Editing Birational geometry (section)

=== 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''.