Editing Birational geometry (section)

==Birational maps==

=== Rational maps ===
A [[rational mapping|rational map]] from one variety (understood to be [[Irreducible component|irreducible]]) <math>X</math> to another variety <math>Y</math>, written as a dashed arrow {{nowrap|''X'' {{font|size=145%|⇢}}''Y''}}, is defined as a [[algebraic geometry#Morphism of affine varieties|morphism]] from a nonempty open subset <math>U \subset X</math> to <math>Y</math>. By definition of the [[Zariski topology]] used in algebraic geometry, a nonempty open subset <math>U</math> is always dense in <math>X</math>, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.

=== Birational maps ===
A '''birational map''' from ''X'' to ''Y'' is a rational map {{nowrap|''f'' : ''X'' ⇢ ''Y''}} such that there is a rational map {{nowrap|''Y'' ⇢ ''X''}} inverse to ''f''. A birational map induces an isomorphism from a nonempty open subset of ''X'' to a nonempty open subset of ''Y'', and vice versa: an isomorphism between nonempty open subsets of ''X'', ''Y'' by definition gives a birational map {{nowrap|''f'' : ''X'' ⇢ ''Y''}}. In this case, ''X'' and ''Y'' are said to be '''birational''', or '''birationally equivalent'''. In algebraic terms, two varieties over a field ''k'' are birational if and only if their [[Function field of an algebraic variety|function fields]] are isomorphic as extension fields of ''k''.

A special case is a '''birational morphism''' {{nowrap|''f'' : ''X'' → ''Y''}}, meaning a morphism which is birational. That is, ''f'' is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of ''X'' to points in ''Y''.

=== Birational equivalence and rationality ===
A variety ''X'' is said to be '''[[Rational variety|rational]]''' if it is birational to [[affine space]] (or equivalently, to [[projective space]]) of some dimension. Rationality is a very natural property: it means that ''X'' minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.

==== Birational equivalence of a plane conic ====
For example, the circle <math>X</math> with equation <math>x^2 + y^2 - 1 = 0</math> in the affine plane is a rational curve, because there is a rational map {{nowrap|''f'' : <math>\mathbb{A}^1</math> ⇢ ''X''}} given by
:<math>f(t) = \left( \frac{2t}{1+t^2}, \frac{1 - t^2}{1 + t^2}\right),</math>
which has a rational inverse ''g'': ''X'' ⇢ <math>\mathbb{A}^1</math> given by
:<math>g(x,y) = \frac{1-y}{x}.</math>
Applying the map ''f'' with ''t'' a [[rational number]] gives a systematic construction of [[Pythagorean triple]]s.

The rational map <math>f</math> is not defined on the locus where <math>1 + t^2 = 0</math>. So, on the complex affine line <math>\mathbb{A}^1_{\Complex}</math>, <math>f</math> is a morphism on the open subset <math>U = \mathbb{A}^1_{\Complex}-\{i, -i\}</math>, <math>f: U \to X</math>. Likewise, the rational map {{nowrap|''g'' : ''X'' ⇢ <math>\mathbb{A}^1</math>}} is not defined at the point (0,−1) in <math>X</math>.

==== Birational equivalence of smooth quadrics and P<sup>n</sup> ====
More generally, a smooth [[quadric (algebraic geometry)|quadric]] (degree 2) hypersurface ''X'' of any dimension ''n'' is rational, by [[stereographic projection]]. (For ''X'' a quadric over a field ''k'', ''X'' must be assumed to have a [[Rational point#Rational or K-rational points on algebraic varieties|''k''-rational point]]; this is automatic if ''k'' is algebraically closed.) To define stereographic projection, let ''p'' be a point in ''X''. Then a birational map from ''X'' to the projective space <math>\mathbb{P}^n</math> of lines through ''p'' is given by sending a point ''q'' in ''X'' to the line through ''p'' and ''q''. This is a birational equivalence but not an isomorphism of varieties, because it fails to be defined where {{nowrap|1=''q'' = ''p''}} (and the inverse map fails to be defined at those lines through ''p'' which are contained in ''X'').

===== Birational equivalence of quadric surface =====
The [[Segre embedding]] gives an embedding <math>\mathbb{P}^1\times\mathbb{P}^1 \to \mathbb{P}^3</math> given by
:<math>([x,y],[z,w]) \mapsto [xz,xw,yz,yw].</math>
The image is the quadric surface <math>x_0x_3=x_1x_2</math> in <math>\mathbb{P}^3</math>. That gives another proof that this quadric surface is rational, since <math>\mathbb{P}^1\times\mathbb{P}^1</math> is obviously rational, having an open subset isomorphic to <math>\mathbb{A}^2</math>.