Difference between revisions of "Lüroth problem"
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The problem of characterizing subfields of a field of rational functions. | 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 | + | In 1876 J. Lüroth [[#References|[1]]] (see also [[#References|[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 | + | 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 | + | 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 [[#References|[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 [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]). It has been proved [[#References|[5]]] that a three-dimensional [[ | + | 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]], 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 [[#References|[4]]] that smooth three-dimensional quadrics are not rational. In [[#References|[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==== | ====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> | + | <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> | ||
+ | |||
+ | [[Category:Algebraic geometry]] | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 17:38, 19 September 2017
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 |
Lüroth problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%BCroth_problem&oldid=22775