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''birational isomorphism''

A rational mapping between algebraic varieties inducing an isomorphism of their fields of rational functions. In a more general setting, a rational mapping of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165001.png" /> is said to be a birational mapping if it satisfies one of the following equivalent conditions: 1) there exist dense open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165003.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165004.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165005.png" /> and realizes an isomorphism of subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165006.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165008.png" /> are the sets of generic points of the irreducible components of the schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650010.png" /> respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650011.png" /> induces a bijective correspondence between the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650012.png" /> and an isomorphism of local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650013.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650014.png" />.

If the schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650016.png" /> are irreducible and reduced, the local rings of their generic points become identical with the fields of rational functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650018.png" />, respectively. In such a case the birational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650019.png" /> induces, in accordance with condition 2), an isomorphism of the fields of rational functions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650020.png" />.

Two schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650022.png" /> are said to be birationally equivalent or birationally isomorphic if a birational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650023.png" /> exists. A [[Birational morphism|birational morphism]] is a special case of a birational mapping.

The simplest birational mapping is a [[Monoidal transformation|monoidal transformation]] with a non-singular centre. For smooth complete varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650024.png" /> any birational mapping may be represented as the composite of such transformations and their inverses. At the time of writing (1986) this question remains open in the general case.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

[[Category:Algebraic geometry]]