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 $k(x)$ of rational functions in one variable, containing $k$ and distinct from $k$, is isomorphic to the field $k(x)$ (Lüroth's theorem). The question of whether a similar assertion is true for subfields $R$ of the field $k(x_1,\ldots,x_n)$, $R \supsetneq k$, $n \ge 2$, is known as the Lüroth problem.

Let $X$ be an algebraic variety that is a model (see Minimal model) of the field $R$; then the imbedding $R = k(X) \subset k(x_1,\ldots,x_n)$ defines a rational mapping $f : P^n \rightarrow X$ whose image is dense in $X$. 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 $P^m$ are said to be rational (cf. Rational variety). In geometrical language Lüroth's problem can be stated as follows: Is any unirational variety $X$ rational? Without loss of generality one may assume that $\dim X = n$, that is, that $R$ has transcendence degree $n$.

In the case $n=1$ an affirmative solution of Lüroth's problem for any ground field $k$ is given by Lüroth's theorem stated above. For $n=2$ and an algebraically closed field $k$ 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 $X$ over an algebraically closed field of arbitrary characteristic for which there is a separable mapping $f : P^2 \rightarrow X$ (see [7]). For non-separable mappings $f$ 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 $k$ such examples are the minimal cubic surfaces in $P^3$ that have $k$-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 $n \ge 3$.

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