563:) 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.
550:
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
851:
154:
54:. 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.
689:
1410:
467:
735:
1631:
893:
723:
545:
1439:
1358:
335:
605:
291:
222:
188:
1768:
1721:
1748:
1701:
1674:
1547:
1520:
1493:
1466:
1329:
1276:
1222:
1169:
1120:
1042:
920:
494:
416:
389:
257:
967:
2118:
1788:
1249:
1195:
1142:
1093:
1065:
1015:
991:
944:
355:
86:
567:). 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.
1068:
94:
2042:
1921:
1851:
625:
2020:
1371:
294:
421:
846:{\displaystyle f^{+}\colon X^{+}=\operatorname {Proj} {\big (}\bigoplus _{m}f_{*}({\mathcal {O}}_{X}(mK)){\big )}\to Y}
2080:
1998:
555:. The existence of log flips, a more general kind of flip, in dimension three and four were proved by Shokurov (
2174:
1566:
2169:
867:
697:
1990:
1770:
of flips of varieties with
Kawamata log terminal singularities, projective over a fixed normal variety
499:
1415:
1334:
300:
1804:
Proceedings of the Royal
Society of London. Series A: Mathematical, Physical and Engineering Sciences
578:
51:
232:), which is the desired result. The major technical problem is that, at some stage, the variety
1172:
262:
193:
159:
1753:
225:
63:
43:
1706:
2129:
2073:
2030:
1971:
1952:
1898:
1870:
1831:
1811:
1726:
1679:
1652:
1525:
1498:
1471:
1444:
1302:
1254:
1200:
1147:
1098:
1020:
898:
472:
394:
367:
235:
89:
8:
2148:
2136:
1874:
1815:
949:
2103:
2084:
2061:
1860:
1773:
1234:
1180:
1127:
1078:
1050:
1000:
976:
929:
340:
71:
27:
2056:
2016:
1994:
1978:
1959:
1945:
Proceedings of the
International Congress of Mathematicians, Vol. I, II (Kyoto, 1990)
1940:
1886:
1842:
229:
68:
The minimal model program can be summarised very briefly as follows: given a variety
2051:
1878:
1819:
2124:, Adv. Stud. Pure Math., vol. 1, Amsterdam: North-Holland, pp. 131–180,
1882:
2125:
2069:
2026:
2012:
1967:
1948:
1909:
1894:
1827:
20:
2037:
1982:
1846:
1838:
1799:
39:
2163:
1905:
1890:
149:{\displaystyle X=X_{1}\rightarrow X_{2}\rightarrow \cdots \rightarrow X_{n}}
2141:
Three-dimensional log flips. With an appendix in
English by Yujiro Kawamata
1823:
864:. If the relative canonical ring is finitely generated (as an algebra over
1849:(2010), "Existence of minimal models for varieties of log general type",
2040:(1988), "Flip theorem and the existence of minimal models for 3-folds",
1865:
2097:
2065:
997:
is relatively trivial. (Sometimes the induced birational morphism from
47:
1368:
at the origin. The exceptional locus of this blowup is isomorphic to
156:, each of which contracts some curves on which the canonical divisor
259:
may become 'too singular', in the sense that the canonical divisor
1649:
More precisely, there is a conjecture stating that every sequence
2143:, vol. 1, Russian Acad. Sci. Izv. Math. 40, pp. 95–202.
418:
is a birational map (in fact an isomorphism in codimension 1)
1071:
of an extremal ray, which implies several extra properties:
469:
to a variety whose singularities are 'better' than those of
684:{\displaystyle \bigoplus _{m}f_{*}({\mathcal {O}}_{X}(mK))}
1837:
564:
1914:
2122:
Algebraic varieties and analytic varieties (Tokyo, 1981)
2092:, Algebraic Geometry and Beyond, RIMS, Kyoto University
1405:{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}}
1964:
Surveys in differential geometry (Cambridge, MA, 1990)
2106:
1776:
1756:
1729:
1709:
1682:
1655:
1569:
1528:
1501:
1474:
1447:
1418:
1374:
1337:
1305:
1257:
1237:
1203:
1183:
1150:
1130:
1101:
1081:
1053:
1023:
1003:
979:
952:
932:
901:
870:
738:
700:
628:
581:
502:
475:
424:
397:
370:
343:
303:
265:
238:
196:
162:
97:
74:
1802:(1958), "On analytic surfaces with double points",
860:along the relative canonical ring is a morphism to
2112:
1782:
1762:
1742:
1715:
1695:
1668:
1625:
1541:
1514:
1487:
1460:
1433:
1404:
1352:
1323:
1270:
1243:
1216:
1189:
1163:
1136:
1114:
1087:
1059:
1036:
1009:
985:
961:
938:
914:
887:
845:
717:
683:
599:
539:
488:
462:{\displaystyle f\colon X_{i}\rightarrow X_{i}^{+}}
461:
410:
383:
360:The (conjectural) solution to this problem is the
349:
329:
285:
251:
216:
182:
148:
80:
2155:, Proc. Steklov Inst. Math. 240, pp. 75–213.
1966:, Bethlehem, PA: Lehigh Univ., pp. 113–199,
694:and is a sheaf of graded algebras over the sheaf
2161:
1559:, a generalization of Atiyah's flop replacing
1962:(1991), "Flips, flops, minimal models, etc",
1947:, Tokyo: Math. Soc. Japan, pp. 709–714,
832:
773:
2043:Journal of the American Mathematical Society
1922:Notices of the American Mathematical Society
1852:Journal of the American Mathematical Society
57:
1987:Birational Geometry of Algebraic Varieties
1977:
1287:The first example of a flop, known as the
16:Surgery operation in minimal model program
2055:
1864:
1421:
1392:
1377:
1340:
2147:
2135:
2079:
1441:in two different ways, giving varieties
560:
556:
2006:
2162:
2086:Flops, flips, and matrix factorization
1958:
1939:
1798:
1292:
1171:only have mild singularities, such as
615:, then the relative canonical ring of
2100:(1983), "Minimal models of canonical
1904:
1790:terminates after finitely many steps.
1626:{\displaystyle xy=(z+w^{k})(z-w^{k})}
228:(at least in the case of nonnegative
2096:
2036:
1552:
552:
13:
2011:, Universitext, Berlin, New York:
1495:. The natural birational map from
1075:The exceptional sets of both maps
888:{\displaystyle {\mathcal {O}}_{Y}}
874:
804:
718:{\displaystyle {\mathcal {O}}_{Y}}
704:
655:
14:
2186:
2057:10.1090/s0894-0347-1988-0924704-x
1228:, which is normal and projective.
540:{\displaystyle X_{i+1}=X_{i}^{+}}
2009:Introduction to the Mori program
1434:{\displaystyle \mathbb {P} ^{1}}
1353:{\displaystyle \mathbb {A} ^{4}}
330:{\displaystyle K_{X_{i}}\cdot C}
1643:
1620:
1601:
1598:
1579:
1224:are birational morphisms onto
837:
827:
824:
815:
798:
678:
675:
666:
649:
600:{\displaystyle f\colon X\to Y}
591:
441:
133:
127:
114:
1:
1883:10.1090/S0894-0347-09-00649-3
1636:
1278:are numerically proportional.
969:is relatively ample, and the
570:
297:, so the intersection number
88:, we construct a sequence of
1231:All curves in the fibers of
1122:have codimension at least 2,
547:, and continue the process.
7:
1412:, and can be blown down to
1282:
1044:is called a flip or flop.)
611:is the canonical bundle of
10:
2191:
1991:Cambridge University Press
61:
42:operations arising in the
18:
1943:(1991), "Flip and flop",
286:{\displaystyle K_{X_{i}}}
217:{\displaystyle K_{X_{n}}}
190:is negative. Eventually,
183:{\displaystyle K_{X_{i}}}
58:The minimal model program
725:of regular functions on
19:Not to be confused with
2149:Shokurov, Vyacheslav V.
2137:Shokurov, Vyacheslav V.
2007:Matsuki, Kenji (2002),
1800:Atiyah, Michael Francis
1763:{\displaystyle \cdots }
52:relative canonical ring
2114:
1824:10.1098/rspa.1958.0181
1784:
1764:
1744:
1717:
1716:{\displaystyle \dots }
1697:
1670:
1627:
1543:
1516:
1489:
1462:
1435:
1406:
1354:
1325:
1272:
1245:
1218:
1191:
1173:terminal singularities
1165:
1138:
1116:
1089:
1061:
1038:
1011:
987:
963:
940:
916:
889:
847:
719:
685:
601:
541:
490:
463:
412:
391:as above, the flip of
385:
364:. Given a problematic
351:
331:
287:
253:
218:
184:
150:
82:
2115:
1843:Hacon, Christopher D.
1785:
1765:
1745:
1743:{\displaystyle X_{n}}
1718:
1698:
1696:{\displaystyle X_{1}}
1671:
1669:{\displaystyle X_{0}}
1628:
1544:
1542:{\displaystyle X_{2}}
1517:
1515:{\displaystyle X_{1}}
1490:
1488:{\displaystyle X_{2}}
1463:
1461:{\displaystyle X_{1}}
1436:
1407:
1355:
1326:
1324:{\displaystyle xy=zw}
1273:
1271:{\displaystyle f^{+}}
1246:
1219:
1217:{\displaystyle f^{+}}
1192:
1166:
1164:{\displaystyle X^{+}}
1139:
1117:
1115:{\displaystyle f^{+}}
1090:
1062:
1039:
1037:{\displaystyle X^{+}}
1012:
988:
964:
941:
917:
915:{\displaystyle f^{+}}
895:) then the morphism
890:
848:
720:
686:
602:
542:
491:
489:{\displaystyle X_{i}}
464:
413:
411:{\displaystyle X_{i}}
386:
384:{\displaystyle X_{i}}
357:is not even defined.
352:
332:
288:
254:
252:{\displaystyle X_{i}}
219:
185:
151:
83:
64:Minimal model program
44:minimal model program
2104:
1774:
1754:
1727:
1707:
1680:
1653:
1567:
1549:is the Atiyah flop.
1526:
1499:
1472:
1445:
1416:
1372:
1335:
1303:
1255:
1235:
1201:
1181:
1148:
1128:
1099:
1079:
1051:
1021:
1001:
977:
950:
930:
899:
868:
736:
698:
626:
579:
500:
473:
422:
395:
368:
341:
301:
263:
236:
194:
160:
95:
72:
2175:Birational geometry
1910:"What Is...a Flip?"
1875:2010JAMS...23..405B
1816:1958RSPSA.247..237A
607:is a morphism, and
536:
458:
38:are codimension-2
2170:Algebraic geometry
2110:
1841:; Cascini, Paolo;
1780:
1760:
1740:
1713:
1693:
1666:
1623:
1539:
1512:
1485:
1458:
1431:
1402:
1350:
1321:
1268:
1241:
1214:
1187:
1161:
1134:
1112:
1085:
1057:
1034:
1007:
983:
962:{\displaystyle -K}
959:
936:
912:
885:
843:
787:
715:
681:
638:
597:
537:
522:
486:
459:
444:
408:
381:
347:
327:
283:
249:
214:
180:
146:
78:
28:algebraic geometry
2153:Prelimiting flips
2113:{\displaystyle 3}
2022:978-0-387-98465-0
1908:(December 2004),
1810:(1249): 237–244,
1783:{\displaystyle Z}
1364:be the blowup of
1244:{\displaystyle f}
1190:{\displaystyle f}
1137:{\displaystyle X}
1088:{\displaystyle f}
1069:small contraction
1060:{\displaystyle f}
1047:In applications,
1010:{\displaystyle X}
986:{\displaystyle f}
939:{\displaystyle f}
778:
629:
350:{\displaystyle C}
230:Kodaira dimension
81:{\displaystyle X}
2182:
2156:
2144:
2132:
2119:
2117:
2116:
2111:
2093:
2091:
2076:
2059:
2033:
2003:
1974:
1955:
1936:
1935:
1934:
1918:
1901:
1868:
1834:
1791:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1749:
1747:
1746:
1741:
1739:
1738:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1692:
1691:
1675:
1673:
1672:
1667:
1665:
1664:
1647:
1632:
1630:
1629:
1624:
1619:
1618:
1597:
1596:
1563:by the zeros of
1548:
1546:
1545:
1540:
1538:
1537:
1521:
1519:
1518:
1513:
1511:
1510:
1494:
1492:
1491:
1486:
1484:
1483:
1467:
1465:
1464:
1459:
1457:
1456:
1440:
1438:
1437:
1432:
1430:
1429:
1424:
1411:
1409:
1408:
1403:
1401:
1400:
1395:
1386:
1385:
1380:
1359:
1357:
1356:
1351:
1349:
1348:
1343:
1330:
1328:
1327:
1322:
1299:be the zeros of
1291:, was found in (
1277:
1275:
1274:
1269:
1267:
1266:
1250:
1248:
1247:
1242:
1223:
1221:
1220:
1215:
1213:
1212:
1196:
1194:
1193:
1188:
1170:
1168:
1167:
1162:
1160:
1159:
1143:
1141:
1140:
1135:
1121:
1119:
1118:
1113:
1111:
1110:
1094:
1092:
1091:
1086:
1066:
1064:
1063:
1058:
1043:
1041:
1040:
1035:
1033:
1032:
1016:
1014:
1013:
1008:
992:
990:
989:
984:
968:
966:
965:
960:
945:
943:
942:
937:
921:
919:
918:
913:
911:
910:
894:
892:
891:
886:
884:
883:
878:
877:
852:
850:
849:
844:
836:
835:
814:
813:
808:
807:
797:
796:
786:
777:
776:
761:
760:
748:
747:
724:
722:
721:
716:
714:
713:
708:
707:
690:
688:
687:
682:
665:
664:
659:
658:
648:
647:
637:
606:
604:
603:
598:
546:
544:
543:
538:
535:
530:
518:
517:
496:. So we can put
495:
493:
492:
487:
485:
484:
468:
466:
465:
460:
457:
452:
440:
439:
417:
415:
414:
409:
407:
406:
390:
388:
387:
382:
380:
379:
356:
354:
353:
348:
336:
334:
333:
328:
320:
319:
318:
317:
292:
290:
289:
284:
282:
281:
280:
279:
258:
256:
255:
250:
248:
247:
223:
221:
220:
215:
213:
212:
211:
210:
189:
187:
186:
181:
179:
178:
177:
176:
155:
153:
152:
147:
145:
144:
126:
125:
113:
112:
87:
85:
84:
79:
2190:
2189:
2185:
2184:
2183:
2181:
2180:
2179:
2160:
2159:
2105:
2102:
2101:
2089:
2081:Morrison, David
2038:Mori, Shigefumi
2023:
2013:Springer-Verlag
2001:
1983:Mori, Shigefumi
1932:
1930:
1929:(11): 1350–1351
1912:
1866:math.AG/0610203
1847:McKernan, James
1839:Birkar, Caucher
1795:
1794:
1775:
1772:
1771:
1755:
1752:
1751:
1734:
1730:
1728:
1725:
1724:
1708:
1705:
1704:
1687:
1683:
1681:
1678:
1677:
1660:
1656:
1654:
1651:
1650:
1648:
1644:
1639:
1614:
1610:
1592:
1588:
1568:
1565:
1564:
1533:
1529:
1527:
1524:
1523:
1506:
1502:
1500:
1497:
1496:
1479:
1475:
1473:
1470:
1469:
1452:
1448:
1446:
1443:
1442:
1425:
1420:
1419:
1417:
1414:
1413:
1396:
1391:
1390:
1381:
1376:
1375:
1373:
1370:
1369:
1344:
1339:
1338:
1336:
1333:
1332:
1304:
1301:
1300:
1285:
1262:
1258:
1256:
1253:
1252:
1236:
1233:
1232:
1208:
1204:
1202:
1199:
1198:
1182:
1179:
1178:
1155:
1151:
1149:
1146:
1145:
1129:
1126:
1125:
1106:
1102:
1100:
1097:
1096:
1080:
1077:
1076:
1052:
1049:
1048:
1028:
1024:
1022:
1019:
1018:
1002:
999:
998:
978:
975:
974:
951:
948:
947:
931:
928:
927:
906:
902:
900:
897:
896:
879:
873:
872:
871:
869:
866:
865:
831:
830:
809:
803:
802:
801:
792:
788:
782:
772:
771:
756:
752:
743:
739:
737:
734:
733:
709:
703:
702:
701:
699:
696:
695:
660:
654:
653:
652:
643:
639:
633:
627:
624:
623:
580:
577:
576:
573:
531:
526:
507:
503:
501:
498:
497:
480:
476:
474:
471:
470:
453:
448:
435:
431:
423:
420:
419:
402:
398:
396:
393:
392:
375:
371:
369:
366:
365:
342:
339:
338:
313:
309:
308:
304:
302:
299:
298:
295:Cartier divisor
293:is no longer a
275:
271:
270:
266:
264:
261:
260:
243:
239:
237:
234:
233:
206:
202:
201:
197:
195:
192:
191:
172:
168:
167:
163:
161:
158:
157:
140:
136:
121:
117:
108:
104:
96:
93:
92:
73:
70:
69:
66:
60:
24:
21:Flip (geometry)
17:
12:
11:
5:
2188:
2178:
2177:
2172:
2158:
2157:
2145:
2133:
2109:
2094:
2077:
2050:(1): 117–253,
2034:
2021:
2004:
1999:
1975:
1956:
1937:
1906:Corti, Alessio
1902:
1859:(2): 405–468,
1835:
1793:
1792:
1779:
1759:
1737:
1733:
1712:
1690:
1686:
1663:
1659:
1641:
1640:
1638:
1635:
1622:
1617:
1613:
1609:
1606:
1603:
1600:
1595:
1591:
1587:
1584:
1581:
1578:
1575:
1572:
1536:
1532:
1509:
1505:
1482:
1478:
1455:
1451:
1428:
1423:
1399:
1394:
1389:
1384:
1379:
1347:
1342:
1320:
1317:
1314:
1311:
1308:
1284:
1281:
1280:
1279:
1265:
1261:
1240:
1229:
1211:
1207:
1186:
1176:
1158:
1154:
1133:
1123:
1109:
1105:
1084:
1056:
1031:
1027:
1006:
982:
958:
955:
935:
922:is called the
909:
905:
882:
876:
854:
853:
842:
839:
834:
829:
826:
823:
820:
817:
812:
806:
800:
795:
791:
785:
781:
775:
770:
767:
764:
759:
755:
751:
746:
742:
729:. The blowup
712:
706:
692:
691:
680:
677:
674:
671:
668:
663:
657:
651:
646:
642:
636:
632:
596:
593:
590:
587:
584:
572:
569:
534:
529:
525:
521:
516:
513:
510:
506:
483:
479:
456:
451:
447:
443:
438:
434:
430:
427:
405:
401:
378:
374:
346:
326:
323:
316:
312:
307:
278:
274:
269:
246:
242:
224:should become
209:
205:
200:
175:
171:
166:
143:
139:
135:
132:
129:
124:
120:
116:
111:
107:
103:
100:
77:
62:Main article:
59:
56:
15:
9:
6:
4:
3:
2:
2187:
2176:
2173:
2171:
2168:
2167:
2165:
2154:
2150:
2146:
2142:
2138:
2134:
2131:
2127:
2123:
2107:
2099:
2095:
2088:
2087:
2082:
2078:
2075:
2071:
2067:
2063:
2058:
2053:
2049:
2045:
2044:
2039:
2035:
2032:
2028:
2024:
2018:
2014:
2010:
2005:
2002:
2000:0-521-63277-3
1996:
1992:
1988:
1984:
1980:
1979:Kollár, János
1976:
1973:
1969:
1965:
1961:
1960:Kollár, János
1957:
1954:
1950:
1946:
1942:
1941:Kollár, János
1938:
1928:
1924:
1923:
1916:
1911:
1907:
1903:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1867:
1862:
1858:
1854:
1853:
1848:
1844:
1840:
1836:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1796:
1777:
1757:
1735:
1731:
1710:
1688:
1684:
1661:
1657:
1646:
1642:
1634:
1615:
1611:
1607:
1604:
1593:
1589:
1585:
1582:
1576:
1573:
1570:
1562:
1558:
1557:Reid's pagoda
1554:
1550:
1534:
1530:
1507:
1503:
1480:
1476:
1453:
1449:
1426:
1397:
1387:
1382:
1367:
1363:
1345:
1318:
1315:
1312:
1309:
1306:
1298:
1294:
1290:
1263:
1259:
1238:
1230:
1227:
1209:
1205:
1184:
1177:
1174:
1156:
1152:
1131:
1124:
1107:
1103:
1082:
1074:
1073:
1072:
1070:
1054:
1045:
1029:
1025:
1004:
996:
980:
972:
956:
953:
933:
925:
907:
903:
880:
863:
859:
840:
821:
818:
810:
793:
789:
783:
779:
768:
765:
762:
757:
753:
749:
744:
740:
732:
731:
730:
728:
710:
672:
669:
661:
644:
640:
634:
630:
622:
621:
620:
618:
614:
610:
594:
588:
585:
582:
568:
566:
562:
558:
554:
548:
532:
527:
523:
519:
514:
511:
508:
504:
481:
477:
454:
449:
445:
436:
432:
428:
425:
403:
399:
376:
372:
363:
358:
344:
337:with a curve
324:
321:
314:
310:
305:
296:
276:
272:
267:
244:
240:
231:
227:
207:
203:
198:
173:
169:
164:
141:
137:
130:
122:
118:
109:
105:
101:
98:
91:
75:
65:
55:
53:
49:
45:
41:
37:
33:
29:
22:
2152:
2140:
2121:
2085:
2047:
2041:
2008:
1986:
1963:
1944:
1931:, retrieved
1926:
1920:
1856:
1850:
1807:
1803:
1645:
1560:
1556:
1551:
1365:
1361:
1296:
1288:
1286:
1225:
1067:is often a
1046:
994:
970:
923:
861:
857:
855:
726:
693:
616:
612:
608:
574:
549:
361:
359:
90:contractions
67:
35:
31:
25:
2098:Reid, Miles
1555:introduced
1553:Reid (1983)
1293:Atiyah 1958
1289:Atiyah flop
553:Mori (1988)
46:, given by
2164:Categories
1933:2008-01-17
1637:References
1360:, and let
571:Definition
48:blowing up
2120:-folds",
1891:0894-0347
1758:⋯
1711:…
1608:−
1388:×
954:−
838:→
794:∗
780:⨁
769:
750::
645:∗
631:⨁
592:→
586::
442:→
429::
322:⋅
134:→
131:⋯
128:→
115:→
2151:(2003),
2139:(1993),
2083:(2005),
1985:(1998),
1283:Examples
50:along a
2130:0715649
2074:0924704
2066:1990969
2031:1875410
1972:1144527
1953:1159257
1899:2601039
1871:Bibcode
1832:0095974
1812:Bibcode
1295:). Let
40:surgery
2128:
2072:
2064:
2029:
2019:
1997:
1970:
1951:
1897:
1889:
1830:
2090:(PDF)
2062:JSTOR
1861:arXiv
36:flops
34:and
32:flips
2017:ISBN
1995:ISBN
1887:ISSN
1468:and
1251:and
1197:and
1144:and
1095:and
971:flop
924:flip
766:Proj
565:2010
561:2003
557:1993
362:flip
2052:doi
1915:PDF
1879:doi
1820:doi
1808:247
1522:to
1331:in
1017:to
993:if
973:of
946:if
926:of
856:of
619:is
575:If
226:nef
26:In
2166::
2126:MR
2070:MR
2068:,
2060:,
2046:,
2027:MR
2025:,
2015:,
1993:,
1989:,
1981:;
1968:MR
1949:MR
1927:51
1925:,
1919:,
1895:MR
1893:,
1885:,
1877:,
1869:,
1857:23
1855:,
1845:;
1828:MR
1826:,
1818:,
1806:,
1750:⇢
1723:⇢
1703:⇢
1676:⇢
1633:.
559:,
30:,
2108:3
2054::
2048:1
1917:)
1913:(
1881::
1873::
1863::
1822::
1814::
1778:Z
1736:n
1732:X
1689:1
1685:X
1662:0
1658:X
1621:)
1616:k
1612:w
1605:z
1602:(
1599:)
1594:k
1590:w
1586:+
1583:z
1580:(
1577:=
1574:y
1571:x
1561:Y
1535:2
1531:X
1508:1
1504:X
1481:2
1477:X
1454:1
1450:X
1427:1
1422:P
1398:1
1393:P
1383:1
1378:P
1366:Y
1362:V
1346:4
1341:A
1319:w
1316:z
1313:=
1310:y
1307:x
1297:Y
1264:+
1260:f
1239:f
1226:Y
1210:+
1206:f
1185:f
1175:.
1157:+
1153:X
1132:X
1108:+
1104:f
1083:f
1055:f
1030:+
1026:X
1005:X
995:K
981:f
957:K
934:f
908:+
904:f
881:Y
875:O
862:Y
858:Y
841:Y
833:)
828:)
825:)
822:K
819:m
816:(
811:X
805:O
799:(
790:f
784:m
774:(
763:=
758:+
754:X
745:+
741:f
727:Y
711:Y
705:O
679:)
676:)
673:K
670:m
667:(
662:X
656:O
650:(
641:f
635:m
617:f
613:X
609:K
595:Y
589:X
583:f
533:+
528:i
524:X
520:=
515:1
512:+
509:i
505:X
482:i
478:X
455:+
450:i
446:X
437:i
433:X
426:f
404:i
400:X
377:i
373:X
345:C
325:C
315:i
311:X
306:K
277:i
273:X
268:K
245:i
241:X
208:n
204:X
199:K
174:i
170:X
165:K
142:n
138:X
123:2
119:X
110:1
106:X
102:=
99:X
76:X
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.