Rational illustration

If and are two irreducible algebraic varieties or schemes , then a rational mapping is a function of an open subset of to . Similar to how mappings of varieties correspond to homomorphisms of the coordinate rings, rational mappings correspond to body homomorphisms of the functional fields of the varieties. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

Rational maps are needed to define birational equivalence , an important term for classifying varieties.

Definitions

Regular functions of algebraic varieties

In the following we assume an irreducible affine variety with a coordinate ring . The co-ordinate ring is a domain of integrity, designate its quotient field . The elements from are referred to as rational functions on . ${\displaystyle V}$ ${\displaystyle K[V]}$ ${\displaystyle K(V)}$ ${\displaystyle K(V)}$ ${\displaystyle V}$

Is and so is regular in called when exist with: ${\displaystyle f\in K(V)}$ ${\displaystyle x\in V}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle g,h\in K[V]}$

${\displaystyle h(x)\neq 0}$

${\displaystyle f={\frac {g}{h}}}$

Is , then the set of elements in which is regular is called the domain of , as . ${\displaystyle f\in K(V)}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \mathrm {dom} (f)}$

Rational mappings of varieties

${\displaystyle \mathbb {A} _{k}^{n}}$denote the n- dimensional affine space over a body k.

Be and varieties over one body . A rational mapping from to is a tuple ${\displaystyle V\subset \mathbb {A} _{k}^{l}}$ ${\displaystyle W\subset \mathbb {A} _{k}^{n}}$ ${\displaystyle k}$ ${\displaystyle V}$ ${\displaystyle W}$

${\displaystyle f=(f_{1},\dots ,f_{n})}$

with and for everyone ${\displaystyle f_{i}\in K(V)}$ ${\displaystyle (f_{1}(x),\dots ,f_{n}(x))\in W}$ ${\displaystyle x\in \bigcap _{i=1}^{n}\mathrm {dom} (f_{i})\subset V}$

The mapping is called in regular if all in are regular. The domain of is ${\displaystyle x\in V}$ ${\displaystyle f_{i}}$ ${\displaystyle x}$ ${\displaystyle f}$

${\displaystyle \mathrm {dom} (f)=\bigcap _{i=1}^{n}\mathrm {dom} (f_{i})}$

A rational mapping from to is therefore not entirely defined, but only on an open subset . ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle U\subset V}$

Therefore they are also noted with a dashed arrow:

${\displaystyle f:V{\text{⇢}}W}$

Dominant rational illustrations

Rational maps cannot always be chained together, as the following example shows:

${\displaystyle f\colon \mathbb {A} _{k}^{1}{\text{⇢}}\mathbb {A} _{k}^{2}}$

${\displaystyle f\colon x\mapsto (x,0)}$

${\displaystyle g\colon \mathbb {A} _{k}^{2}{\text{⇢}}\mathbb {A} _{k}^{1}}$

${\displaystyle g\colon (x,y)\mapsto ({\frac {x}{y}})}$ so ${\displaystyle \mathrm {dom} (g)=\{(x,y)\in \mathbb {A} _{k}^{2}|y\neq 0\}}$

because

${\displaystyle f(\mathbb {A} _{k}^{1})\cap \mathrm {dom} (g)=\emptyset }$

Concatenation, however, is always possible with dominant rational mappings:

A rational figure

${\displaystyle f\colon V{\text{⇢}}W}$

means dominant when there is one in dense crowd. ${\displaystyle f(\mathrm {dom} (f))}$ ${\displaystyle W}$

Birational illustrations

A birational figure

${\displaystyle \phi \colon X{\text{⇢}}Y}$

is a rational mapping to which there is a rational mapping

${\displaystyle \psi \colon Y{\text{⇢}}X}$

gives with

${\displaystyle \psi \circ \phi =id_{X}}$

and

${\displaystyle \phi \circ \psi =id_{Y}}$

The varieties are then called birational equivalent .

Relation to body homomorphisms

Be

${\displaystyle f\colon V{\text{⇢}}W}$

${\displaystyle f=(f_{1},\ldots ,f_{n})}$

a rational figure. be defined by the ideal . Because of ${\displaystyle W\subset \mathbb {A} _{k}^{l}}$ ${\displaystyle I}$

${\displaystyle f(x)\in W}$

applies to everyone ${\displaystyle h\in I}$

${\displaystyle h(f_{1}(x),\ldots ,f_{n}(x))=0}$

Is so

${\displaystyle h\in k[x_{1},\ldots ,x_{n}]}$ so ${\displaystyle {\bar {h}}\in K[W]=k[x_{1},\dots ,x_{n}]/I}$

so is well defined. A rational mapping therefore induces a mapping ${\displaystyle f^{*}({\bar {h}}):=h(f_{1},\ldots ,f_{n})}$ ${\displaystyle f}$

${\displaystyle f^{*}\colon k[W]\to K(V)}$

Is

${\displaystyle f^{*}(g)=0}$

so that's equivalent to

${\displaystyle f(\mathrm {dom} (f))\subset V(g)}$

If dominant, then in this case it must be because no function can vanish on a dense set. It therefore applies: ${\displaystyle f}$ ${\displaystyle g=0}$

${\displaystyle f^{*}\colon K[W]\to K[V]}$is injective is dominant. ${\displaystyle \Leftrightarrow f}$

In this case induces a linear body homomorphism ${\displaystyle f}$ ${\displaystyle k}$

${\displaystyle f^{*}\colon k(W)\to k(V)}$

Conversely, a linear body homomorphism can be applied to each ${\displaystyle k}$

${\displaystyle \phi \colon k(W)\to k(V)}$

a (thereby uniquely determined) dominant rational mapping

${\displaystyle f\colon V{\text{⇢}}W}$

find with

${\displaystyle \phi =f^{*}}$

It can even be shown that the star mapping is a contravariant functor that establishes an equivalence between certain categories. ${\displaystyle ^{*}}$

Generalizations

The above definition can be generalized to quasi-affine, quasi-projective and projective varieties through equivalence classes. Now be and affine, quasi-affine, quasi-projective or projective varieties. ${\displaystyle X}$ ${\displaystyle Y}$

Are open sets and and morphisms from and to . ${\displaystyle U,V\subset X}$ ${\displaystyle \phi _{U}}$ ${\displaystyle \phi _{V}}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle W}$

The equivalence relation is defined as follows: is equivalent to if and on match. ${\displaystyle (U,\phi _{U})}$ ${\displaystyle (V,\phi _{V})}$ ${\displaystyle \phi _{U}}$ ${\displaystyle \phi _{V}}$ ${\displaystyle U\cap V}$

A rational figure

${\displaystyle \phi \colon V{\text{⇢}}W}$

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

A rational mapping is said to be dominant if a (and therefore every) representative has a dense picture. ${\displaystyle (U,\phi _{U})}$

Examples

Neil's parable

Let be the Neil parabola , which is replaced by the polynomial ${\displaystyle V\subset \mathbb {A} _{k}^{2}}$

${\displaystyle y^{2}=x^{3}}$

is defined. The morphism

${\displaystyle \phi \colon A_{k}^{1}\to V}$

${\displaystyle x\mapsto (x^{2},x^{3})}$

is bijective , but not an isomorphism, since the inverse mapping is not a morphism. On lets through ${\displaystyle V}$

${\displaystyle \psi \colon (x,y)\mapsto ({\frac {y}{x}})}$

define a rational mapping with

${\displaystyle \mathrm {dom} (\psi )=V\setminus (0,0)}$

for which applies:

${\displaystyle \psi \circ \phi =id_{A_{k}^{1}}}$and . ${\displaystyle \phi \circ \psi =id_{V}}$

The two varieties are therefore birationally equivalent.

Projection in projective space

The projection

${\displaystyle p\colon \mathbb {P} _{k}^{n}{\text{⇢}}\mathbb {P} _{k}^{n-1}}$

${\displaystyle p\colon (a_{0}:\ldots :a_{n})\mapsto (a_{1}:\ldots :a_{n})}$

is a rational figure. For n> 1 it is only in the point

${\displaystyle (1:0:\ldots :0)}$

not regular.

If n = 1, the figure appears in the point

${\displaystyle (1:0)}$

not to be regular because by definition is

${\displaystyle p\colon (a_{0}:a_{1})\mapsto (a_{1}),}$

and

${\displaystyle (0)\notin P_{k}^{0}}$

But the illustration can be continued at this point, namely the illustration can also be written as

${\displaystyle p\colon (a_{0}:a_{1})\mapsto (1),}$

In general, every rational mapping from a smooth curve into projective space is a morphism.

literature

• Klaus Hulek: Elementary Algebraic Geometry . Vieweg, Braunschweig / Wiesbaden 2000, ISBN 3-528-03156-5 .
• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9