Birational geometry

Birational geometry

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, birational geometry is a part of the subject of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, that deals with the geometry of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

that is dependent only on its function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...

. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...

in the years 1890–1910. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension three.

Birational geometry is largely a geometry of transformations, but it doesn't fit exactly with the Erlangen programme. One reason is that its nature is to deal with transformations that are only defined on an open, dense subset of an algebraic variety. Such transformations, given by rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s in the co-ordinates, can be undefined not just at isolated points on curves, but on entire curves on a surface, and so on.

Birational mapping

A formal definition of birational mapping from one algebraic variety V to another is that it is a rational mapping
Rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.-Formal definition:...

with a rational inverse mapping. This has to be understood in the extended sense that the composition, in either order, is only in fact defined on a non-empty Zariski open
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...

subset.

One of the first results in the subject is the birational isomorphism of the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

, and a non-singular quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

Q in projective 3-space. Already in this example whole sets have ill-defined mappings: taking a point P on Q as origin, we can use lines through P, intersecting Q at one other point, to project to a plane — but this definition breaks down with all lines tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to Q at P, which in a certain sense 'blow up' P into the intersection of the tangent plane with the plane to which we project.

The Cremona group

That is, quite generally, birational mappings act like relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

s, with graphs containing parts that are not functional. On an open dense set they do behave like functions, but the Zariski closures of their graphs are more complex correspondences on the product showing 'blowing up' and 'blowing down'. Detailed descriptions of those, in terms of projective spaces associated to tangent spaces can be given and justified by the theory.

An example is the Cremona group
Cremona group
In mathematics, in birational geometry, the Cremona group of order n over a field k is the group of birational automorphisms of the n-dimensional projective space over k...

of birational automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

s of the projective plane. In purely algebraic terms, for a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

K, this is the automorphism group over K of the field K(XY) of rational functions in two variables. Its structure has been analysed since the nineteenth century, but it is 'large' (while the corresponding group for the projective line consists only of Möbius transformations determined by three parameters). It is still the subject of research.