Rational variety

In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K. This is a question on its function field: is it up to isomorphism

$K(x_1, \dots , x_n),$

the field of all rational functions for some set $\{x_1, \dots, x_n\}$ of indeterminates?

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 K < L is this: is L isomorphic to a rational function field over K 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 K and L are constructed.

For example, let K be a field, and let

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

be indeterminates over K and let L be the field generated over K by them. Consider a finite group G permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of L, typically denoted L^G. The rationality question for K < L^G 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 (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 (Noether 1913) where she attributed the problem to E. Fischer.) She showed this was true for n = 2, 3, or 4. R. G. Swan (1969) found a counter-example to the Noether's problem, with n = 47 and G a cyclic group of order 47.

Classical results

A celebrated case is Lüroth's problem, which Jacob Lüroth solved in the nineteenth century, and its generalisations to higher dimensions which lie much deeper. 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 birational mapping from the projective line to a curve C can only occur when C also has genus 0. That fact can be read off geometrically from the Riemann–Hurwitz formula.

Unirationality

A unirational variety is one covered by a rational variety, so that on the function field level it has a function field that lies in a pure transcendental field that has finite degree over it. 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 surfaces) in characteristic p > 0 that are unirational but not rational. 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. Iskovskih & Manin (1971) showed that all non-singular quartic 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 the field of complex numbers, János Kollár proved in 2000 that a smooth cubic hypersurface is unirational over any field K for which it has a point defined. This is an improvement of many classical results, beginning with the case of cubic surfaces (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^[1] .

See also

• Rational surface
• Severi–Brauer variety
• Birational geometry
• Rationally connected variety

Notes

1. ^ [http://journals.cambridge.org/abstract_S1474748002000117 MR1956057 (2003m:14082) Kollár, János(1-PRIN) Unirationality of cubic hypersurfaces. (English summary) J. Inst. Math. Jussieu (2002), no. 3, 467–476. 14M20 (11G25 14G05) Retrieved From Google Scholar 05-12-2008]

References

• 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, doi:10.1112/plms/s3-25.1.75, ISSN 0024-6115, MR0321934 
• Clemens, C. Herbert; Griffiths, Phillip A. (1972), "The intermediate Jacobian of the cubic threefold", Annals of Mathematics. Second Series (The Annals of Mathematics, Vol. 95, No. 2) 95 (2): 281–356, doi:10.2307/1970801, ISSN 0003-486X, JSTOR 1970801, MR0302652 
• Iskovskih, V. A.; Manin, Ju. I. (1971), "Three-dimensional quartics and counterexamples to the Lüroth problem", Matematicheskii Sbornik. Novaya Seriya 86: 140–166, doi:10.1070/SM1971v015n01ABEH001536, MR0291172 
• Kollár, János; Smith, Karen E.; Corti, Alessio (2004), Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, 92, Cambridge University Press, ISBN 978-0-521-83207-6, MR2062787, http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521832076 
• Noether, Emmy (1913), "Rationale Funkionenkorper", J. Ber. D. DMV 22: 316–319 .
• Noether, Emmy (1918), "Gleichungen mit vorgeschriebener Gruppe", Mathematische Annalen 78: 221–229, doi:10.1007/BF01457099 .
• Swan, R. G. (1969), "Invariant rational functions and a problem of Steenrod", Inventiones Mathematicae 7 (2): 148–158, doi:10.1007/BF01389798 
• 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, 189, Berlin, New York: Springer-Verlag, MR0272580