View source for Lüroth problem

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

In 1876 J. Lüroth [[#References|[1]]] (see also [[#References|[2]]]) proved that any subfield of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609801.png" /> of rational functions in one variable, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609802.png" /> and distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609803.png" />, is isomorphic to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609804.png" /> (Lüroth's theorem). The question of whether a similar assertion is true for subfields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609805.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l0609809.png" />, is known as the Lüroth problem.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098010.png" /> be an [[Algebraic variety|algebraic variety]] that is a model (see [[Minimal model|Minimal model]]) of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098011.png" />; then the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098012.png" /> defines a rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098013.png" /> whose image is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098014.png" />. Varieties for which there is such a mapping of projective space onto them are said to be unirational (cf. [[Unirational variety|Unirational variety]]). Varieties that are birationally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098015.png" /> are said to be rational (cf. [[Rational variety|Rational variety]]). In geometrical language Lüroth's problem can be stated as follows: Is any unirational variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098016.png" /> rational? Without loss of generality one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098017.png" />, that is, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098018.png" /> has transcendence degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098019.png" />.

In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098020.png" /> an affirmative solution of Lüroth's problem for any ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098021.png" /> is given by Lüroth's theorem stated above. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098022.png" /> and an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098023.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098024.png" /> over an algebraically closed field of arbitrary characteristic for which there is a [[Separable mapping|separable mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098025.png" /> (see [[#References|[7]]]). For non-separable mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098027.png" /> such examples are the minimal cubic surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098028.png" /> that have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098029.png" />-points.

For three-dimensional varieties Lüroth's problem has also been solved negatively (see [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]). It has been proved [[#References|[5]]] that a three-dimensional [[Cubic hypersurface|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|intermediate Jacobian]] of the cubic with the Jacobians of curves. It has been proved [[#References|[4]]] that smooth three-dimensional quadrics are not rational. In [[#References|[6]]], for the construction of counter-examples the [[Brauer group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060980/l06098030.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Lüroth,   ''Math. Ann.'' , '''9'''  (1876)  pp. 163–165</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.I. Manin,   "Cubic forms. Algebra, geometry, arithmetic" , North-Holland  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.H. Clemens,   P. Griffiths,   "The intermediate Jacobian of the cubic threefold"  ''Ann. of Math.'' , '''95'''  (1972)  pp. 281–356</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Artin,   D. Mumford,   "Some elementary examples of unirational varieties which are not rational"  ''Proc. London Math. Soc.'' , '''25''' :  1  (1972)  pp. 75–95</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O. Zariski,   "The problem of minimal models in the theory of algebraic surfaces"  ''Amer. J. Math.'' , '''80'''  (1958)  pp. 146–184</TD></TR></table>