51:
The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant
841:
is not unique, though there is a unique one isomorphic to the product of the projective line and a curve. A somewhat subtle point is that even though a surface might have infinitely many -1-curves, one need only contract finitely many of them to obtain a surface with no -1-curves.
689:, then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also. The singularities that appear are called
707:
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the
Italian school around 1900; the
902:. The major conceptual advance of the 1970s and early 1980s was that the construction of minimal models is still feasible, provided one is careful about the types of singularities which occur. (For example, we want to decide if
1249:
relying on earlier work of
Shokurov and Hacon, and McKernan. They also proved several other problems including finite generation of log canonical rings and existence of minimal models for varieties of log general type.
837:
nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface over a curve). In the second case, the ruled surface birational to
387:
487:
321:
239:
110:
970:
566:
746:
1005:
785:
932:
820:
195:
1141:
1311:
1224:
896:
647:
594:
512:
264:
135:
420:
1199:
1168:
1107:
1080:
1060:
1029:
871:
687:
667:
618:
532:
440:
341:
284:
162:
70:
1533:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 32, Berlin: Springer-Verlag,
748:
must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve
1457:
716:
essentially describes the process of constructing a minimal model of any surface. The theorem states that any nontrivial birational morphism
1651:
1339:
1629:
1591:
1546:
1201:. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model
709:
40:
702:
73:
1699:
1709:
669:
appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth
457:
1730:
144:
In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety
203:
1725:
1704:
346:
1575:
1253:
The problem of termination of log flips in higher dimensions remains the subject of active research.
1171:
537:
289:
78:
719:
1267:
937:
755:
790:
825:
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply
171:
1112:
850:
In dimensions greater than 2, the theory becomes far more involved. In particular, there exist
690:
1293:
975:
568:
905:
395:
1688:
1639:
1601:
1556:
1519:
1488:
1441:
1413:
1378:
1358:
1262:
1177:
1146:
1085:
826:
8:
1234:
36:
28:
1499:
1362:
1237:
in dimensions three and four. This was subsequently generalized to higher dimensions by
1204:
876:
627:
574:
492:
244:
115:
1676:
1348:
1065:
1045:
1014:
856:
672:
652:
603:
517:
425:
326:
269:
147:
55:
32:
20:
1668:
1625:
1587:
1563:
1542:
1526:
1507:
1495:
1476:
1448:
1429:
1401:
1389:
1330:
1242:
713:
443:
165:
1471:
1735:
1660:
1617:
1579:
1534:
1466:
1366:
1452:
1370:
822:
which shows that if the canonical class is nef then the surface has no −1-curves.
1695:
1684:
1635:
1613:
1597:
1552:
1515:
1484:
1437:
1409:
1385:
1374:
1008:
899:
390:
164:, which for simplicity is assumed non-singular. There are two cases based on its
1143:
nef. However, the process may encounter difficulties: at some point the variety
1646:
1567:
1393:
1334:
1326:
1246:
1238:
1039:
1035:
448:
1621:
1538:
1719:
1672:
1583:
1511:
1480:
1433:
1405:
851:
35:
which is as simple as possible. The subject has its origins in the classical
1170:
may become "too singular". The conjectural solution to this problem is the
1453:"The structure of algebraic threefolds: an introduction to Mori's program"
1420:
Fujino, Osamu (2009), "New developments in the theory of minimal models",
1337:(2010), "Existence of minimal models for varieties of log general type",
138:
1649:(1988), "Flip theorem and the existence of minimal models for 3-folds",
1680:
43:, and is currently an active research area within algebraic geometry.
1353:
1664:
972:
must be defined. Hence, at the very least, our varieties must have
1233:
The existence of the more general log flips was established by
31:. Its goal is to construct a birational model of any complex
1109:, each of which is "closer" than the previous one to having
829:
all the −1-curves on the surface, and the resulting variety
1325:
1500:"Minimal models of algebraic threefolds: Mori's program"
1082:, one can inductively construct a sequence of varieties
1296:
1207:
1180:
1149:
1115:
1088:
1068:
1048:
1017:
978:
940:
908:
879:
859:
793:
758:
722:
675:
655:
630:
606:
577:
540:
520:
495:
460:
428:
398:
349:
329:
292:
272:
247:
206:
174:
150:
118:
81:
58:
1042:, describing the structure of the cone of curves of
845:
1230:showed that flips exist in the 3-dimensional case.
16:
Effort to birationally classify algebraic varieties
1574:, Cambridge Tracts in Mathematics, vol. 134,
1305:
1218:
1193:
1162:
1135:
1101:
1074:
1054:
1023:
999:
964:
926:
890:
865:
814:
779:
740:
681:
661:
641:
612:
588:
560:
526:
506:
481:
434:
414:
381:
335:
315:
278:
258:
233:
189:
156:
129:
104:
64:
1384:
1717:
1062:. Briefly, the theorem shows that starting with
1396:(1988), "Higher-dimensional complex geometry",
1174:, a kind of codimension-2 surgery operation on
873:which are not birational to any smooth variety
1659:(1), American Mathematical Society: 117–253,
1458:Bulletin of the American Mathematical Society
1428:(2), Mathematical Society of Japan: 162–186,
696:
1652:Journal of the American Mathematical Society
1340:Journal of the American Mathematical Society
27:is part of the birational classification of
1572:Birational geometry of algebraic varieties
1562:
1470:
1352:
1694:
833:is either a (unique) minimal model with
1607:
1718:
1531:Rational curves on algebraic varieties
1525:
1494:
1447:
1419:
1286:Note that the Kodaira dimension of an
624:The question of whether the varieties
482:{\displaystyle \kappa (X)\geqslant 0.}
1645:
1227:
234:{\displaystyle \kappa (X)=-\infty .}
13:
1612:, Universitext, Berlin, New York:
1300:
551:
382:{\displaystyle \dim Y<\dim X',}
225:
14:
1747:
846:Higher-dimensional minimal models
1610:Introduction to the Mori program
1313:or an integer in the range 0 to
934:is nef, so intersection numbers
1472:10.1090/S0273-0979-1987-15548-0
1290:-dimensional variety is either
703:Enriques–Kodaira classification
561:{\displaystyle K_{X^{\prime }}}
316:{\displaystyle f\colon X'\to Y}
105:{\displaystyle f\colon X\to X'}
1700:"Mori theory of extremal rays"
1280:
741:{\displaystyle f\colon X\to Y}
732:
470:
464:
446:. Such a morphism is called a
307:
216:
210:
184:
178:
91:
1:
1371:10.1090/S0894-0347-09-00649-3
1273:
965:{\displaystyle K_{X'}\cdot C}
1034:The first key result is the
780:{\displaystyle C\cdot C=-1.}
7:
1705:Encyclopedia of Mathematics
1256:
815:{\displaystyle K\cdot C=-1}
534:, with the canonical class
39:of surfaces studied by the
10:
1752:
1576:Cambridge University Press
1011:for some positive integer
700:
697:Minimal models of surfaces
241:We want to find a variety
190:{\displaystyle \kappa (X)}
46:
1622:10.1007/978-1-4757-5602-9
1539:10.1007/978-3-662-03276-3
1226:in finitely many steps).
1136:{\displaystyle K_{X_{i}}}
787:Any such curve must have
72:for which any birational
52:finding a smooth variety
1584:10.1017/CBO9780511662560
1306:{\displaystyle -\infty }
1268:Minimal rational surface
323:to a projective variety
1608:Matsuki, Kenji (2002),
1400:(166): 144 pp. (1989),
1000:{\displaystyle nK_{X'}}
752:with self-intersection
1307:
1220:
1195:
1164:
1137:
1103:
1076:
1056:
1025:
1001:
966:
928:
927:{\displaystyle K_{X'}}
892:
867:
816:
781:
742:
691:terminal singularities
683:
663:
643:
614:
590:
562:
528:
508:
483:
436:
416:
415:{\displaystyle -K_{F}}
383:
337:
317:
280:
260:
235:
191:
158:
131:
112:with a smooth surface
106:
66:
1308:
1221:
1196:
1194:{\displaystyle X_{i}}
1165:
1163:{\displaystyle X_{i}}
1138:
1104:
1102:{\displaystyle X_{i}}
1077:
1057:
1026:
1002:
967:
929:
893:
868:
817:
782:
743:
684:
664:
644:
615:
591:
563:
529:
509:
484:
437:
417:
384:
338:
318:
281:
261:
236:
192:
159:
132:
107:
67:
25:minimal model program
1294:
1263:Abundance conjecture
1205:
1178:
1147:
1113:
1086:
1066:
1046:
1015:
976:
938:
906:
877:
857:
791:
756:
720:
673:
653:
628:
604:
575:
538:
518:
493:
458:
426:
396:
347:
327:
290:
270:
245:
204:
172:
148:
116:
79:
56:
1731:Birational geometry
1363:2010JAMS...23..405B
1235:Vyacheslav Shokurov
900:nef canonical class
710:contraction theorem
422:of a general fibre
391:anticanonical class
37:birational geometry
29:algebraic varieties
1726:Algebraic geometry
1331:Hacon, Christopher
1329:; Cascini, Paolo;
1303:
1219:{\displaystyle X'}
1216:
1191:
1160:
1133:
1099:
1072:
1052:
1021:
997:
962:
924:
891:{\displaystyle X'}
888:
863:
812:
777:
738:
679:
659:
642:{\displaystyle X'}
639:
610:
589:{\displaystyle X'}
586:
558:
524:
507:{\displaystyle X'}
504:
479:
432:
412:
379:
333:
313:
276:
259:{\displaystyle X'}
256:
231:
187:
154:
130:{\displaystyle X'}
127:
102:
62:
33:projective variety
21:algebraic geometry
1631:978-0-387-98465-0
1593:978-0-521-63277-5
1548:978-3-642-08219-1
1243:Christopher Hacon
1241:, Paolo Cascini,
1075:{\displaystyle X}
1055:{\displaystyle X}
1024:{\displaystyle n}
866:{\displaystyle X}
714:Guido Castelnuovo
682:{\displaystyle X}
662:{\displaystyle X}
613:{\displaystyle X}
527:{\displaystyle X}
435:{\displaystyle F}
336:{\displaystyle Y}
286:, and a morphism
279:{\displaystyle X}
166:Kodaira dimension
157:{\displaystyle X}
65:{\displaystyle X}
1743:
1712:
1696:Kawamata, Yujiro
1691:
1642:
1604:
1559:
1522:
1506:(177): 303–326,
1491:
1474:
1444:
1416:
1386:Clemens, Herbert
1381:
1356:
1318:
1312:
1310:
1309:
1304:
1284:
1225:
1223:
1222:
1217:
1215:
1200:
1198:
1197:
1192:
1190:
1189:
1169:
1167:
1166:
1161:
1159:
1158:
1142:
1140:
1139:
1134:
1132:
1131:
1130:
1129:
1108:
1106:
1105:
1100:
1098:
1097:
1081:
1079:
1078:
1073:
1061:
1059:
1058:
1053:
1030:
1028:
1027:
1022:
1006:
1004:
1003:
998:
996:
995:
994:
971:
969:
968:
963:
955:
954:
953:
933:
931:
930:
925:
923:
922:
921:
897:
895:
894:
889:
887:
872:
870:
869:
864:
852:smooth varieties
821:
819:
818:
813:
786:
784:
783:
778:
747:
745:
744:
739:
688:
686:
685:
680:
668:
666:
665:
660:
648:
646:
645:
640:
638:
619:
617:
616:
611:
595:
593:
592:
587:
585:
571:. In this case,
567:
565:
564:
559:
557:
556:
555:
554:
533:
531:
530:
525:
513:
511:
510:
505:
503:
489:We want to find
488:
486:
485:
480:
449:Fano fibre space
441:
439:
438:
433:
421:
419:
418:
413:
411:
410:
388:
386:
385:
380:
375:
342:
340:
339:
334:
322:
320:
319:
314:
306:
285:
283:
282:
277:
265:
263:
262:
257:
255:
240:
238:
237:
232:
196:
194:
193:
188:
163:
161:
160:
155:
136:
134:
133:
128:
126:
111:
109:
108:
103:
101:
71:
69:
68:
63:
1751:
1750:
1746:
1745:
1744:
1742:
1741:
1740:
1716:
1715:
1665:10.2307/1990969
1647:Mori, Shigefumi
1632:
1614:Springer-Verlag
1594:
1568:Mori, Shigefumi
1549:
1394:Mori, Shigefumi
1335:McKernan, James
1327:Birkar, Caucher
1322:
1321:
1295:
1292:
1291:
1285:
1281:
1276:
1259:
1208:
1206:
1203:
1202:
1185:
1181:
1179:
1176:
1175:
1154:
1150:
1148:
1145:
1144:
1125:
1121:
1120:
1116:
1114:
1111:
1110:
1093:
1089:
1087:
1084:
1083:
1067:
1064:
1063:
1047:
1044:
1043:
1016:
1013:
1012:
1009:Cartier divisor
987:
986:
982:
977:
974:
973:
946:
945:
941:
939:
936:
935:
914:
913:
909:
907:
904:
903:
880:
878:
875:
874:
858:
855:
854:
848:
792:
789:
788:
757:
754:
753:
721:
718:
717:
705:
699:
674:
671:
670:
654:
651:
650:
631:
629:
626:
625:
605:
602:
601:
578:
576:
573:
572:
550:
546:
545:
541:
539:
536:
535:
519:
516:
515:
496:
494:
491:
490:
459:
456:
455:
427:
424:
423:
406:
402:
397:
394:
393:
368:
348:
345:
344:
328:
325:
324:
299:
291:
288:
287:
271:
268:
267:
248:
246:
243:
242:
205:
202:
201:
173:
170:
169:
149:
146:
145:
119:
117:
114:
113:
94:
80:
77:
76:
57:
54:
53:
49:
17:
12:
11:
5:
1749:
1739:
1738:
1733:
1728:
1714:
1713:
1692:
1643:
1630:
1605:
1592:
1560:
1547:
1523:
1492:
1465:(2): 211–273,
1461:, New Series,
1445:
1417:
1382:
1347:(2): 405–468,
1320:
1319:
1302:
1299:
1278:
1277:
1275:
1272:
1271:
1270:
1265:
1258:
1255:
1247:James McKernan
1239:Caucher Birkar
1214:
1211:
1188:
1184:
1157:
1153:
1128:
1124:
1119:
1096:
1092:
1071:
1051:
1040:Shigefumi Mori
1020:
993:
990:
985:
981:
961:
958:
952:
949:
944:
920:
917:
912:
886:
883:
862:
847:
844:
811:
808:
805:
802:
799:
796:
776:
773:
770:
767:
764:
761:
737:
734:
731:
728:
725:
701:Main article:
698:
695:
678:
658:
637:
634:
622:
621:
609:
584:
581:
553:
549:
544:
523:
514:birational to
502:
499:
478:
475:
472:
469:
466:
463:
453:
431:
409:
405:
401:
378:
374:
371:
367:
364:
361:
358:
355:
352:
332:
312:
309:
305:
302:
298:
295:
275:
266:birational to
254:
251:
230:
227:
224:
221:
218:
215:
212:
209:
186:
183:
180:
177:
153:
125:
122:
100:
97:
93:
90:
87:
84:
61:
48:
45:
41:Italian school
15:
9:
6:
4:
3:
2:
1748:
1737:
1734:
1732:
1729:
1727:
1724:
1723:
1721:
1711:
1707:
1706:
1701:
1697:
1693:
1690:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1653:
1648:
1644:
1641:
1637:
1633:
1627:
1623:
1619:
1615:
1611:
1606:
1603:
1599:
1595:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1564:Kollár, János
1561:
1558:
1554:
1550:
1544:
1540:
1536:
1532:
1528:
1527:Kollár, János
1524:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1496:Kollár, János
1493:
1490:
1486:
1482:
1478:
1473:
1468:
1464:
1460:
1459:
1454:
1450:
1449:Kollár, János
1446:
1443:
1439:
1435:
1431:
1427:
1423:
1418:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1390:Kollár, János
1387:
1383:
1380:
1376:
1372:
1368:
1364:
1360:
1355:
1350:
1346:
1342:
1341:
1336:
1332:
1328:
1324:
1323:
1316:
1297:
1289:
1283:
1279:
1269:
1266:
1264:
1261:
1260:
1254:
1251:
1248:
1244:
1240:
1236:
1231:
1229:
1212:
1209:
1186:
1182:
1173:
1155:
1151:
1126:
1122:
1117:
1094:
1090:
1069:
1049:
1041:
1037:
1032:
1018:
1010:
991:
988:
983:
979:
959:
956:
950:
947:
942:
918:
915:
910:
901:
884:
881:
860:
853:
843:
840:
836:
832:
828:
823:
809:
806:
803:
800:
797:
794:
774:
771:
768:
765:
762:
759:
751:
735:
729:
726:
723:
715:
711:
704:
694:
692:
676:
656:
635:
632:
607:
599:
598:minimal model
582:
579:
570:
547:
542:
521:
500:
497:
476:
473:
467:
461:
454:
451:
450:
445:
429:
407:
403:
399:
392:
376:
372:
369:
365:
362:
359:
356:
353:
350:
330:
310:
303:
300:
296:
293:
273:
252:
249:
228:
222:
219:
213:
207:
200:
199:
198:
181:
175:
167:
151:
142:
140:
123:
120:
98:
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88:
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1421:
1397:
1354:math/0610203
1344:
1338:
1314:
1287:
1282:
1252:
1232:
1036:cone theorem
1033:
849:
838:
834:
830:
824:
749:
706:
623:
597:
447:
143:
50:
24:
18:
1228:Mori (1988)
139:isomorphism
1720:Categories
1504:Astérisque
1398:Astérisque
1274:References
343:such that
1710:EMS Press
1698:(2001) ,
1673:0894-0347
1512:0303-1179
1481:0002-9904
1434:0039-470X
1406:0303-1179
1301:∞
1298:−
957:⋅
807:−
798:⋅
772:−
763:⋅
733:→
727::
552:′
474:⩾
462:κ
400:−
389:with the
366:
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297::
226:∞
223:−
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176:κ
92:→
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1570:(1998),
1529:(1996),
1498:(1989),
1451:(1987),
1257:See also
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1007:to be a
992:′
951:′
919:′
885:′
827:contract
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74:morphism
1736:3-folds
1689:0924704
1681:1990969
1640:1875410
1602:1658959
1557:1440180
1520:1040578
1489:0903730
1442:2560253
1414:1004926
1379:2601039
1359:Bibcode
47:Outline
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1600:
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1245:, and
442:being
137:is an
23:, the
1677:JSTOR
1349:arXiv
898:with
596:is a
444:ample
1669:ISSN
1626:ISBN
1588:ISBN
1543:ISBN
1508:ISSN
1477:ISSN
1430:ISSN
1402:ISSN
1172:flip
649:and
600:for
360:<
1661:doi
1618:doi
1580:doi
1535:doi
1467:doi
1367:doi
1038:of
1031:.)
712:of
569:nef
363:dim
351:dim
19:In
1722::
1708:,
1702:,
1685:MR
1683:,
1675:,
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477:0.
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1317:.
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465:(
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331:Y
311:Y
301:X
294:f
274:X
250:X
229:.
220:=
217:)
214:X
211:(
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179:(
152:X
121:X
96:X
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83:f
60:X
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