Lüroth problem

The problem of characterizing subfields of a field of rational functions.

In 1876 J. Lüroth [1] (see also [2]) proved that any subfield of a field of rational functions in one variable, containing and distinct from , is isomorphic to the field (Lüroth's theorem). The question of whether a similar assertion is true for subfields of the field , , , , is known as the Lüroth problem.

Let be an algebraic variety that is a model (see Minimal model) of the field ; then the imbedding defines a rational mapping whose image is dense in . Varieties for which there is such a mapping of projective space onto them are said to be unirational (cf. Unirational variety). Varieties that are birationally isomorphic to are said to be rational (cf. Rational variety). In geometrical language Lüroth's problem can be stated as follows: Is any unirational variety rational? Without loss of generality one may assume that , that is, that has transcendence degree .

In the case an affirmative solution of Lüroth's problem for any ground field is given by Lüroth's theorem stated above. For and an algebraically closed field of characteristic 0 the problem was solved affirmatively by G. Castelnuovo in 1893. Castelnuovo's rationality criterion implies also an affirmative solution of Lüroth's problem for surfaces over an algebraically closed field of arbitrary characteristic for which there is a separable mapping (see [7]). For non-separable mappings there are examples that give a negative solution of Lüroth's problem for fields of prime characteristic. In the case of an algebraically non-closed field such examples are the minimal cubic surfaces in that have -points.

For three-dimensional varieties Lüroth's problem has also been solved negatively (see [4], [5], [6]). It has been proved [5] that a three-dimensional cubic hypersurface, which is known to be unirational, is not rational. For the proof a new method was found, based on the comparison of the intermediate Jacobian of the cubic with the Jacobians of curves. It has been proved [4] that smooth three-dimensional quadrics are not rational. In [6], for the construction of counter-examples the Brauer group of the variety (the torsion subgroup of the three-dimensional cohomology group) was used as an invariant. This birational invariant has also been used in the construction of counter-examples in all dimensions .

References

[1]	J. Lüroth, Math. Ann. , 9 (1876) pp. 163–165
[2]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[3]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)
[4]	V.A. Iskovskikh, Yu.I. Manin, "Three-dimensional quartics and counterexamples to the Lüroth problem" Math. USSR Sb. , 15 : 1 (1971) pp. 141–166 Mat. Sb. , 86 : 1 (1971) pp. 140–166
[5]	C.H. Clemens, P. Griffiths, "The intermediate Jacobian of the cubic threefold" Ann. of Math. , 95 (1972) pp. 281–356
[6]	M. Artin, D. Mumford, "Some elementary examples of unirational varieties which are not rational" Proc. London Math. Soc. , 25 : 1 (1972) pp. 75–95
[7]	O. Zariski, "The problem of minimal models in the theory of algebraic surfaces" Amer. J. Math. , 80 (1958) pp. 146–184