Birational equivalence

One goal of algebraic geometry is to classify varieties down to isomorphism . In general, this is too difficult a problem. With the weaker concept of birational equivalence , on the other hand, there are better possibilities for classification. Two varieties and are called birational equivalent if they contain isomorphic dense open subsets. ${\displaystyle X}$ ${\displaystyle Y}$

Definitions

If and are varieties, they are called birational equivalent if there are rational mappings ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle \phi \colon X\ -\to Y}$

${\displaystyle \psi \colon Y\ -\to X}$

gives with

${\displaystyle \psi \circ \phi ={\text{id}}_{X}}$

and

${\displaystyle \phi \circ \psi ={\text{id}}_{Y}}$

The varieties can be affine, quasi-affine, projective, quasi-projective or abstract varieties .

${\displaystyle \phi }$and in this case they are called birational maps . ${\displaystyle \psi }$

It is for varieties and equivalent: ${\displaystyle X}$ ${\displaystyle Y}$

• ${\displaystyle X}$and are birationally equivalent. ${\displaystyle Y}$
• ${\displaystyle X}$and have isomorphic dense open sets. ${\displaystyle Y}$
• There are in and points with isomorphic local rings. ${\displaystyle X}$ ${\displaystyle Y}$
• ${\displaystyle X}$and have isomorphic function fields . ${\displaystyle Y}$

A birational morphism is a morphism of algebraic varieties that is also a birational mapping.

Rational varieties

A variety that is birationally equivalent to a projective space is called rational. For example, an irreducible cubic curve is rational if and only if it is singular. Examples are the Neil parabola or the Newtonian knot .

Examples

• The inflation of a point or, more generally, a closed sub- variety , is birationally equivalent to the initial variety .
• Each variety is birationally equivalent to a hypersurface.
• Every curve is birationally equivalent to a plane curve that has only very simple singularities ( colons ).
• Every variety over a field of characteristic 0 is a non- singular variety with an actual birational morphism . (This is called a resolution of the singularities.) This is a deep sentence by Heisuke Hironaka ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \phi \colon Y\to X}$
• It is easier to show that for every curve there is a clearly defined, non-singular curve with an actual birational morphism ${\displaystyle C}$ ${\displaystyle C'}$ ${\displaystyle \phi \colon C'\to C.}$
• A birational mapping from to is called the Cremona Transformation . One example is the quadratic transformation ${\displaystyle P_{k}^{2}}$ ${\displaystyle P_{k}^{2}}$

${\displaystyle \phi \colon P_{k}^{2}-\to P_{k}^{2}}$

${\displaystyle \phi \colon (a_{0}:a_{1}:a_{2})\mapsto (a_{1}a_{2}:a_{0}a_{1}:a_{1}:a_{0}a_{1})}$

This mapping is defined everywhere except for the points (1: 0: 0), (0: ​​1: 0) and (0: 0: 1). The image of the straight line connecting these points is a point, outside the straight line the image is an isomorphism. This mapping is self-inverse, so ${\displaystyle \phi ^{2}=id}$

classification

The classification program, the classification of varieties, is a master program of algebraic geometry. It can be broken down into several tasks. The first part is the classification down to birational equivalence. That means classifying the finitely generated extension bodies of the basic body down to isomorphism. The next step is then to find a good subset like the non-singular varieties within a birational equivalence class and then to classify them down to isomorphism. The third part is then to determine how far a general variety is from the good.

The program is well implemented for the algebraic curves. There is a birational invariant, gender. Gender is a natural number, and any natural number is assumed to be the gender of a curve. For g = 0 there is exactly one birational equivalence class, that of the rational curves. For every g> 0 there is a continuous family of birational equivalence classes that are parameterized by an irreducible algebraic variety. This variety has dimension 1 if g equals 1 (these are elliptic curves) and dimension 3g-3 for g> 1. Part one is solved for curves: An equivalence class of an algebraic curve is determined by a natural number, the gender (a discrete invariant), and then a point on a variety (a continuous invariant). The second part has a simple solution: In every birational equivalence class there is exactly one non-singular curve. And for the third part it should be added that a finite number of points must be adjoint to every curve in order to arrive at a non-singular curve.

Birational invariants

A birational invariant is an invariant that does not change under birational mappings. The simplest birational invariant is of course the dimension.

The arithmetic gender is a birational invariant of curves, surfaces and non-singular varieties over algebraically closed fields. The geometric gender is a birational invariant of non-singular projective varieties. In particular, irrational varieties of any dimension can easily be found in this way.

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