Inmathematics, arational variety is analgebraic variety, over a givenfieldK, which isbirationally equivalent to aprojective space of some dimension overK. This means that itsfunction field is isomorphic to

${\displaystyle K(U_1,\dots ,U_d),}$

the field of allrational functions for some set${\displaystyle \{U_1,\dots ,U_d\}}$ ofindeterminates, whered is thedimension of the variety.

LetV be anaffine algebraic variety of dimensiond defined by a prime idealI = ⟨f₁, ...,f_k⟩ in${\displaystyle K[X_1,\dots ,X_n]}$. IfV is rational, then there aren + 1 polynomialsg₀, ...,g_n in${\displaystyle K(U_1,\dots ,U_d)}$ such that${\displaystyle f_i(g_1/g_0,\dots ,g_n/g_0)=0.}$ In other words, we have arational parameterization${\displaystyle x_i=\frac{g_i}{g_0}(u_1,\dots ,u_d)}$ of the variety.

Conversely, such a rational parameterization induces afield homomorphism of the field of functions ofV into${\displaystyle K(U_1,\dots ,U_d)}$. But this homomorphism is not necessarilyonto. If such a parameterization exists, the variety is said to beunirational. 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.

Arationality question asks whether a givenfield extension isrational, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described aspurely transcendental. More precisely, the rationality question for the field extension${\displaystyle K\subset L}$ is this: is${\displaystyle L}$isomorphic to arational function field over${\displaystyle K}$ in the number of indeterminates given by thetranscendence degree?

There are several different variations of this question, arising from the way in which the fields${\displaystyle K}$ and${\displaystyle L}$ are constructed.

For example, let${\displaystyle K}$ be a field, and let

${\displaystyle \{y_1,\dots ,y_n\}}$

be indeterminates overK and letL be the field generated overK by them. Consider afinite group${\displaystyle G}$ permuting thoseindeterminates overK. By standardGalois theory, the set offixed points of thisgroup action is asubfield of${\displaystyle L}$, typically denoted${\displaystyle L^G}$. The rationality question for${\displaystyle K\subset L^G}$ is calledNoether's problem and asks if this field of fixed points is or is not a purely transcendental extension ofK.In the paper (Noether 1918) on Galois theory she studied the problem of parameterizing the equations with givenGalois group, which she reduced to "Noether's problem". (She first mentioned this problem in (Noether 1913) where she attributed the problem to E. Fischer.) She showed this was true forn = 2, 3, or 4.R. G. Swan (1969) found a counter-example to the Noether's problem, withn = 47 andG acyclic group of order 47.

A celebrated case isLüroth's problem, whichJacob Lüroth solved in the nineteenth century. Lüroth's problem concerns subextensionsL ofK(X), the rational functions in the single indeterminateX. Any such field is either equal toK or is also rational, i.e.L =K(F) for some rational functionF. In geometrical terms this states that a non-constantrational map from theprojective line to a curveC can only occur whenC also hasgenus 0. That fact can be read off geometrically from theRiemann–Hurwitz formula.

Aunirational varietyV over a fieldK is one dominated by a rational variety, so that its function fieldK(V) lies in a pure transcendental field of finite type (which can be chosen to be of finite degree overK(V) ifK is infinite). The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, andCastelnuovo's theorem implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both thearithmetic genus and the secondplurigenus. Zariski found some examples (Zariski surfaces) in characteristicp > 0 that are unirational but not rational.Clemens & Griffiths (1972) showed that a cubicthree-fold is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used anintermediate Jacobian.Iskovskih & Manin (1971) showed that all non-singularquartic threefolds are irrational, though some of them are unirational.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 fieldK,János Kollár proved in 2000 that a smoothcubic hypersurface of dimension at least 2 is unirational if it has a point defined overK. This is an improvement of many classical results, beginning with the case ofcubic surfaces (which are rational varieties over analgebraic closure). Other examples of varieties that are shown to be unirational are many cases of themoduli space of curves.^[1]

Arationally connected varietyV is aprojective algebraic variety over analgebraically closed field such that through every two points there passes the image of aregular map from theprojective line intoV. Equivalently, a variety is rationally connected if every two points are connected by arational curve contained in the variety.^[2]

This definition differs from that ofpath 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 theprojective spaces, 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.

A varietyV is calledstably rational if${\displaystyle V\times {\mathbf{P}}^m}$ is rational for some${\displaystyle m\ge 0}$. Any rational variety is thus, by definition, stably rational. Examples constructed byBeauville et al. (1985) show, that the converse is false however.

Schreieder (2019) showed that very generalhypersurfaces${\displaystyle V\subset {\mathbf{P}}^{N+1}}$ are not stably rational, provided that thedegree ofV is at least${\displaystyle {\log }_{2}N+2}$.

• Rational curve
• Rational surface
• Severi–Brauer variety
• Birational geometry

• Artin, Michael;Mumford, David (1972), "Some elementary examples of unirational varieties which are not rational",Proceedings of the London Mathematical Society, Third Series,25:75–95,CiteSeerX 10.1.1.121.2765,doi:10.1112/plms/s3-25.1.75,ISSN 0024-6115,MR 0321934
• Beauville, Arnaud; Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter (1985), "Variétés stablement rationnelles non rationnelles",Annals of Mathematics, Second Series,121 (2):283–318,doi:10.2307/1971174,JSTOR 1971174,MR 0786350
• Clemens, C. Herbert;Griffiths, Phillip A. (1972), "The intermediate Jacobian of the cubic threefold",Annals of Mathematics, Second Series,95 (2):281–356,CiteSeerX 10.1.1.401.4550,doi:10.2307/1970801,ISSN 0003-486X,JSTOR 1970801,MR 0302652
• Iskovskih, V. A.; Manin, Ju. I. (1971), "Three-dimensional quartics and counterexamples to the Lüroth problem",Matematicheskii Sbornik, Novaya Seriya,86 (1):140–166,Bibcode:1971SbMat..15..141I,doi:10.1070/SM1971v015n01ABEH001536,MR 0291172
• Kollár, János;Smith, Karen E.;Corti, Alessio (2004),Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92,Cambridge University Press,doi:10.1017/CBO9780511734991,ISBN 978-0-521-83207-6,MR 2062787
• Noether, Emmy (1913), "Rationale Funktionenkörper",J. Ber. D. DMV,22:316–319.
• Noether, Emmy (1918), "Gleichungen mit vorgeschriebener Gruppe",Mathematische Annalen,78 (1–4):221–229,doi:10.1007/BF01457099,S2CID 122353858.
• Swan, R. G. (1969), "Invariant rational functions and a problem of Steenrod",Inventiones Mathematicae,7 (2):148–158,Bibcode:1969InMat...7..148S,doi:10.1007/BF01389798,S2CID 121951942
• Martinet, J. (1971), "Exp. 372 Un contre-exemple à une conjecture d'E. Noether (d'après R. Swan);",Séminaire Bourbaki. Vol. 1969/70: Exposés 364–381, Lecture Notes in Mathematics, vol. 189, Berlin, New York:Springer-Verlag,MR 0272580
• Schreieder, Stefan (2019), "Stably irrational hypersurfaces of small slopes",Journal of the American Mathematical Society,32 (4):1171–1199,arXiv:1801.05397,doi:10.1090/jams/928,S2CID 119326067

• Algebraic varieties
• Birational geometry

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