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{{short description|Algebraic variety}}
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In [[mathematics]], a '''rational variety''' is an [[algebraic variety]], over a given [[field (mathematics)|field]] ''K'', which is [[birationally equivalent]] to a [[projective space]] of some dimension over ''K''. This means that its [[function field of an algebraic variety|function field]] is isomorphic to
:<math>K(U_1, \dots , U_d),</math>
the field of all [[rational function]]s for some set <math>\{U_1, \dots, U_d\}</math> of [[Indeterminate (variable)|indeterminate]]s, where ''d'' is the [[dimension of an algebraic variety|dimension]] of the variety.

==Rationality and parameterization==

Let ''V'' be an [[affine algebraic variety]] of dimension ''d'' defined by a prime ideal ''I''&nbsp;=&nbsp;⟨''f''<sub>1</sub>, ..., ''f''<sub>''k''</sub>⟩  in <math>K[X_1, \dots , X_n]</math>. If ''V'' is rational, then there are ''n''&nbsp;+&nbsp;1  polynomials ''g''<sub>0</sub>, ..., ''g''<sub>''n''</sub> in <math>K(U_1, \dots , U_d)</math> such that <math>f_i(g_1/g_0, \ldots, g_n/g_0)=0. </math> In other words, we have a '''{{vanchor|rational parameterization}}''' <math>x_i=\frac{g_i}{g_0}(u_1,\ldots,u_d)</math> of the variety.

Conversely, such a rational parameterization induces a [[field homomorphism]] of the field of functions of ''V'' into <math>K(U_1, \dots , U_d)</math>. But this homomorphism is not necessarily [[onto]]. If such a parameterization exists, the variety is said to be [[#Unirationality|unirational]]. Lüroth's theorem (see below) implies that unirational curves are rational.  [[Castelnuovo's theorem]] implies also that, in characteristic zero, every unirational surface is rational.

==Rationality questions==
A '''rationality question''' asks whether a given [[field extension]] is ''rational'', in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as [[purely transcendental]]. More precisely, the rationality question for the [[field extension]] <math>K \subset L</math> is this: is <math>L</math> [[isomorphic]] to a [[rational function field]] over <math>K</math> in the number of indeterminates given by the [[transcendence degree]]?

There are several different variations of this question, arising from the way in which the fields <math>K</math> and <math>L</math> are constructed.

For example, let <math>K</math> be a field, and let

:<math>\{y_1, \dots, y_n \}</math>

be indeterminates over ''K'' and let ''L'' be the field generated over ''K'' by them.  Consider a [[finite group]] <math>G</math> permuting those [[Indeterminate (variable)|indeterminates]] over ''K''.  By standard [[Galois theory]], the set of [[Fixed point (mathematics)|fixed points]] of this [[Group action (mathematics)|group action]] is a [[Field extension|subfield]] of <math>L</math>, typically denoted <math>L^G</math>. The rationality question for <math>K \subset L^G</math> is called '''Noether's problem''' and asks if this field of fixed points is or is not a purely transcendental extension of ''K''.
In the paper {{harv|Noether|1918}} on [[Galois theory]] she studied the problem of parameterizing the equations with given Galois group, which she reduced  to "Noether's problem". (She first mentioned this problem in {{harv|Noether|1913}} where she attributed the problem to E. Fischer.) She showed this was true for ''n'' = 2, 3, or 4. {{harvs|first=R. G.|last= Swan|authorlink = Richard Swan|year=1969|txt}} found a counter-example to the Noether's problem, with ''n'' = 47 and ''G'' a cyclic group of order 47.

==Lüroth's theorem==
{{main|Lüroth's theorem}}
A celebrated case is '''Lüroth's problem''', which [[Jacob Lüroth]] solved in the nineteenth century. Lüroth's problem concerns subextensions ''L'' of ''K''(''X''), the rational functions in the single indeterminate ''X''. Any such field is either equal to ''K'' or is also rational, i.e. ''L'' = ''K''(''F'') for some rational function ''F''. In geometrical terms this states that a non-constant [[rational map]] from the [[projective line]] to a curve ''C'' can only occur when ''C'' also has [[genus of a curve|genus]] 0. That fact can be read off geometrically from the [[Riemann–Hurwitz formula]].

==Unirationality==

A '''unirational variety''' ''V'' over a field ''K'' is one dominated by a rational variety, so that its function field ''K''(''V'') lies in a pure transcendental field of finite type (which can be chosen to be of finite degree over ''K''(''V'') if ''K'' is infinite). The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and [[Castelnuovo's theorem]] implies that for complex surfaces  unirational implies rational, because both are characterized by the vanishing of both the [[arithmetic genus]] and the second [[plurigenus]]. Zariski found some examples ([[Zariski surface]]s) in characteristic ''p''&nbsp;>&nbsp;0 that are unirational but not rational.  {{harvtxt|Clemens|Griffiths|1972}} showed  that a cubic [[three-fold]] is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an [[intermediate Jacobian]]. 
{{harvtxt|Iskovskih|Manin|1971}} showed that all non-singular [[quartic threefold]]s are irrational, though some of them are unirational. {{harvtxt|Artin|Mumford|1972}} found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational.

For any field ''K'', [[János Kollár]] proved in 2000 that a smooth [[cubic hypersurface]] of dimension at least 2 is unirational if it has a point defined over ''K''. This is an improvement of many classical results, beginning with the case of [[cubic surface]]s (which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the [[moduli space]] of curves.<ref>{{cite journal |author=János Kollár |title=Unirationality of cubic hypersurfaces |year=2002 |journal=Journal of the Institute of Mathematics of Jussieu |volume=1 |issue=3 |pages=467–476 |doi=10.1017/S1474748002000117 |mr=1956057|arxiv=math/0005146 |s2cid=6775041 }}</ref>

==Rationally connected variety ==
A '''rationally connected variety''' ''V'' is a [[Algebraic variety#Projective variety|projective algebraic variety]] over an algebraically closed field such that through every two points there passes the image of a [[Regular map (algebraic geometry)|regular map]] from the [[projective line]] into ''V''. Equivalently, a variety is rationally connected if every two points are connected by a [[rational curve]] contained in the variety.<ref>{{Citation | last1=Kollár | first1=János | title=Rational Curves on Algebraic Varieties | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1996}}.</ref>

This definition differs from that of [[path connectedness]] only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones.

Every rational variety, including the [[projective space]]s, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties.  Unirational varieties are rationally connected, but it is not known if the converse holds.

==Stably rational varieties==

A variety ''V'' is called ''stably rational'' if <math>V \times \mathbf P^m</math> is rational for some <math>m \ge 0</math>. Any rational variety is thus, by definition, stably rational. Examples constructed by {{harvtxt|Beauville|Colliot-Thélène|Sansuc|Swinnerton-Dyer|1985}} show, that the converse is false however.

{{harvtxt|Schreieder|2019}} showed that very general [[hypersurface]]s <math>V \subset \mathbf P^{N+1}</math> are not stably rational, provided that the [[degree (algebraic geometry)|degree]] of ''V'' is at least <math>\log_2 N+2</math>.

==See also==

* [[Rational curve]]
*[[Rational surface]]
*[[Severi–Brauer variety]]
*[[Birational geometry]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | last2=Mumford | first2=David | author2-link=David Mumford | title=Some elementary examples of unirational varieties which are not rational | doi=10.1112/plms/s3-25.1.75  |mr=0321934 | year=1972 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=25 | pages=75–95| citeseerx=10.1.1.121.2765 }}
*{{Citation|last1=Beauville|first1=Arnaud|last2=Colliot-Thélène|first2=Jean-Louis|last3=Sansuc|first3=Jean-Jacques|last4=Swinnerton-Dyer|first4=Peter|title=Variétés stablement rationnelles non rationnelles|journal=Annals of Mathematics |series=Second Series|volume=121|year=1985|issue=2|pages=283–318|mr=786350|doi=10.2307/1971174|jstor=1971174}}
*{{Citation | last1=Clemens | first1=C. Herbert |author1-link=Herbert Clemens| last2=Griffiths | first2=Phillip A. |author2-link=Phillip Griffiths| title=The intermediate Jacobian of the cubic threefold |mr=0302652 | year=1972 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=95 | pages=281–356 | doi=10.2307/1970801 | issue=2 | jstor=1970801| citeseerx=10.1.1.401.4550 }}
*{{Citation | last1=Iskovskih | first1=V. A. | last2=Manin | first2=Ju. I. | title=Three-dimensional quartics and counterexamples to the Lüroth problem | doi= 10.1070/SM1971v015n01ABEH001536 |mr=0291172 | year=1971 | journal=[[Matematicheskii Sbornik]] |series=Novaya Seriya | volume=86 | issue=1 | pages=140–166| bibcode=1971SbMat..15..141I }}
*{{Citation | last1=Kollár | first1=János |author1-link=János Kollár| last2=Smith | first2=Karen E. |author2-link=Karen Smith (mathematician)| last3=Corti | first3=Alessio |author3-link=Alessio Corti| title=Rational and nearly rational varieties | url=http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521832076 | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-83207-6 |mr=2062787 | year=2004 | volume=92 | doi=10.1017/CBO9780511734991}}
*{{citation|last=Noether|first=Emmy|author1-link=Emmy Noether|title=Rationale Funktionenkörper|journal=J. Ber. D. DMV|volume=22|year=1913|pages=316–319}}.
*{{citation|last=Noether|first=Emmy|author1-link=Emmy Noether|title=Gleichungen mit vorgeschriebener Gruppe|journal=[[Mathematische Annalen]] |volume=78|issue=1–4|year=1918|pages=221–229|doi=10.1007/BF01457099|s2cid=122353858 }}.
*{{citation|first=R. G. |last=Swan| title=Invariant rational functions and a problem of Steenrod|journal=Inventiones Mathematicae |volume=7|year=1969|pages=148–158|doi=10.1007/BF01389798|issue=2|bibcode=1969InMat...7..148S|s2cid=121951942 }}
*{{Citation | last1=Martinet | first1=J. | title=Séminaire Bourbaki. Vol. 1969/70: Exposés 364–381 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics |mr=0272580 | year=1971 | volume=189 | chapter=Exp. 372 Un contre-exemple à une conjecture d'E. Noether (d'après R. Swan);}}
*{{Citation|last1=Schreieder|first1=Stefan|title=Stably irrational hypersurfaces of small slopes|journal=Journal of the American Mathematical Society|year=2019|volume=32|issue=4|pages=1171–1199|doi=10.1090/jams/928|arxiv=1801.05397|s2cid=119326067 }}

[[Category:Field (mathematics)]]
[[Category:Algebraic varieties]]
[[Category:Birational geometry]]