Rational mapping

Are two irreducible algebraic varieties and schemes or so is a rational map is a function from an open subset of after. Similar to match pictures of varieties of homomorphisms coordinate rings corresponding rational pictures Körperhomomorphismen the function body of the varieties.

Rational pictures are required to define the birational equivalence, an important concept for the classification of varieties.

• 4.1 Neilsche parabola
• 4.2 projection in projective space

Definitions

Regular functions of algebraic varieties

Below is an irreducible affine Variätet with coordinate ring. The coordinate ring is an integral domain, denote its quotient field. The elements are as rational functions on designated.

If and so is regularly called in if there are with:

, So the amount of the elements in which is regular when the domain of called.

Rational pictures of varieties

K designates the n-dimensional space over a body affinity

Be and varieties over a field. A rational map from to is a tuple

With and for all

The mapping is called in regular if all are in regular. The domain of

A rational map from to is thus not defined at all, but only on an open subset.

Therefore, they are also listed with a dashed arrow:

Dominant rational pictures

Rational illustrations may not always be chained together, as the following example shows:

Because

A concatenation is possible, however, always with dominant rational pictures:

A rational map

Is called dominant, if one is in dense crowd.

Birational pictures

A birational mapping

Is a rational map to which it is a rational map

Are with

And

The varieties are then referred to as birationally equivalent.

Connection with Körperhomomorphismen

Be

A rational map. is defined by the ideal. because of

Applies to all

So is

As is well-defined. Therefore, a rational map induces a map

Is

Thus, the equivalent to

Is dominant, it must be in this case, since no function on a dense set can disappear. It is therefore:

In this case, induces a linear Körperhomomorphismus

Conversely, to every - linear Körperhomomorphismus

A (this single) dominant rational map

Find with

It can be shown even that the star image is a contravariant functor establishing an equivalence between certain categories.

Generalizations

The above definition can be generalized to quasi- affine, and projective varieties quasiprojektive by equivalence classes. Be now and affine, quasi- affine, or projective varieties quasiprojektive.

Are open sets and and morphisms of or after.

The equivalence relation is defined as follows: is equivalent to if and to match.

A rational map

Is now an equivalence class with respect to this equivalence relation.

A rational map is called dominant if one (and hence any ) representative of a dense picture.

Examples

Neilsche parabola

Be the Neilsche parabola by the polynomial

Is defined. The morphism

Is bijective, but no isomorphism, since the inverse map is not a morphism. In can be extended by

A rational map defined by

In which:

The two varieties are birationally equivalent, therefore.

Projection in projective space

The projection

Is a rational map. It is, for n> 1 only at the point

Not regular.

If n = 1, it seems the picture at the point

Not to be regular, because by definition

And

But the picture can continue on this point, the picture may be written namely as

In general, any rational map from a smooth curve in a projective space is a morphism.