine variety.

We shall show it is locally an affine variety.

Many properties of the affine varieties depend on their behaviour "at infinity".

Since , a morphism between affine varieties in general would have this form.

In fact, a projective subvariety of an affine variety must have dimension zero.

In complex geometry, an affine variety is an analog of a Stein manifold.

An affine variety is almost never projective.

This theorem applies in particular to any smooth affine variety of dimension .

Notice that every affine variety is quasi-projective.

Affine varieties of non-zero dimension are never proper.

The following theorem characterizes the category of affine varieties:

If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition.

A Zariski open subvariety of an affine variety is called a quasi-affine variety.

Let be the affine variety.

The affine varieties is a subcategory of the category of the algebraic sets.

Varieties isomorphic to open subsets of affine varieties are called quasi-affine.

The normalization of an affine variety is affine.

The conclusion is that any affine variety is a branched covering of affine space.

In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials.

An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section.

See affine varieties.

More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.

In algebraic geometry, an affine variety is the zero set of a polynomial, or collection of polynomials.

This establishes that the above equation, clearly a generalization of the previous one, defines the Zariski topology on any affine variety.

The analogous result in algebraic geometry, due to applies to vector bundles in the category of affine varieties.