22:
1702:
1516:
2188:
623:
1378:
2113:
1937:
2044:
2255:
406:
314:
1850:
1570:
930:
252:
869:
1386:
1073:
1539:
51:
1748:
742:
489:
1126:
1014:
363:
1967:
2287:
2071:
1775:
1291:
1185:
1153:
962:
774:
653:
279:
1777:
is nonnegative, we know nothing, but in the complementary half-space, the cone is spanned by some countable collection of curves which are quite special: they are
2228:
2208:
1990:
1870:
1799:
1562:
1311:
1251:
1101:
1038:
989:
889:
833:
813:
673:
516:
449:
429:
338:
203:
175:
147:
119:
706:
2122:
524:
1316:
968:
1199:
The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the
2325:
2311:
73:
2079:
1875:
2345:
409:
34:
2340:
971:
of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety
44:
38:
30:
2002:
1969:
is ample. So the cone theorem shows that the cone of curves of a Fano variety is generated by rational curves.
1188:
55:
2233:
1697:{\displaystyle {\overline {NE(X)}}={\overline {NE(X)}}_{K_{X}+\epsilon H\geq 0}+\sum \mathbf {R} _{\geq 0},}
122:
1992:
is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the
284:
1804:
894:
1076:
777:
368:
1187:. This process encounters difficulties, however, whose resolution necessitates the introduction of the
211:
1204:
2290:
1511:{\displaystyle {\overline {NE(X)}}={\overline {NE(X)}}_{K_{X}\geq 0}+\sum _{i}\mathbf {R} _{\geq 0}.}
1224:
842:
126:
1254:
206:
1216:
1524:
2321:
2307:
1717:
1711:
1041:
836:
711:
458:
452:
99:
1942:
1778:
1046:
2260:
2049:
1753:
1269:
1158:
1131:
935:
747:
631:
257:
1212:
178:
1106:
994:
343:
2213:
2193:
1975:
1855:
1784:
1547:
1296:
1264:
1261:
1236:
1208:
1086:
1080:
1023:
974:
874:
818:
798:
790:
658:
501:
434:
414:
323:
188:
160:
132:
104:
678:
2334:
1542:
2183:{\displaystyle (\operatorname {cont} _{F})_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{Z}}
1801:. The second assertion then tells us more: it says that, away from the hyperplane
1083:
and others) was to construct (at least morally) the necessary birational map from
967:
A more involved example is the role played by the cone of curves in the theory of
87:
1220:
1017:
932:, the closure of the cone of curves in the usual real topology. (In general,
618:{\displaystyle NE(X)=\left\{\sum a_{i},\ 0\leq a_{i}\in \mathbb {R} \right\}}
1128:
as a sequence of steps, each of which can be thought of as contraction of a
2074:
1373:{\displaystyle 0<-K_{X}\cdot C_{i}\leq \operatorname {dim} X+1}
788:
One useful application of the notion of the cone of curves is the
1781:, and their 'degree' is bounded very tightly by the dimension of
964:
need not be closed, so taking the closure here is important.)
15:
1211:; it was later generalised to a larger class of varieties by
2169:
2152:
1227:, and others. Mori's version of the theorem is as follows:
2108:{\displaystyle \operatorname {cont} _{F}:X\rightarrow Z}
320:
of 1-cycles is defined by intersections: two 1-cycles
2263:
2236:
2216:
2196:
2125:
2082:
2052:
2005:
1978:
1945:
1878:
1858:
1807:
1787:
1756:
1720:
1573:
1550:
1527:
1389:
1319:
1299:
1272:
1239:
1161:
1134:
1109:
1089:
1049:
1026:
997:
977:
938:
897:
877:
845:
821:
801:
750:
714:
681:
661:
634:
527:
504:
461:
437:
417:
371:
346:
326:
287:
260:
214:
191:
163:
135:
107:
1852:, extremal rays of the cone cannot accumulate. When
1079:. The great breakthrough of the early 1980s (due to
2046:be an extremal face of the cone of curves on which
1932:{\displaystyle {\overline {NE(X)}}_{K_{X}\geq 0}=0}
2281:
2249:
2222:
2202:
2182:
2107:
2065:
2038:
1984:
1961:
1931:
1864:
1844:
1793:
1769:
1742:
1696:
1556:
1533:
1510:
1372:
1305:
1285:
1245:
1179:
1147:
1120:
1095:
1067:
1032:
1008:
983:
956:
924:
883:
863:
827:
807:
768:
736:
700:
667:
647:
617:
510:
483:
443:
423:
400:
357:
332:
308:
273:
246:
197:
169:
141:
113:
43:but its sources remain unclear because it lacks
8:
2039:{\displaystyle F\subset {\overline {NE(X)}}}
1839:
1808:
455:of 1-cycles modulo numerical equivalence by
655:are irreducible, reduced, proper curves on
2318:Birational Geometry of Algebraic Varieties
1707:where the sum in the last term is finite.
254:of irreducible, reduced and proper curves
2262:
2250:{\displaystyle \operatorname {cont} _{F}}
2241:
2235:
2215:
2195:
2174:
2168:
2167:
2157:
2151:
2150:
2143:
2133:
2124:
2087:
2081:
2057:
2051:
2012:
2004:
1977:
1953:
1944:
1909:
1904:
1880:
1877:
1857:
1821:
1806:
1786:
1761:
1755:
1725:
1719:
1682:
1666:
1661:
1631:
1626:
1602:
1574:
1572:
1549:
1526:
1496:
1480:
1475:
1468:
1447:
1442:
1418:
1390:
1388:
1346:
1333:
1318:
1298:
1277:
1271:
1238:
1203:. The first version of this theorem, for
1160:
1139:
1133:
1108:
1088:
1054:
1048:
1025:
996:
976:
937:
898:
896:
876:
844:
820:
800:
749:
719:
713:
689:
680:
660:
639:
633:
606:
605:
596:
571:
558:
526:
503:
466:
460:
436:
416:
370:
345:
325:
302:
301:
292:
286:
265:
259:
238:
228:
213:
190:
162:
134:
106:
74:Learn how and when to remove this message
1710:The first assertion says that, in the
795:, which says that a (Cartier) divisor
309:{\displaystyle a_{i}\in \mathbb {R} }
7:
2320:, Cambridge University Press, 1998.
2073:is negative. Then there is a unique
1845:{\displaystyle \{C:K_{X}\cdot C=0\}}
991:, find a (mildly singular) variety
925:{\displaystyle {\overline {NE(X)}}}
2304:Positivity in Algebraic Geometry I
744:. It is not difficult to see that
401:{\displaystyle C\cdot D=C'\cdot D}
14:
780:in the sense of convex geometry.
247:{\displaystyle C=\sum a_{i}C_{i}}
181:variety. By definition, a (real)
1662:
1521:2. For any positive real number
1476:
20:
2270:
2264:
2140:
2126:
2099:
2027:
2021:
1895:
1889:
1737:
1731:
1688:
1675:
1617:
1611:
1589:
1583:
1502:
1489:
1433:
1427:
1405:
1399:
1174:
1168:
951:
945:
913:
907:
763:
757:
731:
725:
695:
682:
577:
564:
540:
534:
478:
472:
365:are numerically equivalent if
1:
864:{\displaystyle D\cdot x>0}
2031:
1899:
1621:
1593:
1437:
1409:
917:
1972:If in addition the variety
2362:
1155:-negative extremal ray of
2316:Kollár, J. and Mori, S.,
2306:, Springer-Verlag, 2004.
2190:and an irreducible curve
1534:{\displaystyle \epsilon }
2230:is mapped to a point by
2115:to a projective variety
1750:where intersection with
1743:{\displaystyle N_{1}(X)}
871:for any nonzero element
737:{\displaystyle N_{1}(X)}
484:{\displaystyle N_{1}(X)}
29:This article includes a
123:combinatorial invariant
58:more precise citations.
2283:
2251:
2224:
2204:
2184:
2109:
2067:
2040:
1986:
1963:
1962:{\displaystyle -K_{X}}
1933:
1866:
1846:
1795:
1771:
1744:
1698:
1558:
1535:
1512:
1374:
1307:
1287:
1247:
1181:
1149:
1122:
1097:
1069:
1068:{\displaystyle K_{X'}}
1034:
1010:
985:
958:
926:
885:
865:
829:
815:on a complete variety
809:
770:
738:
702:
669:
649:
619:
512:
485:
445:
425:
402:
359:
334:
310:
275:
248:
199:
171:
143:
115:
2284:
2282:{\displaystyle \in F}
2252:
2225:
2205:
2185:
2110:
2068:
2066:{\displaystyle K_{X}}
2041:
1987:
1964:
1934:
1867:
1847:
1796:
1772:
1770:{\displaystyle K_{X}}
1745:
1699:
1559:
1536:
1513:
1375:
1308:
1288:
1286:{\displaystyle C_{i}}
1248:
1182:
1180:{\displaystyle NE(X)}
1150:
1148:{\displaystyle K_{X}}
1123:
1098:
1070:
1035:
1011:
986:
959:
957:{\displaystyle NE(X)}
927:
886:
866:
830:
810:
771:
769:{\displaystyle NE(X)}
739:
703:
670:
650:
648:{\displaystyle C_{i}}
620:
513:
486:
446:
426:
403:
360:
335:
318:Numerical equivalence
311:
276:
274:{\displaystyle C_{i}}
249:
200:
172:
144:
125:of importance to the
116:
2291:contraction morphism
2261:
2234:
2214:
2194:
2123:
2080:
2050:
2003:
1976:
1943:
1876:
1872:is a Fano variety,
1856:
1805:
1785:
1754:
1718:
1571:
1548:
1525:
1387:
1317:
1297:
1270:
1237:
1159:
1132:
1107:
1087:
1047:
1024:
995:
975:
936:
895:
875:
843:
819:
799:
748:
712:
679:
659:
632:
525:
502:
459:
435:
415:
369:
344:
324:
285:
281:, with coefficients
258:
212:
189:
161:
133:
105:
2346:Birational geometry
1994:Contraction Theorem
1195:A structure theorem
127:birational geometry
2341:Algebraic geometry
2279:
2247:
2220:
2200:
2180:
2105:
2063:
2036:
1982:
1959:
1929:
1862:
1842:
1791:
1767:
1740:
1694:
1554:
1531:
1508:
1473:
1370:
1303:
1283:
1255:projective variety
1243:
1177:
1145:
1121:{\displaystyle X'}
1118:
1093:
1065:
1030:
1009:{\displaystyle X'}
1006:
981:
954:
922:
881:
861:
825:
805:
766:
734:
698:
665:
645:
615:
508:
481:
441:
421:
408:for every Cartier
398:
358:{\displaystyle C'}
355:
330:
306:
271:
244:
207:linear combination
195:
167:
139:
111:
31:list of references
2223:{\displaystyle X}
2203:{\displaystyle C}
2034:
1985:{\displaystyle X}
1902:
1865:{\displaystyle X}
1794:{\displaystyle X}
1712:closed half-space
1624:
1596:
1557:{\displaystyle H}
1464:
1440:
1412:
1306:{\displaystyle X}
1246:{\displaystyle X}
1096:{\displaystyle X}
1042:canonical divisor
1033:{\displaystyle X}
984:{\displaystyle X}
920:
884:{\displaystyle x}
828:{\displaystyle X}
808:{\displaystyle D}
708:their classes in
668:{\displaystyle X}
585:
511:{\displaystyle X}
453:real vector space
444:{\displaystyle X}
424:{\displaystyle D}
333:{\displaystyle C}
198:{\displaystyle X}
170:{\displaystyle X}
142:{\displaystyle X}
114:{\displaystyle X}
100:algebraic variety
84:
83:
76:
2353:
2302:Lazarsfeld, R.,
2288:
2286:
2285:
2280:
2256:
2254:
2253:
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2229:
2227:
2226:
2221:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2179:
2178:
2173:
2172:
2162:
2161:
2156:
2155:
2148:
2147:
2138:
2137:
2114:
2112:
2111:
2106:
2092:
2091:
2072:
2070:
2069:
2064:
2062:
2061:
2045:
2043:
2042:
2037:
2035:
2030:
2013:
1991:
1989:
1988:
1983:
1968:
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1741:
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1703:
1701:
1700:
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1603:
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1408:
1391:
1379:
1377:
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1371:
1351:
1350:
1338:
1337:
1312:
1310:
1309:
1304:
1292:
1290:
1289:
1284:
1282:
1281:
1252:
1250:
1249:
1244:
1205:smooth varieties
1186:
1184:
1183:
1178:
1154:
1152:
1151:
1146:
1144:
1143:
1127:
1125:
1124:
1119:
1117:
1102:
1100:
1099:
1094:
1074:
1072:
1071:
1066:
1064:
1063:
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1039:
1037:
1036:
1031:
1015:
1013:
1012:
1007:
1005:
990:
988:
987:
982:
963:
961:
960:
955:
931:
929:
928:
923:
921:
916:
899:
890:
888:
887:
882:
870:
868:
867:
862:
834:
832:
831:
826:
814:
812:
811:
806:
775:
773:
772:
767:
743:
741:
740:
735:
724:
723:
707:
705:
704:
701:{\displaystyle }
699:
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693:
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672:
671:
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654:
652:
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72:
68:
65:
59:
54:this article by
45:inline citations
24:
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1265:rational curves
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1234:
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839:if and only if
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94:(sometimes the
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35:related reading
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494:We define the
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1200:
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784:Applications
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96:Kleiman-Mori
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50:Please help
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778:convex cone
88:mathematics
56:introducing
2335:Categories
2297:References
1018:birational
628:where the
153:Definition
2274:∈
2145:∗
2100:→
2032:¯
2010:⊂
1947:−
1916:≥
1900:¯
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1016:which is
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850:⋅
793:condition
603:∈
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376:⋅
299:∈
222:∑
64:June 2020
2075:morphism
1939:because
1779:rational
1541:and any
1225:Shokurov
1213:Kawamata
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1999:3. Let
1380:, and
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791:Kleiman
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183:1-cycle
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675:, and
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179:proper
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177:be a
121:is a
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2322:ISBN
2308:ISBN
2239:cont
2131:cont
2085:cont
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