574:) whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension. The existence of log flips in higher dimensions has been settled by (Caucher Birkar, Paolo Cascini & Christopher D. Hacon et al.
561:
Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved, then the minimal model program can be carried out. The existence of flips for 3-folds was proved by
862:
165:
65:. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.
700:
1421:
478:
746:
1642:
904:
734:
556:
1450:
1369:
346:
616:
302:
233:
199:
1779:
1732:
1759:
1712:
1685:
1558:
1531:
1504:
1477:
1340:
1287:
1233:
1180:
1131:
1053:
931:
505:
427:
400:
268:
978:
2129:
1799:
1260:
1206:
1153:
1104:
1076:
1026:
1002:
955:
366:
97:
2053:
1932:
1862:
578:). On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.
105:
2031:
2091:
2009:
636:
1382:
2185:
432:
305:
857:{\displaystyle f^{+}\colon X^{+}=\operatorname {Proj} {\big (}\bigoplus _{m}f_{*}({\mathcal {O}}_{X}(mK)){\big )}\to Y}
2180:
566:. The existence of log flips, a more general kind of flip, in dimension three and four were proved by Shokurov (
2001:
62:
1577:
1183:
878:
708:
236:
74:
54:
1781:
of flips of varieties with
Kawamata log terminal singularities, projective over a fixed normal variety
1079:
510:
1426:
1345:
311:
1881:
1822:
1815:
Proceedings of the Royal
Society of London. Series A: Mathematical, Physical and Engineering Sciences
100:
589:
2159:
2147:
2072:
1871:
38:
1989:
1970:
1951:
243:), which is the desired result. The major technical problem is that, at some stage, the variety
273:
204:
170:
2027:
2005:
1897:
1853:
240:
1764:
2062:
1889:
1830:
1717:
2140:
2084:
2041:
1982:
1963:
1909:
1842:
1737:
1690:
1663:
1536:
1509:
1482:
1455:
1313:
1265:
1211:
1158:
1109:
1031:
909:
483:
405:
378:
246:
2136:
2080:
2037:
2023:
1978:
1959:
1905:
1838:
31:
17:
1885:
1826:
960:
2114:
2095:
2048:
1993:
1857:
1849:
1810:
1784:
1245:
1191:
1138:
1089:
1061:
1011:
987:
940:
351:
82:
50:
2067:
2174:
1956:
Proceedings of the
International Congress of Mathematicians, Vol. I, II (Kyoto, 1990)
1916:
79:
The minimal model program can be summarised very briefly as follows: given a variety
2135:, Adv. Stud. Pure Math., vol. 1, Amsterdam: North-Holland, pp. 131–180,
1893:
1920:
2108:
58:
1901:
160:{\displaystyle X=X_{1}\rightarrow X_{2}\rightarrow \cdots \rightarrow X_{n}}
2152:
Three-dimensional log flips. With an appendix in
English by Yujiro Kawamata
1834:
875:. If the relative canonical ring is finitely generated (as an algebra over
1860:(2010), "Existence of minimal models for varieties of log general type",
2051:(1988), "Flip theorem and the existence of minimal models for 3-folds",
1876:
2076:
1008:
is relatively trivial. (Sometimes the induced birational morphism from
1379:
at the origin. The exceptional locus of this blowup is isomorphic to
167:, each of which contracts some curves on which the canonical divisor
270:
may become 'too singular', in the sense that the canonical divisor
1660:
More precisely, there is a conjecture stating that every sequence
2154:, vol. 1, Russian Acad. Sci. Izv. Math. 40, pp. 95–202.
1925:
429:
is a birational map (in fact an isomorphism in codimension 1)
1082:
of an extremal ray, which implies several extra properties:
885:
815:
715:
666:
480:
to a variety whose singularities are 'better' than those of
695:{\displaystyle \bigoplus _{m}f_{*}({\mathcal {O}}_{X}(mK))}
575:
2133:
Algebraic varieties and analytic varieties (Tokyo, 1981)
2103:, Algebraic Geometry and Beyond, RIMS, Kyoto University
1416:{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}}
1975:
Surveys in differential geometry (Cambridge, MA, 1990)
2117:
1787:
1767:
1740:
1720:
1693:
1666:
1580:
1539:
1512:
1485:
1458:
1429:
1385:
1348:
1316:
1268:
1248:
1214:
1194:
1161:
1141:
1112:
1092:
1064:
1034:
1014:
990:
963:
943:
912:
881:
749:
711:
639:
592:
513:
486:
435:
408:
381:
354:
314:
276:
249:
207:
173:
108:
85:
1813:(1958), "On analytic surfaces with double points",
871:along the relative canonical ring is a morphism to
2123:
1793:
1773:
1753:
1726:
1706:
1679:
1636:
1552:
1525:
1498:
1471:
1444:
1415:
1363:
1334:
1281:
1254:
1227:
1200:
1174:
1147:
1125:
1098:
1070:
1047:
1020:
996:
972:
949:
925:
898:
856:
728:
694:
610:
550:
499:
473:{\displaystyle f\colon X_{i}\rightarrow X_{i}^{+}}
472:
421:
394:
371:The (conjectural) solution to this problem is the
360:
340:
296:
262:
227:
193:
159:
91:
2166:, Proc. Steklov Inst. Math. 240, pp. 75–213.
1977:, Bethlehem, PA: Lehigh Univ., pp. 113–199,
705:and is a sheaf of graded algebras over the sheaf
1570:, a generalization of Atiyah's flop replacing
1973:(1991), "Flips, flops, minimal models, etc",
1958:, Tokyo: Math. Soc. Japan, pp. 709–714,
843:
784:
8:
2054:Journal of the American Mathematical Society
1933:Notices of the American Mathematical Society
1863:Journal of the American Mathematical Society
1998:Birational Geometry of Algebraic Varieties
1298:The first example of a flop, known as the
27:Surgery operation in minimal model program
2116:
2066:
1875:
1786:
1766:
1745:
1739:
1719:
1698:
1692:
1671:
1665:
1625:
1603:
1579:
1544:
1538:
1517:
1511:
1490:
1484:
1463:
1457:
1436:
1432:
1431:
1428:
1407:
1403:
1402:
1392:
1388:
1387:
1384:
1355:
1351:
1350:
1347:
1315:
1273:
1267:
1247:
1219:
1213:
1193:
1166:
1160:
1140:
1117:
1111:
1091:
1063:
1039:
1033:
1013:
989:
962:
942:
917:
911:
890:
884:
883:
880:
842:
841:
820:
814:
813:
803:
793:
783:
782:
767:
754:
748:
720:
714:
713:
710:
671:
665:
664:
654:
644:
638:
591:
542:
537:
518:
512:
491:
485:
464:
459:
446:
434:
413:
407:
386:
380:
353:
324:
319:
313:
286:
281:
275:
254:
248:
217:
212:
206:
183:
178:
172:
151:
132:
119:
107:
84:
1452:in two different ways, giving varieties
571:
567:
1653:
2097:Flops, flips, and matrix factorization
1303:
1182:only have mild singularities, such as
626:, then the relative canonical ring of
2111:(1983), "Minimal models of canonical
1801:terminates after finitely many steps.
1637:{\displaystyle xy=(z+w^{k})(z-w^{k})}
239:(at least in the case of nonnegative
7:
1563:
563:
2022:, Universitext, Berlin, New York:
1506:. The natural birational map from
1086:The exceptional sets of both maps
899:{\displaystyle {\mathcal {O}}_{Y}}
729:{\displaystyle {\mathcal {O}}_{Y}}
25:
2068:10.1090/s0894-0347-1988-0924704-x
1239:, which is normal and projective.
551:{\displaystyle X_{i+1}=X_{i}^{+}}
2020:Introduction to the Mori program
1445:{\displaystyle \mathbb {P} ^{1}}
1364:{\displaystyle \mathbb {A} ^{4}}
341:{\displaystyle K_{X_{i}}\cdot C}
1631:
1612:
1609:
1590:
1235:are birational morphisms onto
848:
838:
835:
826:
809:
689:
686:
677:
660:
611:{\displaystyle f\colon X\to Y}
602:
452:
144:
138:
125:
1:
1894:10.1090/S0894-0347-09-00649-3
1289:are numerically proportional.
980:is relatively ample, and the
308:, so the intersection number
99:, we construct a sequence of
1242:All curves in the fibers of
1133:have codimension at least 2,
558:, and continue the process.
1423:, and can be blown down to
1055:is called a flip or flop.)
622:is the canonical bundle of
2202:
2002:Cambridge University Press
72:
53:operations arising in the
29:
1954:(1991), "Flip and flop",
297:{\displaystyle K_{X_{i}}}
228:{\displaystyle K_{X_{n}}}
201:is negative. Eventually,
194:{\displaystyle K_{X_{i}}}
69:The minimal model program
18:Flop (algebraic geometry)
736:of regular functions on
30:Not to be confused with
2160:Shokurov, Vyacheslav V.
2148:Shokurov, Vyacheslav V.
2018:Matsuki, Kenji (2002),
1811:Atiyah, Michael Francis
1774:{\displaystyle \cdots }
63:relative canonical ring
2125:
1835:10.1098/rspa.1958.0181
1795:
1775:
1755:
1728:
1727:{\displaystyle \dots }
1708:
1681:
1638:
1554:
1527:
1500:
1473:
1446:
1417:
1365:
1336:
1283:
1256:
1229:
1202:
1184:terminal singularities
1176:
1149:
1127:
1100:
1072:
1049:
1022:
998:
974:
951:
927:
900:
858:
730:
696:
612:
552:
501:
474:
423:
402:as above, the flip of
396:
375:. Given a problematic
362:
342:
298:
264:
229:
195:
161:
93:
2126:
1854:Hacon, Christopher D.
1796:
1776:
1756:
1754:{\displaystyle X_{n}}
1729:
1709:
1707:{\displaystyle X_{1}}
1682:
1680:{\displaystyle X_{0}}
1639:
1555:
1553:{\displaystyle X_{2}}
1528:
1526:{\displaystyle X_{1}}
1501:
1499:{\displaystyle X_{2}}
1474:
1472:{\displaystyle X_{1}}
1447:
1418:
1366:
1337:
1335:{\displaystyle xy=zw}
1284:
1282:{\displaystyle f^{+}}
1257:
1230:
1228:{\displaystyle f^{+}}
1203:
1177:
1175:{\displaystyle X^{+}}
1150:
1128:
1126:{\displaystyle f^{+}}
1101:
1073:
1050:
1048:{\displaystyle X^{+}}
1023:
999:
975:
952:
928:
926:{\displaystyle f^{+}}
906:) then the morphism
901:
859:
731:
697:
613:
553:
502:
500:{\displaystyle X_{i}}
475:
424:
422:{\displaystyle X_{i}}
397:
395:{\displaystyle X_{i}}
368:is not even defined.
363:
343:
299:
265:
263:{\displaystyle X_{i}}
230:
196:
162:
94:
75:Minimal model program
55:minimal model program
2115:
1785:
1765:
1738:
1718:
1691:
1664:
1578:
1560:is the Atiyah flop.
1537:
1510:
1483:
1456:
1427:
1383:
1346:
1314:
1266:
1246:
1212:
1192:
1159:
1139:
1110:
1090:
1062:
1032:
1012:
988:
961:
941:
910:
879:
747:
709:
637:
590:
511:
484:
433:
406:
379:
352:
312:
274:
247:
205:
171:
106:
83:
2186:Birational geometry
1921:"What Is...a Flip?"
1886:2010JAMS...23..405B
1827:1958RSPSA.247..237A
618:is a morphism, and
547:
469:
49:are codimension-2
2181:Algebraic geometry
2121:
1852:; Cascini, Paolo;
1791:
1771:
1751:
1724:
1704:
1677:
1634:
1550:
1523:
1496:
1469:
1442:
1413:
1361:
1332:
1279:
1252:
1225:
1198:
1172:
1145:
1123:
1096:
1068:
1045:
1018:
994:
973:{\displaystyle -K}
970:
947:
923:
896:
854:
798:
726:
692:
649:
608:
548:
533:
497:
470:
455:
419:
392:
358:
338:
294:
260:
225:
191:
157:
89:
39:algebraic geometry
2164:Prelimiting flips
2124:{\displaystyle 3}
2033:978-0-387-98465-0
1919:(December 2004),
1821:(1249): 237–244,
1794:{\displaystyle Z}
1375:be the blowup of
1255:{\displaystyle f}
1201:{\displaystyle f}
1148:{\displaystyle X}
1099:{\displaystyle f}
1080:small contraction
1071:{\displaystyle f}
1058:In applications,
1021:{\displaystyle X}
997:{\displaystyle f}
950:{\displaystyle f}
789:
640:
361:{\displaystyle C}
241:Kodaira dimension
92:{\displaystyle X}
16:(Redirected from
2193:
2167:
2155:
2143:
2130:
2128:
2127:
2122:
2104:
2102:
2087:
2070:
2044:
2014:
1985:
1966:
1947:
1946:
1945:
1929:
1912:
1879:
1845:
1802:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1760:
1758:
1757:
1752:
1750:
1749:
1733:
1731:
1730:
1725:
1713:
1711:
1710:
1705:
1703:
1702:
1686:
1684:
1683:
1678:
1676:
1675:
1658:
1643:
1641:
1640:
1635:
1630:
1629:
1608:
1607:
1574:by the zeros of
1559:
1557:
1556:
1551:
1549:
1548:
1532:
1530:
1529:
1524:
1522:
1521:
1505:
1503:
1502:
1497:
1495:
1494:
1478:
1476:
1475:
1470:
1468:
1467:
1451:
1449:
1448:
1443:
1441:
1440:
1435:
1422:
1420:
1419:
1414:
1412:
1411:
1406:
1397:
1396:
1391:
1370:
1368:
1367:
1362:
1360:
1359:
1354:
1341:
1339:
1338:
1333:
1310:be the zeros of
1302:, was found in (
1288:
1286:
1285:
1280:
1278:
1277:
1261:
1259:
1258:
1253:
1234:
1232:
1231:
1226:
1224:
1223:
1207:
1205:
1204:
1199:
1181:
1179:
1178:
1173:
1171:
1170:
1154:
1152:
1151:
1146:
1132:
1130:
1129:
1124:
1122:
1121:
1105:
1103:
1102:
1097:
1077:
1075:
1074:
1069:
1054:
1052:
1051:
1046:
1044:
1043:
1027:
1025:
1024:
1019:
1003:
1001:
1000:
995:
979:
977:
976:
971:
956:
954:
953:
948:
932:
930:
929:
924:
922:
921:
905:
903:
902:
897:
895:
894:
889:
888:
863:
861:
860:
855:
847:
846:
825:
824:
819:
818:
808:
807:
797:
788:
787:
772:
771:
759:
758:
735:
733:
732:
727:
725:
724:
719:
718:
701:
699:
698:
693:
676:
675:
670:
669:
659:
658:
648:
617:
615:
614:
609:
557:
555:
554:
549:
546:
541:
529:
528:
507:. So we can put
506:
504:
503:
498:
496:
495:
479:
477:
476:
471:
468:
463:
451:
450:
428:
426:
425:
420:
418:
417:
401:
399:
398:
393:
391:
390:
367:
365:
364:
359:
347:
345:
344:
339:
331:
330:
329:
328:
303:
301:
300:
295:
293:
292:
291:
290:
269:
267:
266:
261:
259:
258:
234:
232:
231:
226:
224:
223:
222:
221:
200:
198:
197:
192:
190:
189:
188:
187:
166:
164:
163:
158:
156:
155:
137:
136:
124:
123:
98:
96:
95:
90:
21:
2201:
2200:
2196:
2195:
2194:
2192:
2191:
2190:
2171:
2170:
2158:
2146:
2113:
2112:
2107:
2100:
2092:Morrison, David
2090:
2049:Mori, Shigefumi
2047:
2034:
2024:Springer-Verlag
2017:
2012:
1994:Mori, Shigefumi
1988:
1969:
1950:
1943:
1941:
1940:(11): 1350–1351
1923:
1915:
1877:math.AG/0610203
1858:McKernan, James
1850:Birkar, Caucher
1848:
1809:
1806:
1805:
1783:
1782:
1763:
1762:
1741:
1736:
1735:
1716:
1715:
1694:
1689:
1688:
1667:
1662:
1661:
1659:
1655:
1650:
1621:
1599:
1576:
1575:
1540:
1535:
1534:
1513:
1508:
1507:
1486:
1481:
1480:
1459:
1454:
1453:
1430:
1425:
1424:
1401:
1386:
1381:
1380:
1349:
1344:
1343:
1312:
1311:
1296:
1269:
1264:
1263:
1244:
1243:
1215:
1210:
1209:
1190:
1189:
1162:
1157:
1156:
1137:
1136:
1113:
1108:
1107:
1088:
1087:
1060:
1059:
1035:
1030:
1029:
1010:
1009:
986:
985:
959:
958:
939:
938:
913:
908:
907:
882:
877:
876:
812:
799:
763:
750:
745:
744:
712:
707:
706:
663:
650:
635:
634:
588:
587:
584:
514:
509:
508:
487:
482:
481:
442:
431:
430:
409:
404:
403:
382:
377:
376:
350:
349:
320:
315:
310:
309:
306:Cartier divisor
304:is no longer a
282:
277:
272:
271:
250:
245:
244:
213:
208:
203:
202:
179:
174:
169:
168:
147:
128:
115:
104:
103:
81:
80:
77:
71:
35:
32:Flip (geometry)
28:
23:
22:
15:
12:
11:
5:
2199:
2197:
2189:
2188:
2183:
2173:
2172:
2169:
2168:
2156:
2144:
2120:
2105:
2088:
2061:(1): 117–253,
2045:
2032:
2015:
2010:
1986:
1967:
1948:
1917:Corti, Alessio
1913:
1870:(2): 405–468,
1846:
1804:
1803:
1790:
1770:
1748:
1744:
1723:
1701:
1697:
1674:
1670:
1652:
1651:
1649:
1646:
1633:
1628:
1624:
1620:
1617:
1614:
1611:
1606:
1602:
1598:
1595:
1592:
1589:
1586:
1583:
1547:
1543:
1520:
1516:
1493:
1489:
1466:
1462:
1439:
1434:
1410:
1405:
1400:
1395:
1390:
1358:
1353:
1331:
1328:
1325:
1322:
1319:
1295:
1292:
1291:
1290:
1276:
1272:
1251:
1240:
1222:
1218:
1197:
1187:
1169:
1165:
1144:
1134:
1120:
1116:
1095:
1067:
1042:
1038:
1017:
993:
969:
966:
946:
933:is called the
920:
916:
893:
887:
865:
864:
853:
850:
845:
840:
837:
834:
831:
828:
823:
817:
811:
806:
802:
796:
792:
786:
781:
778:
775:
770:
766:
762:
757:
753:
740:. The blowup
723:
717:
703:
702:
691:
688:
685:
682:
679:
674:
668:
662:
657:
653:
647:
643:
607:
604:
601:
598:
595:
583:
580:
545:
540:
536:
532:
527:
524:
521:
517:
494:
490:
467:
462:
458:
454:
449:
445:
441:
438:
416:
412:
389:
385:
357:
337:
334:
327:
323:
318:
289:
285:
280:
257:
253:
235:should become
220:
216:
211:
186:
182:
177:
154:
150:
146:
143:
140:
135:
131:
127:
122:
118:
114:
111:
88:
73:Main article:
70:
67:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2198:
2187:
2184:
2182:
2179:
2178:
2176:
2165:
2161:
2157:
2153:
2149:
2145:
2142:
2138:
2134:
2118:
2110:
2106:
2099:
2098:
2093:
2089:
2086:
2082:
2078:
2074:
2069:
2064:
2060:
2056:
2055:
2050:
2046:
2043:
2039:
2035:
2029:
2025:
2021:
2016:
2013:
2011:0-521-63277-3
2007:
2003:
1999:
1995:
1991:
1990:Kollár, János
1987:
1984:
1980:
1976:
1972:
1971:Kollár, János
1968:
1965:
1961:
1957:
1953:
1952:Kollár, János
1949:
1939:
1935:
1934:
1927:
1922:
1918:
1914:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1878:
1873:
1869:
1865:
1864:
1859:
1855:
1851:
1847:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1807:
1788:
1768:
1746:
1742:
1721:
1699:
1695:
1672:
1668:
1657:
1654:
1647:
1645:
1626:
1622:
1618:
1615:
1604:
1600:
1596:
1593:
1587:
1584:
1581:
1573:
1569:
1568:Reid's pagoda
1565:
1561:
1545:
1541:
1518:
1514:
1491:
1487:
1464:
1460:
1437:
1408:
1398:
1393:
1378:
1374:
1356:
1329:
1326:
1323:
1320:
1317:
1309:
1305:
1301:
1293:
1274:
1270:
1249:
1241:
1238:
1220:
1216:
1195:
1188:
1185:
1167:
1163:
1142:
1135:
1118:
1114:
1093:
1085:
1084:
1083:
1081:
1065:
1056:
1040:
1036:
1015:
1007:
991:
983:
967:
964:
944:
936:
918:
914:
891:
874:
870:
851:
832:
829:
821:
804:
800:
794:
790:
779:
776:
773:
768:
764:
760:
755:
751:
743:
742:
741:
739:
721:
683:
680:
672:
655:
651:
645:
641:
633:
632:
631:
629:
625:
621:
605:
599:
596:
593:
581:
579:
577:
573:
569:
565:
559:
543:
538:
534:
530:
525:
522:
519:
515:
492:
488:
465:
460:
456:
447:
443:
439:
436:
414:
410:
387:
383:
374:
369:
355:
348:with a curve
335:
332:
325:
321:
316:
307:
287:
283:
278:
255:
251:
242:
238:
218:
214:
209:
184:
180:
175:
152:
148:
141:
133:
129:
120:
116:
112:
109:
102:
86:
76:
68:
66:
64:
60:
56:
52:
48:
44:
40:
33:
19:
2163:
2151:
2132:
2096:
2058:
2052:
2019:
1997:
1974:
1955:
1942:, retrieved
1937:
1931:
1867:
1861:
1818:
1814:
1656:
1571:
1567:
1562:
1376:
1372:
1307:
1299:
1297:
1236:
1078:is often a
1057:
1005:
981:
934:
872:
868:
866:
737:
704:
627:
623:
619:
585:
560:
372:
370:
101:contractions
78:
46:
42:
36:
2109:Reid, Miles
1566:introduced
1564:Reid (1983)
1304:Atiyah 1958
1300:Atiyah flop
564:Mori (1988)
57:, given by
2175:Categories
1944:2008-01-17
1648:References
1371:, and let
582:Definition
59:blowing up
2131:-folds",
1902:0894-0347
1769:⋯
1722:…
1619:−
1399:×
965:−
849:→
805:∗
791:⨁
780:
761::
656:∗
642:⨁
603:→
597::
453:→
440::
333:⋅
145:→
142:⋯
139:→
126:→
2162:(2003),
2150:(1993),
2094:(2005),
1996:(1998),
1294:Examples
61:along a
2141:0715649
2085:0924704
2077:1990969
2042:1875410
1983:1144527
1964:1159257
1910:2601039
1882:Bibcode
1843:0095974
1823:Bibcode
1306:). Let
51:surgery
2139:
2083:
2075:
2040:
2030:
2008:
1981:
1962:
1908:
1900:
1841:
2101:(PDF)
2073:JSTOR
1872:arXiv
47:flops
45:and
43:flips
2028:ISBN
2006:ISBN
1898:ISSN
1479:and
1262:and
1208:and
1155:and
1106:and
982:flop
935:flip
777:Proj
576:2010
572:2003
568:1993
373:flip
2063:doi
1926:PDF
1890:doi
1831:doi
1819:247
1533:to
1342:in
1028:to
1004:if
984:of
957:if
937:of
867:of
630:is
586:If
237:nef
37:In
2177::
2137:MR
2081:MR
2079:,
2071:,
2057:,
2038:MR
2036:,
2026:,
2004:,
2000:,
1992:;
1979:MR
1960:MR
1938:51
1936:,
1930:,
1906:MR
1904:,
1896:,
1888:,
1880:,
1868:23
1866:,
1856:;
1839:MR
1837:,
1829:,
1817:,
1761:⇢
1734:⇢
1714:⇢
1687:⇢
1644:.
570:,
41:,
2119:3
2065::
2059:1
1928:)
1924:(
1892::
1884::
1874::
1833::
1825::
1789:Z
1747:n
1743:X
1700:1
1696:X
1673:0
1669:X
1632:)
1627:k
1623:w
1616:z
1613:(
1610:)
1605:k
1601:w
1597:+
1594:z
1591:(
1588:=
1585:y
1582:x
1572:Y
1546:2
1542:X
1519:1
1515:X
1492:2
1488:X
1465:1
1461:X
1438:1
1433:P
1409:1
1404:P
1394:1
1389:P
1377:Y
1373:V
1357:4
1352:A
1330:w
1327:z
1324:=
1321:y
1318:x
1308:Y
1275:+
1271:f
1250:f
1237:Y
1221:+
1217:f
1196:f
1186:.
1168:+
1164:X
1143:X
1119:+
1115:f
1094:f
1066:f
1041:+
1037:X
1016:X
1006:K
992:f
968:K
945:f
919:+
915:f
892:Y
886:O
873:Y
869:Y
852:Y
844:)
839:)
836:)
833:K
830:m
827:(
822:X
816:O
810:(
801:f
795:m
785:(
774:=
769:+
765:X
756:+
752:f
738:Y
722:Y
716:O
690:)
687:)
684:K
681:m
678:(
673:X
667:O
661:(
652:f
646:m
628:f
624:X
620:K
606:Y
600:X
594:f
544:+
539:i
535:X
531:=
526:1
523:+
520:i
516:X
493:i
489:X
466:+
461:i
457:X
448:i
444:X
437:f
415:i
411:X
388:i
384:X
356:C
336:C
326:i
322:X
317:K
288:i
284:X
279:K
256:i
252:X
219:n
215:X
210:K
185:i
181:X
176:K
153:n
149:X
134:2
130:X
121:1
117:X
113:=
110:X
87:X
34:.
20:)
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