1287:, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.
552:
1124:
Even though LĂĽroth's theorem is often thought as a non elementary result, several elementary short proofs have been known for a long time. These simple proofs use only the basics of field theory and
456:
1416:
1336:
855:
172:
113:
618:
358:
997:
1463:
683:
1362:
964:
1015:
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 (
937:
893:
800:
777:
757:
730:
703:
279:
1677:
Beauville, Arnaud; Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter (1985), "Variétés stablement rationnelles non rationnelles",
1195:
is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an
51:
1848:
1234:(which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the
1280:
only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones.
468:
183:
1211:
found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational.
652:, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as
1258:
363:
1171:
implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both the
1125:
1167:
is infinite). The solution of LĂĽroth's problem shows that for algebraic curves, rational and unirational are the same, and
2029:
2024:
2019:
1118:
1818:
1419:
1510:
1832:
912:
1484:
1380:
908:
1776:
Iskovskih, V. A.; Manin, Ju. I. (1971), "Three-dimensional quartics and counterexamples to the LĂĽroth problem",
896:
199:
175:
1306:
1168:
629:
1778:
808:
709:
125:
60:
39:
568:
308:
1731:
1640:
1048:
1887:
1722:
1521:
1196:
653:
1826:
1155:) lies in a pure transcendental field of finite type (which can be chosen to be of finite degree over
1925:
1787:
733:
1952:
Martinet, J. (1971), "Exp. 372 Un contre-exemple à une conjecture d'E. Noether (d'après R. Swan);",
1736:
1645:
969:
1489:
32:
1429:
1257:
over an algebraically closed field such that through every two points there passes the image of a
739:
There are several different variations of this question, arising from the way in which the fields
1999:
1981:
1941:
1904:
1757:
1694:
1583:
1557:
1277:
1223:
558:
1814:
1219:
662:
1844:
1749:
1717:
1658:
1254:
1204:
119:
28:
1341:
1991:
1933:
1896:
1836:
1795:
1741:
1686:
1650:
1567:
1479:
1284:
1172:
1114:
43:
1965:
1858:
1807:
1769:
1706:
1670:
1579:
942:
1961:
1957:
1854:
1803:
1765:
1713:
1702:
1666:
1606:
1575:
1534:
1262:
1180:
1102:
916:
657:
645:
1269:. Equivalently, a variety is rationally connected if every two points are connected by a
1058:
1003:
and asks if this field of fixed points is or is not a purely transcendental extension of
1929:
1791:
1364:. Any rational variety is thus, by definition, stably rational. Examples constructed by
1474:
1270:
922:
878:
785:
762:
742:
715:
688:
229:
2013:
2003:
1945:
1908:
1822:
1628:
1624:
1231:
1012:
904:
1799:
1631:(1972), "Some elementary examples of unirational varieties which are not rational",
1882:
1865:
1587:
1375:
1235:
1098:
1024:
873:
632:
implies also that, in characteristic zero, every unirational surface is rational.
1019:) where she attributed the problem to E. Fischer.) She showed this was true for
20:
1230:. This is an improvement of many classical results, beginning with the case of
1972:
Schreieder, Stefan (2019), "Stably irrational hypersurfaces of small slopes",
1916:
Swan, R. G. (1969), "Invariant rational functions and a problem of
Steenrod",
1654:
1571:
1192:
1176:
705:
628:. LĂĽroth's theorem (see below) implies that unirational curves are rational.
1840:
1753:
1662:
1365:
1061:
solved in the nineteenth century. LĂĽroth's problem concerns subextensions
1995:
1937:
1900:
1761:
1698:
1226:
of dimension at least 2 is unirational if it has a point defined over
1562:
1745:
1690:
1986:
1147:
is one dominated by a rational variety, so that its function field
1956:, Lecture Notes in Mathematics, vol. 189, Berlin, New York:
547:{\displaystyle x_{i}={\frac {g_{i}}{g_{0}}}(u_{1},\ldots ,u_{d})}
621:
624:. If such a parameterization exists, the variety is said to be
1548:
János Kollár (2002). "Unirationality of cubic hypersurfaces".
1720:(1972), "The intermediate Jacobian of the cubic threefold",
1831:, Cambridge Studies in Advanced Mathematics, vol. 92,
1187: > 0 that are unirational but not rational.
1031:) found a counter-example to the Noether's problem, with
451:{\displaystyle f_{i}(g_{1}/g_{0},\ldots ,g_{n}/g_{0})=0.}
1097:. In geometrical terms this states that a non-constant
557:
Conversely, such a rational parameterization induces a
1073:), the rational functions in the single indeterminate
1432:
1383:
1344:
1309:
1207:
are irrational, though some of them are unirational.
972:
945:
925:
881:
811:
788:
765:
745:
718:
691:
665:
571:
471:
366:
311:
232:
128:
63:
1117:
0. That fact can be read off geometrically from the
1885:(1918), "Gleichungen mit vorgeschriebener Gruppe",
656:. More precisely, the rationality question for the
1550:Journal of the Institute of Mathematics of Jussieu
1457:
1410:
1356:
1330:
991:
958:
931:
887:
849:
794:
771:
751:
724:
697:
677:
612:
546:
450:
352:
273:
166:
107:
1954:Séminaire Bourbaki. Vol. 1969/70: Exposés 364–381
1188:
1633:Proceedings of the London Mathematical Society
1511:"Another elementary proof of Luroth's theorem"
1200:
732:in the number of indeterminates given by the
8:
1974:Journal of the American Mathematical Society
1208:
844:
812:
161:
129:
1418:are not stably rational, provided that the
1411:{\displaystyle V\subset \mathbf {P} ^{N+1}}
620:. But this homomorphism is not necessarily
1371:
1368:show, that the converse is false however.
1985:
1735:
1644:
1561:
1437:
1431:
1396:
1391:
1382:
1343:
1322:
1317:
1308:
983:
971:
950:
944:
924:
880:
838:
819:
810:
787:
764:
744:
717:
690:
664:
601:
582:
570:
535:
516:
501:
491:
485:
476:
470:
433:
424:
418:
399:
390:
384:
371:
365:
341:
322:
310:
262:
243:
231:
155:
136:
127:
93:
74:
62:
1331:{\displaystyle V\times \mathbf {P} ^{m}}
1501:
1126:Gauss's lemma for primitive polynomials
1016:
1008:
1868:(1913), "Rationale Funktionenkörper",
1828:Rational and nearly rational varieties
1603:Rational Curves on Algebraic Varieties
1530:
1519:
1283:Every rational variety, including the
850:{\displaystyle \{y_{1},\dots ,y_{n}\}}
167:{\displaystyle \{U_{1},\dots ,U_{d}\}}
108:{\displaystyle K(U_{1},\dots ,U_{d}),}
1276:This definition differs from that of
613:{\displaystyle K(U_{1},\dots ,U_{d})}
353:{\displaystyle K(U_{1},\dots ,U_{d})}
7:
1077:. Any such field is either equal to
1028:
625:
14:
1509:Bensimhoun, Michael (May 2004).
1392:
1318:
190:Rationality and parameterization
1800:10.1070/SM1971v015n01ABEH001536
1179:. Zariski found some examples (
966:. The rationality question for
1189:Clemens & Griffiths (1972)
992:{\displaystyle K\subset L^{G}}
607:
575:
541:
509:
439:
377:
347:
315:
268:
236:
99:
67:
1:
1222:proved in 2000 that a smooth
1203:showed that all non-singular
1093:) for some rational function
561:of the field of functions of
1458:{\displaystyle \log _{2}N+2}
1255:projective algebraic variety
1248:rationally connected variety
1242:Rationally connected variety
1201:Iskovskih & Manin (1971)
1039:a cyclic group of order 47.
868:be the field generated over
289: + 1 polynomials
285:is rational, then there are
2046:
1833:Cambridge University Press
1273:contained in the variety.
1209:Artin & Mumford (1972)
1081:or is also rational, i.e.
1046:
678:{\displaystyle K\subset L}
458:In other words, we have a
1572:10.1017/S1474748002000117
1374:showed that very general
1291:Stably rational varieties
462:rational parameterization
206:defined by a prime ideal
1918:Inventiones Mathematicae
1841:10.1017/CBO9780511734991
200:affine algebraic variety
1779:Matematicheskii Sbornik
1655:10.1112/plms/s3-25.1.75
1366:Beauville et al. (1985)
1357:{\displaystyle m\geq 0}
1119:Riemann–Hurwitz formula
1042:
860:be indeterminates over
710:rational function field
46:of some dimension over
40:birationally equivalent
1601:Kollár, János (1996),
1529:Cite journal requires
1459:
1412:
1358:
1332:
993:
960:
933:
889:
851:
796:
773:
753:
726:
699:
679:
614:
548:
452:
354:
275:
168:
109:
50:. This means that its
1888:Mathematische Annalen
1723:Annals of Mathematics
1718:Griffiths, Phillip A.
1679:Annals of Mathematics
1485:Severi–Brauer variety
1460:
1413:
1359:
1338:is rational for some
1333:
1197:intermediate Jacobian
1191:showed that a cubic
1169:Castelnuovo's theorem
1053:A celebrated case is
994:
961:
959:{\displaystyle L^{G}}
934:
890:
872:by them. Consider a
852:
797:
774:
754:
727:
700:
680:
654:purely transcendental
644:asks whether a given
636:Rationality questions
630:Castelnuovo's theorem
615:
549:
453:
355:
276:
169:
110:
1605:, Berlin, New York:
1430:
1381:
1342:
1307:
1183:) in characteristic
1109:can only occur when
970:
943:
939:, typically denoted
923:
879:
809:
802:be a field, and let
786:
763:
743:
734:transcendence degree
716:
689:
663:
642:rationality question
569:
469:
364:
309:
230:
126:
61:
2030:Birational geometry
2025:Algebraic varieties
2020:Field (mathematics)
1930:1969InMat...7..148S
1792:1971SbMat..15..141I
1714:Clemens, C. Herbert
1490:Birational geometry
1138:unirational variety
1938:10.1007/BF01389798
1901:10.1007/BF01457099
1455:
1408:
1354:
1328:
1278:path connectedness
1224:cubic hypersurface
1205:quartic threefolds
989:
956:
929:
885:
847:
792:
769:
749:
722:
695:
675:
610:
559:field homomorphism
544:
448:
350:
271:
164:
120:rational functions
105:
1850:978-0-521-83207-6
1782:, Novaya Seriya,
1726:, Second Series,
1681:, Second Series,
1372:Schreieder (2019)
1285:projective spaces
1001:Noether's problem
932:{\displaystyle L}
888:{\displaystyle G}
795:{\displaystyle K}
782:For example, let
779:are constructed.
772:{\displaystyle L}
752:{\displaystyle K}
725:{\displaystyle K}
698:{\displaystyle L}
507:
274:{\displaystyle K}
118:the field of all
54:is isomorphic to
29:algebraic variety
16:Algebraic variety
2037:
2006:
1996:10.1090/jams/928
1989:
1980:(4): 1171–1199,
1968:
1948:
1911:
1895:(1–4): 221–229,
1877:
1861:
1810:
1772:
1739:
1709:
1673:
1648:
1635:, Third Series,
1611:
1609:
1598:
1592:
1591:
1565:
1545:
1539:
1538:
1532:
1527:
1525:
1517:
1515:
1506:
1480:Rational surface
1464:
1462:
1461:
1456:
1442:
1441:
1417:
1415:
1414:
1409:
1407:
1406:
1395:
1363:
1361:
1360:
1355:
1337:
1335:
1334:
1329:
1327:
1326:
1321:
1181:Zariski surfaces
1173:arithmetic genus
1055:LĂĽroth's problem
1049:LĂĽroth's theorem
1043:LĂĽroth's theorem
1007:. In the paper (
998:
996:
995:
990:
988:
987:
965:
963:
962:
957:
955:
954:
938:
936:
935:
930:
895:permuting those
894:
892:
891:
886:
856:
854:
853:
848:
843:
842:
824:
823:
801:
799:
798:
793:
778:
776:
775:
770:
758:
756:
755:
750:
731:
729:
728:
723:
704:
702:
701:
696:
684:
682:
681:
676:
619:
617:
616:
611:
606:
605:
587:
586:
554:of the variety.
553:
551:
550:
545:
540:
539:
521:
520:
508:
506:
505:
496:
495:
486:
481:
480:
464:
463:
457:
455:
454:
449:
438:
437:
428:
423:
422:
404:
403:
394:
389:
388:
376:
375:
359:
357:
356:
351:
346:
345:
327:
326:
280:
278:
277:
272:
267:
266:
248:
247:
186:of the variety.
173:
171:
170:
165:
160:
159:
141:
140:
114:
112:
111:
106:
98:
97:
79:
78:
44:projective space
25:rational variety
2045:
2044:
2040:
2039:
2038:
2036:
2035:
2034:
2010:
2009:
1971:
1958:Springer-Verlag
1951:
1915:
1881:
1864:
1851:
1819:Smith, Karen E.
1813:
1775:
1746:10.2307/1970801
1737:10.1.1.401.4550
1712:
1691:10.2307/1971174
1676:
1646:10.1.1.121.2765
1623:
1620:
1615:
1614:
1607:Springer-Verlag
1600:
1599:
1595:
1547:
1546:
1542:
1528:
1518:
1513:
1508:
1507:
1503:
1498:
1471:
1433:
1428:
1427:
1390:
1379:
1378:
1340:
1339:
1316:
1305:
1304:
1301:stably rational
1293:
1263:projective line
1244:
1175:and the second
1134:
1103:projective line
1051:
1045:
979:
968:
967:
946:
941:
940:
921:
920:
903:. By standard
877:
876:
834:
815:
807:
806:
784:
783:
761:
760:
741:
740:
714:
713:
687:
686:
661:
660:
658:field extension
646:field extension
638:
597:
578:
567:
566:
531:
512:
497:
487:
472:
467:
466:
461:
460:
429:
414:
395:
380:
367:
362:
361:
337:
318:
307:
306:
304:
295:
258:
239:
228:
227:
225:
216:
192:
151:
132:
124:
123:
89:
70:
59:
58:
31:, over a given
17:
12:
11:
5:
2043:
2041:
2033:
2032:
2027:
2022:
2012:
2011:
2008:
2007:
1969:
1949:
1924:(2): 148–158,
1913:
1879:
1870:J. Ber. d. DMV
1862:
1849:
1823:Corti, Alessio
1811:
1786:(1): 140–166,
1773:
1730:(2): 281–356,
1710:
1685:(2): 283–318,
1674:
1629:Mumford, David
1625:Artin, Michael
1619:
1616:
1613:
1612:
1593:
1556:(3): 467–476.
1540:
1531:|journal=
1500:
1499:
1497:
1494:
1493:
1492:
1487:
1482:
1477:
1475:Rational curve
1470:
1467:
1454:
1451:
1448:
1445:
1440:
1436:
1405:
1402:
1399:
1394:
1389:
1386:
1353:
1350:
1347:
1325:
1320:
1315:
1312:
1292:
1289:
1271:rational curve
1243:
1240:
1232:cubic surfaces
1214:For any field
1133:
1132:Unirationality
1130:
1047:Main article:
1044:
1041:
1025:R. G. Swan
1023:= 2, 3, or 4.
986:
982:
978:
975:
953:
949:
928:
897:indeterminates
884:
858:
857:
846:
841:
837:
833:
830:
827:
822:
818:
814:
791:
768:
748:
721:
694:
674:
671:
668:
637:
634:
609:
604:
600:
596:
593:
590:
585:
581:
577:
574:
543:
538:
534:
530:
527:
524:
519:
515:
511:
504:
500:
494:
490:
484:
479:
475:
447:
444:
441:
436:
432:
427:
421:
417:
413:
410:
407:
402:
398:
393:
387:
383:
379:
374:
370:
349:
344:
340:
336:
333:
330:
325:
321:
317:
314:
300:
293:
270:
265:
261:
257:
254:
251:
246:
242:
238:
235:
221:
214:
210: = ⟨
191:
188:
176:indeterminates
163:
158:
154:
150:
147:
144:
139:
135:
131:
116:
115:
104:
101:
96:
92:
88:
85:
82:
77:
73:
69:
66:
52:function field
15:
13:
10:
9:
6:
4:
3:
2:
2042:
2031:
2028:
2026:
2023:
2021:
2018:
2017:
2015:
2005:
2001:
1997:
1993:
1988:
1983:
1979:
1975:
1970:
1967:
1963:
1959:
1955:
1950:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1914:
1910:
1906:
1902:
1898:
1894:
1890:
1889:
1884:
1883:Noether, Emmy
1880:
1875:
1871:
1867:
1866:Noether, Emmy
1863:
1860:
1856:
1852:
1846:
1842:
1838:
1834:
1830:
1829:
1824:
1820:
1816:
1815:Kollár, János
1812:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1780:
1774:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1738:
1733:
1729:
1725:
1724:
1719:
1715:
1711:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1675:
1672:
1668:
1664:
1660:
1656:
1652:
1647:
1642:
1638:
1634:
1630:
1626:
1622:
1621:
1617:
1608:
1604:
1597:
1594:
1589:
1585:
1581:
1577:
1573:
1569:
1564:
1559:
1555:
1551:
1544:
1541:
1536:
1523:
1512:
1505:
1502:
1495:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1472:
1468:
1466:
1452:
1449:
1446:
1443:
1438:
1434:
1425:
1421:
1403:
1400:
1397:
1387:
1384:
1377:
1376:hypersurfaces
1373:
1369:
1367:
1351:
1348:
1345:
1323:
1313:
1310:
1302:
1298:
1290:
1288:
1286:
1281:
1279:
1274:
1272:
1268:
1264:
1260:
1256:
1252:
1249:
1241:
1239:
1237:
1233:
1229:
1225:
1221:
1217:
1212:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
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1050:
1040:
1038:
1034:
1030:
1026:
1022:
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1014:
1013:Galois theory
1010:
1006:
1002:
984:
980:
976:
973:
951:
947:
926:
918:
914:
910:
907:, the set of
906:
905:Galois theory
902:
898:
882:
875:
871:
867:
863:
839:
835:
831:
828:
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820:
816:
805:
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719:
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288:
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259:
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249:
244:
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233:
224:
220:
213:
209:
205:
202:of dimension
201:
197:
189:
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185:
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156:
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142:
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121:
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75:
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56:
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53:
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41:
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22:
1977:
1973:
1953:
1921:
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1892:
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1873:
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1777:
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1563:math/0005146
1553:
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1543:
1522:cite journal
1516:. Jerusalem.
1504:
1426:is at least
1423:
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1296:
1294:
1282:
1275:
1266:
1250:
1247:
1245:
1236:moduli space
1227:
1220:János Kollár
1215:
1213:
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1164:
1160:
1156:
1152:
1148:
1144:
1140:
1137:
1135:
1128:(see e.g.).
1123:
1110:
1106:
1099:rational map
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1059:Jacob LĂĽroth
1054:
1052:
1036:
1032:
1020:
1017:Noether 1913
1009:Noether 1918
1004:
1000:
913:group action
909:fixed points
900:
874:finite group
869:
865:
861:
859:
781:
738:
685:is this: is
649:
641:
639:
562:
556:
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290:
286:
282:
222:
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211:
207:
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195:
193:
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117:
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1259:regular map
1238:of curves.
1105:to a curve
626:unirational
38:, which is
21:mathematics
2014:Categories
1987:1801.05397
1618:References
1299:is called
1295:A variety
1193:three-fold
1177:plurigenus
999:is called
706:isomorphic
360:such that
2004:119326067
1946:121951942
1909:122353858
1876:: 316–319
1754:0003-486X
1732:CiteSeerX
1663:0024-6115
1641:CiteSeerX
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184:dimension
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1825:(2004),
1469:See also
1057:, which
917:subfield
911:of this
864:and let
650:rational
178:, where
1966:0272580
1926:Bibcode
1859:2062787
1808:0291172
1788:Bibcode
1770:0302652
1762:1970801
1707:0786350
1699:1971174
1671:0321934
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1942:S2CID
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1695:JSTOR
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1558:arXiv
1514:(PDF)
1496:Notes
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