Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.Let G be a unipotent group acting on an affine variety.Then every G-orbit in the variety is closed.…

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

Using regular functions from an affine variety to A 1, we can define regular maps from one affine variety to another.

This correspondence is the specificity of algebraic geometry among the other subareas of geometry.An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section.Next, one can define projective and quasi-projective varieties in a similar way.…

For that, it is enough to show: are affine varieties.For simplicity, we consider only the case i = 0.…

For example, it can be shown that the scheme obtained by contracting two disjoint projective lines in some P 3 to one is a proper, but non-projective variety.Affine varieties of non-zero dimension are never complete.More generally, it can be shown that affine proper morphisms are necessarily finite.…

Two affine varieties are birationally equivalent if there two rational functions between them which are inverse one to the other in the regions where both are defined.Equivalently, they are birationally equivalent if their function fields are isomorphic.…

Equivalently, they are birationally equivalent if their function fields are isomorphic.An affine variety is a rational variety if it is birationally equivalent to an affine space.This means that the variety admits a rational parameterization.…

Regular maps are, by definition, morphisms in the category of algebraic varieties.In particular, a regular map between affine varieties corresponds contravariantly in one-to-one to a ring homomorphism between the coordinate rings.A regular map whose inverse is also regular is called biregular, and are isomorphisms in the category of algebraic varieties.…

If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set.A Zariski open subspace of an affine variety is called a quasi-affine variety.…

Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets.

Basically, a variety over is a scheme whose structure sheaf is a sheaf of -algebras with the property that the rings R that occur above are all integral domains and are all finitely generated -algebras, that is to say, they are quotients of polynomial algebras by prime ideals.This definition works over any field, It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space.This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled.…

Like in the non graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.In a similar way one can define an affine algebraic supergroup as a group object in the category of super algebraic affine varieties.An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of super Polynomials.…

An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety).

More generally, given a commutative ring (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and subvarieties of the affine variety Spectrum of a ring.More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.Suppose and Y are arbitrary sets and a binary relation over and Y is given.…

A quasi-projective variety is a Zariski open subset of a projective variety.Notice that every affine variety is quasi-projective.Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a constructible set.…

In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials.To be explicit, suppose X and Y are subvarieties (or algebraic subsets) of A n resp.…

Contrarily to the preceding ones, this section concerns only varieties and not algebraic sets.On the other hand the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.…

Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces.An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring.In complex geometry, an affine variety is an analog of a Stein manifold.…

As affine spaces can be embedded in projective spaces, all affine varieties can be embedded in projective spaces too.Any choice of a finite system of nonsimultaneously vanishing global sections of a globally generated line bundle defines a morphism to a projective space.…