225:
is basepoint-free. It follows that a semi-ample line bundle is nef. Semi-ample line bundles can be considered the main geometric source of nef line bundles, although the two concepts are not equivalent; see the examples below.
1912:
933:
1952:. (This correspondence can also be formulated dually, in terms of faces of the cone of curves.) Knowing which nef line bundles are semi-ample would determine which faces correspond to contractions. The
2080:
1719:
1427:
2218:
2175:
1010:
1165:
2276:
2247:
2035:
1994:
1548:
1759:
1490:
839:
870:
223:
1335:
1064:
1109:
149:, definition 7.6) and "numerically effective", as well as for the phrase "numerically eventually free". The older terms were misleading, in view of the examples below.
2120:
1803:
1375:
1201:
731:
675:
612:
576:
540:
487:
428:
349:
286:
794:
681:-divisor is called ample if it can be written as a positive linear combination of ample Cartier divisors.) It follows from Kleiman's criterion that, for
2122:
has dimension 2. By the geometry of convex cones of dimension 2, the nef cone must be spanned by two rays; explicitly, these are the rays spanned by
51:, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and
1835:
1275:
Every semi-ample line bundle is nef, but not every nef line bundle is even numerically equivalent to a semi-ample line bundle. For example,
106:
883:
506:
is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space
2511:
17:
2044:
2587:
2548:
1650:
629:, since that amounts to describing the different ways a variety can be embedded into projective space. One answer is
1284:
193:
2652:
441:-divisors are said to be numerically equivalent if they have the same intersection number with all curves in
388:
1380:
2188:
2145:
241:
52:
2607:(1962), "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface",
973:
2495:
1138:
2252:
2223:
2011:
1970:
1503:
2457:
774:
434:
1732:
1463:
818:
431:
1567:
848:
766:
198:
1297:
1036:
877:
801:
1081:
1377:
is numerically equivalent to an effective divisor. In particular, the space of global sections
2089:
1772:
1344:
1170:
700:
644:
581:
545:
509:
456:
397:
318:
265:
779:
97:
2636:
2597:
2558:
2521:
2476:
2453:
1957:
1444:
797:
79:
8:
1591:
1254:
260:
86:
2624:
376:
312:
36:
28:
2572:, Advanced Studies in Pure Mathematics, vol. 1, North-Holland, pp. 131–180,
2583:
2544:
2528:
2507:
2483:
1826:
1242:
1215:
626:
2616:
2573:
2536:
2499:
1956:
describes a significant class of faces that do correspond to contractions, and the
1261:
770:
618:
to each other by the intersection pairing, and the nef cone is (by definition) the
135:
2632:
2593:
2554:
2517:
2472:
1645:
1555:
1219:
1211:
625:
A significant problem in algebraic geometry is to analyze which line bundles are
230:
76:
44:
2487:
502:
169:
165:
2540:
1944:
As a result, there is a one-to-one correspondence between the contractions of
2646:
2604:
2503:
2286:
has no other nontrivial faces, these are the only nontrivial contractions of
1449:
1276:
1207:
2578:
449:-divisor is called nef if it has nonnegative degree on every curve. The nef
1953:
1907:{\displaystyle Y={\text{Proj }}\bigoplus _{a\geq 0}H^{0}(X,L^{\otimes a}).}
392:
304:
1260:
at least 1, most line bundles of degree 0 are not torsion, using that the
380:
56:
48:
32:
2628:
2565:
1124:
842:
615:
142:
2458:"Compact complex manifolds with numerically effective tangent bundles"
619:
2620:
2290:; that would be harder to see without the relation to convex cones.
168:
that is not identically zero has nonnegative degree. As a result, a
47:
in the variety. The classes of nef line bundles are described by a
145:
as a replacement for the older terms "arithmetically effective" (
928:{\displaystyle \Theta _{h_{\epsilon }}(L)\geq -\epsilon \omega }
1294:
has positive degree on all curves, but the intersection number
1761:
up to isomorphism. Indeed, there is a semi-ample line bundle
1078:
curves on a surface have nonnegative intersection number. If
1012:, which explains the more complicated definition just given.
542:
of 1-cycles modulo numerical equivalence. The vector spaces
2134:. In this example, both rays correspond to contractions of
2452:
2440:
Kollár & Mori (1998), Lemma 1.22 and
Example 1.23(1).
2570:
Algebraic varieties and analytic varieties (Tokyo, 1981)
2086:
has Picard number 2, meaning that the real vector space
756:
371:
To work with inequalities, it is convenient to consider
943:, this is equivalent to the previous definition (that
2255:
2226:
2191:
2148:
2092:
2047:
2014:
1973:
1838:
1775:
1735:
1653:
1506:
1466:
1383:
1347:
1300:
1173:
1141:
1084:
1039:
976:
886:
851:
821:
782:
703:
647:
584:
548:
512:
459:
400:
321:
268:
201:
59:-1 subvarieties), there is an equivalent notion of a
1598:
could be the blow-up of a smooth projective surface
184:; that is, it is nef. More generally, a line bundle
1616:means a convex subcone such that any two points of
2270:
2241:
2212:
2169:
2114:
2074:
2029:
1988:
1906:
1797:
1753:
1713:
1542:
1484:
1421:
1369:
1329:
1195:
1159:
1103:
1058:
1004:
927:
864:
833:
788:
725:
669:
606:
570:
534:
481:
422:
343:
280:
217:
1229:of degree 0 on a smooth complex projective curve
2644:
2456:; Peternell, Thomas; Schneider, Michael (1994),
2075:{\displaystyle \pi \colon X\to \mathbb {P} ^{2}}
1437:
2568:(1983), "Minimal models of canonical 3-folds",
1967:be the blow-up of the complex projective plane
1714:{\displaystyle f^{*}(N^{1}(Y))\subset N^{1}(X)}
641:-divisor) is ample if and only if its class in
299:To go back from line bundles to divisors, the
96:if it has nonnegative degree on every (closed
800:, Thomas Peternell and Michael Schneider, a
1729:of the nef cone determines the contraction
2527:
2492:Birational geometry of algebraic varieties
2482:
1214:is nef, using that these varieties have a
677:lies in the interior of the nef cone. (An
2577:
2258:
2229:
2200:
2157:
2062:
2017:
1976:
1948:and some of the faces of the nef cone of
180:has nonnegative degree on every curve in
2535:, vol. 1, Berlin: Springer-Verlag,
2041:be the exceptional curve of the blow-up
947:has nonnegative degree on all curves in
437:with the real numbers. (Explicitly: two
311:to the group of Cartier divisors modulo
2603:
1131:at a point, then the exceptional curve
453:-divisors form a closed convex cone in
146:
134:.) A line bundle may also be called an
14:
2645:
2431:Kollár & Mori (1998), Remark 1.26.
1422:{\displaystyle H^{0}(X,L^{\otimes a})}
240:over a field is said to be nef if the
43:if it has nonnegative degree on every
2404:Lazarsfeld (2004), Definition 2.1.11.
2350:Lazarsfeld (2004), Definition 1.4.25.
2213:{\displaystyle X\to \mathbb {P} ^{1}}
2170:{\displaystyle X\to \mathbb {P} ^{2}}
1550:. (The latter condition implies that
757:Metric definition of nef line bundles
2564:
2377:Demailly et al. (1994), Example 1.7.
2305:Lazarsfeld (2004), Definition 1.4.1.
1341:is nef, but no positive multiple of
1005:{\displaystyle \Theta _{h}(L)\geq 0}
315:. Explicitly, the first Chern class
1821:). Any such line bundle determines
1268:is an abelian variety of dimension
1025:is a smooth projective surface and
24:
2422:Lazarsfeld (2004), Theorem 2.1.27.
2413:Lazarsfeld (2004), Example 2.1.12.
2368:Demailly et al. (1994), section 1.
2359:Lazarsfeld (2004), Theorem 1.4.23.
2341:Lazarsfeld (2004), Example 1.3.10.
1429:is zero for all positive integers
978:
888:
355:) of any nonzero rational section
126:) of any nonzero rational section
25:
2664:
2395:Lazarsfeld (2004), Example 1.5.2.
2386:Lazarsfeld (2004), Example 1.4.7.
2332:Lazarsfeld (2004), Example 1.1.5.
2323:Lazarsfeld (2004), Example 1.4.5.
1640:, namely the intersection of Nef(
1570:zero.) A contraction is called a
1496:a normal projective variety over
1160:{\displaystyle \pi \colon X\to Y}
966:need not have a Hermitian metric
141:The term "nef" was introduced by
2533:Positivity in algebraic geometry
2271:{\displaystyle \mathbb {P} ^{2}}
2242:{\displaystyle \mathbb {P} ^{1}}
2030:{\displaystyle \mathbb {P} ^{2}}
1989:{\displaystyle \mathbb {P} ^{2}}
1721:. Conversely, given the variety
1558:fibers, and it is equivalent to
1543:{\displaystyle f_{*}O_{X}=O_{Y}}
637:over a field, a line bundle (or
633:(1966): for a projective scheme
2434:
2425:
2416:
2407:
2398:
2389:
2380:
2249:(corresponding to the lines in
1127:of a smooth projective surface
366:
288:is nonnegative for every curve
172:line bundle on a proper scheme
2371:
2362:
2353:
2344:
2335:
2326:
2317:
2308:
2299:
2195:
2152:
2142:gives the birational morphism
2109:
2103:
2057:
1898:
1876:
1792:
1786:
1754:{\displaystyle f\colon X\to Y}
1745:
1708:
1702:
1686:
1683:
1677:
1664:
1485:{\displaystyle f\colon X\to Y}
1476:
1416:
1394:
1364:
1358:
1318:
1311:
1151:
1033:with self-intersection number
993:
987:
910:
904:
834:{\displaystyle \epsilon >0}
749:is ample for all real numbers
720:
714:
664:
658:
601:
595:
565:
559:
529:
523:
476:
470:
417:
411:
338:
332:
122:is the degree of the divisor (
71:More generally, a line bundle
13:
1:
2465:Journal of Algebraic Geometry
2446:
1438:Contractions and the nef cone
1237:is semi-ample if and only if
1206:Every effective divisor on a
1029:is an (irreducible) curve in
865:{\displaystyle h_{\epsilon }}
307:of line bundles on a variety
218:{\displaystyle L^{\otimes a}}
66:
1921:in geometric terms: a curve
1817:of any ample line bundle on
1330:{\displaystyle c_{1}(L)^{2}}
1115:is effective but not nef on
303:is the isomorphism from the
7:
2314:Reid (1983), section 0.12f.
1562:having connected fibers if
1059:{\displaystyle C^{2}\geq 0}
1015:
10:
2669:
2496:Cambridge University Press
2220:with fibers isomorphic to
1582:). A contraction with dim(
1279:constructed a line bundle
1104:{\displaystyle C^{2}<0}
375:-divisors, meaning finite
2541:10.1007/978-3-642-18808-4
2282:). Since the nef cone of
1460:is a surjective morphism
1337:is zero. It follows that
430:of finite dimension, the
379:of Cartier divisors with
2504:10.1017/CBO9780511662560
2293:
2115:{\displaystyle N^{1}(X)}
1798:{\displaystyle N^{1}(X)}
1370:{\displaystyle c_{1}(L)}
1196:{\displaystyle E^{2}=-1}
767:compact complex manifold
726:{\displaystyle N^{1}(X)}
670:{\displaystyle N^{1}(X)}
607:{\displaystyle N_{1}(X)}
571:{\displaystyle N^{1}(X)}
535:{\displaystyle N_{1}(X)}
482:{\displaystyle N^{1}(X)}
423:{\displaystyle N^{1}(X)}
344:{\displaystyle c_{1}(L)}
281:{\displaystyle D\cdot C}
1245:in the Picard group of
802:holomorphic line bundle
789:{\displaystyle \omega }
773:, viewed as a positive
622:of the cone of curves.
2272:
2243:
2214:
2171:
2116:
2076:
2031:
1990:
1908:
1813:to be the pullback to
1805:is in the interior of
1799:
1755:
1715:
1624:must themselves be in
1544:
1486:
1423:
1371:
1331:
1197:
1161:
1105:
1060:
1006:
929:
866:
835:
790:
727:
685:projective, every nef
671:
608:
572:
536:
483:
424:
345:
282:
242:associated line bundle
219:
2609:Annals of Mathematics
2579:10.2969/aspm/00110131
2454:Demailly, Jean-Pierre
2273:
2244:
2215:
2172:
2117:
2077:
2032:
1991:
1909:
1800:
1756:
1716:
1590:) is automatically a
1545:
1487:
1424:
1372:
1332:
1198:
1162:
1106:
1061:
1007:
930:
867:
836:
791:
728:
672:
609:
573:
537:
484:
425:
389:numerical equivalence
346:
283:
220:
18:Numerically effective
2653:Geometry of divisors
2253:
2224:
2189:
2146:
2090:
2045:
2012:
1971:
1958:abundance conjecture
1836:
1773:
1733:
1651:
1504:
1464:
1381:
1345:
1298:
1171:
1139:
1082:
1037:
974:
962:, a nef line bundle
884:
849:
819:
798:Jean-Pierre Demailly
780:
701:
693:is a limit of ample
645:
582:
546:
510:
457:
398:
319:
266:
199:
2004:be the pullback to
1937:has degree zero on
1929:maps to a point in
1809:(for example, take
1636:of the nef cone of
1628:. A contraction of
1592:birational morphism
1452:projective variety
939:is projective over
631:Kleiman's criterion
377:linear combinations
261:intersection number
236:on a proper scheme
2529:Lazarsfeld, Robert
2278:through the point
2268:
2239:
2210:
2185:gives a fibration
2167:
2112:
2072:
2027:
1986:
1904:
1865:
1795:
1751:
1711:
1632:determines a face
1540:
1482:
1419:
1367:
1327:
1225:Every line bundle
1193:
1157:
1119:. For example, if
1101:
1074:, because any two
1056:
1002:
925:
862:
831:
786:
723:
667:
604:
568:
532:
479:
432:Néron–Severi group
420:
383:coefficients. The
341:
313:linear equivalence
278:
215:
156:on a proper curve
152:Every line bundle
114:on a proper curve
37:projective variety
29:algebraic geometry
2513:978-0-521-63277-5
1960:would give more.
1850:
1848:
1827:Proj construction
1612:of a convex cone
1216:transitive action
845:Hermitian metric
387:-divisors modulo
301:first Chern class
192:if some positive
110:of a line bundle
16:(Redirected from
2660:
2639:
2600:
2581:
2561:
2524:
2479:
2462:
2441:
2438:
2432:
2429:
2423:
2420:
2414:
2411:
2405:
2402:
2396:
2393:
2387:
2384:
2378:
2375:
2369:
2366:
2360:
2357:
2351:
2348:
2342:
2339:
2333:
2330:
2324:
2321:
2315:
2312:
2306:
2303:
2277:
2275:
2274:
2269:
2267:
2266:
2261:
2248:
2246:
2245:
2240:
2238:
2237:
2232:
2219:
2217:
2216:
2211:
2209:
2208:
2203:
2176:
2174:
2173:
2168:
2166:
2165:
2160:
2121:
2119:
2118:
2113:
2102:
2101:
2081:
2079:
2078:
2073:
2071:
2070:
2065:
2036:
2034:
2033:
2028:
2026:
2025:
2020:
1995:
1993:
1992:
1987:
1985:
1984:
1979:
1913:
1911:
1910:
1905:
1897:
1896:
1875:
1874:
1864:
1849:
1846:
1804:
1802:
1801:
1796:
1785:
1784:
1760:
1758:
1757:
1752:
1720:
1718:
1717:
1712:
1701:
1700:
1676:
1675:
1663:
1662:
1620:whose sum is in
1594:. (For example,
1549:
1547:
1546:
1541:
1539:
1538:
1526:
1525:
1516:
1515:
1491:
1489:
1488:
1483:
1428:
1426:
1425:
1420:
1415:
1414:
1393:
1392:
1376:
1374:
1373:
1368:
1357:
1356:
1336:
1334:
1333:
1328:
1326:
1325:
1310:
1309:
1202:
1200:
1199:
1194:
1183:
1182:
1166:
1164:
1163:
1158:
1110:
1108:
1107:
1102:
1094:
1093:
1065:
1063:
1062:
1057:
1049:
1048:
1011:
1009:
1008:
1003:
986:
985:
958:projective over
934:
932:
931:
926:
903:
902:
901:
900:
871:
869:
868:
863:
861:
860:
840:
838:
837:
832:
795:
793:
792:
787:
771:Hermitian metric
732:
730:
729:
724:
713:
712:
676:
674:
673:
668:
657:
656:
613:
611:
610:
605:
594:
593:
577:
575:
574:
569:
558:
557:
541:
539:
538:
533:
522:
521:
488:
486:
485:
480:
469:
468:
429:
427:
426:
421:
410:
409:
351:is the divisor (
350:
348:
347:
342:
331:
330:
287:
285:
284:
279:
255:. Equivalently,
224:
222:
221:
216:
214:
213:
136:invertible sheaf
21:
2668:
2667:
2663:
2662:
2661:
2659:
2658:
2657:
2643:
2642:
2621:10.2307/1970376
2590:
2551:
2514:
2488:Mori, Shigefumi
2460:
2449:
2444:
2439:
2435:
2430:
2426:
2421:
2417:
2412:
2408:
2403:
2399:
2394:
2390:
2385:
2381:
2376:
2372:
2367:
2363:
2358:
2354:
2349:
2345:
2340:
2336:
2331:
2327:
2322:
2318:
2313:
2309:
2304:
2300:
2296:
2262:
2257:
2256:
2254:
2251:
2250:
2233:
2228:
2227:
2225:
2222:
2221:
2204:
2199:
2198:
2190:
2187:
2186:
2161:
2156:
2155:
2147:
2144:
2143:
2097:
2093:
2091:
2088:
2087:
2066:
2061:
2060:
2046:
2043:
2042:
2021:
2016:
2015:
2013:
2010:
2009:
1980:
1975:
1974:
1972:
1969:
1968:
1933:if and only if
1889:
1885:
1870:
1866:
1854:
1845:
1837:
1834:
1833:
1780:
1776:
1774:
1771:
1770:
1769:whose class in
1734:
1731:
1730:
1696:
1692:
1671:
1667:
1658:
1654:
1652:
1649:
1648:
1534:
1530:
1521:
1517:
1511:
1507:
1505:
1502:
1501:
1465:
1462:
1461:
1440:
1407:
1403:
1388:
1384:
1382:
1379:
1378:
1352:
1348:
1346:
1343:
1342:
1321:
1317:
1305:
1301:
1299:
1296:
1295:
1220:algebraic group
1218:of a connected
1212:abelian variety
1178:
1174:
1172:
1169:
1168:
1140:
1137:
1136:
1135:of the blow-up
1089:
1085:
1083:
1080:
1079:
1044:
1040:
1038:
1035:
1034:
1018:
981:
977:
975:
972:
971:
970:with curvature
896:
892:
891:
887:
885:
882:
881:
856:
852:
850:
847:
846:
820:
817:
816:
781:
778:
777:
759:
708:
704:
702:
699:
698:
652:
648:
646:
643:
642:
589:
585:
583:
580:
579:
553:
549:
547:
544:
543:
517:
513:
511:
508:
507:
464:
460:
458:
455:
454:
405:
401:
399:
396:
395:
369:
326:
322:
320:
317:
316:
267:
264:
263:
231:Cartier divisor
206:
202:
200:
197:
196:
69:
23:
22:
15:
12:
11:
5:
2666:
2656:
2655:
2641:
2640:
2615:(3): 560–615,
2605:Zariski, Oscar
2601:
2588:
2562:
2549:
2525:
2512:
2480:
2448:
2445:
2443:
2442:
2433:
2424:
2415:
2406:
2397:
2388:
2379:
2370:
2361:
2352:
2343:
2334:
2325:
2316:
2307:
2297:
2295:
2292:
2265:
2260:
2236:
2231:
2207:
2202:
2197:
2194:
2164:
2159:
2154:
2151:
2111:
2108:
2105:
2100:
2096:
2069:
2064:
2059:
2056:
2053:
2050:
2024:
2019:
1983:
1978:
1915:
1914:
1903:
1900:
1895:
1892:
1888:
1884:
1881:
1878:
1873:
1869:
1863:
1860:
1857:
1853:
1844:
1841:
1794:
1791:
1788:
1783:
1779:
1750:
1747:
1744:
1741:
1738:
1710:
1707:
1704:
1699:
1695:
1691:
1688:
1685:
1682:
1679:
1674:
1670:
1666:
1661:
1657:
1568:characteristic
1537:
1533:
1529:
1524:
1520:
1514:
1510:
1481:
1478:
1475:
1472:
1469:
1439:
1436:
1435:
1434:
1418:
1413:
1410:
1406:
1402:
1399:
1396:
1391:
1387:
1366:
1363:
1360:
1355:
1351:
1324:
1320:
1316:
1313:
1308:
1304:
1283:on a suitable
1273:
1223:
1204:
1192:
1189:
1186:
1181:
1177:
1156:
1153:
1150:
1147:
1144:
1100:
1097:
1092:
1088:
1055:
1052:
1047:
1043:
1017:
1014:
1001:
998:
995:
992:
989:
984:
980:
924:
921:
918:
915:
912:
909:
906:
899:
895:
890:
859:
855:
830:
827:
824:
811:is said to be
785:
758:
755:
733:. Indeed, for
722:
719:
716:
711:
707:
666:
663:
660:
655:
651:
603:
600:
597:
592:
588:
567:
564:
561:
556:
552:
531:
528:
525:
520:
516:
503:cone of curves
478:
475:
472:
467:
463:
419:
416:
413:
408:
404:
368:
365:
340:
337:
334:
329:
325:
277:
274:
271:
259:is nef if the
212:
209:
205:
170:basepoint-free
166:global section
92:is said to be
68:
65:
9:
6:
4:
3:
2:
2665:
2654:
2651:
2650:
2648:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2599:
2595:
2591:
2589:0-444-86612-4
2585:
2580:
2575:
2571:
2567:
2563:
2560:
2556:
2552:
2550:3-540-22533-1
2546:
2542:
2538:
2534:
2530:
2526:
2523:
2519:
2515:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2484:Kollár, János
2481:
2478:
2474:
2470:
2466:
2459:
2455:
2451:
2450:
2437:
2428:
2419:
2410:
2401:
2392:
2383:
2374:
2365:
2356:
2347:
2338:
2329:
2320:
2311:
2302:
2298:
2291:
2289:
2285:
2281:
2263:
2234:
2205:
2192:
2184:
2180:
2162:
2149:
2141:
2137:
2133:
2129:
2125:
2106:
2098:
2094:
2085:
2067:
2054:
2051:
2048:
2040:
2022:
2008:of a line on
2007:
2003:
1999:
1981:
1966:
1963:Example: Let
1961:
1959:
1955:
1951:
1947:
1942:
1940:
1936:
1932:
1928:
1924:
1920:
1901:
1893:
1890:
1886:
1882:
1879:
1871:
1867:
1861:
1858:
1855:
1851:
1842:
1839:
1832:
1831:
1830:
1828:
1824:
1820:
1816:
1812:
1808:
1789:
1781:
1777:
1768:
1764:
1748:
1742:
1739:
1736:
1728:
1724:
1705:
1697:
1693:
1689:
1680:
1672:
1668:
1659:
1655:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1608:
1603:
1602:at a point.)
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1557:
1553:
1535:
1531:
1527:
1522:
1518:
1512:
1508:
1499:
1495:
1479:
1473:
1470:
1467:
1459:
1456:over a field
1455:
1451:
1447:
1446:
1432:
1411:
1408:
1404:
1400:
1397:
1389:
1385:
1361:
1353:
1349:
1340:
1322:
1314:
1306:
1302:
1293:
1289:
1286:
1285:ruled surface
1282:
1278:
1277:David Mumford
1274:
1271:
1267:
1263:
1259:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1221:
1217:
1213:
1209:
1208:flag manifold
1205:
1190:
1187:
1184:
1179:
1175:
1154:
1148:
1145:
1142:
1134:
1130:
1126:
1122:
1118:
1114:
1098:
1095:
1090:
1086:
1077:
1073:
1069:
1053:
1050:
1045:
1041:
1032:
1028:
1024:
1020:
1019:
1013:
999:
996:
990:
982:
969:
965:
961:
957:
952:
950:
946:
942:
938:
922:
919:
916:
913:
907:
897:
893:
879:
875:
857:
853:
844:
828:
825:
822:
815:if for every
814:
810:
806:
803:
799:
783:
776:
772:
769:with a fixed
768:
764:
754:
752:
748:
744:
740:
736:
717:
709:
705:
697:-divisors in
696:
692:
688:
684:
680:
661:
653:
649:
640:
636:
632:
628:
623:
621:
617:
598:
590:
586:
562:
554:
550:
526:
518:
514:
505:
504:
498:
496:
492:
473:
465:
461:
452:
448:
444:
440:
436:
433:
414:
406:
402:
394:
390:
386:
382:
378:
374:
364:
362:
358:
354:
335:
327:
323:
314:
310:
306:
302:
297:
295:
291:
275:
272:
269:
262:
258:
254:
250:
246:
243:
239:
235:
232:
227:
210:
207:
203:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
150:
148:
144:
139:
137:
133:
129:
125:
121:
117:
113:
109:
108:
103:
99:
95:
91:
88:
84:
81:
78:
74:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
2612:
2608:
2569:
2532:
2491:
2468:
2464:
2436:
2427:
2418:
2409:
2400:
2391:
2382:
2373:
2364:
2355:
2346:
2337:
2328:
2319:
2310:
2301:
2287:
2283:
2279:
2182:
2178:
2139:
2135:
2131:
2127:
2123:
2083:
2038:
2005:
2001:
1997:
1964:
1962:
1954:cone theorem
1949:
1945:
1943:
1938:
1934:
1930:
1926:
1922:
1918:
1917:To describe
1916:
1822:
1818:
1814:
1810:
1806:
1766:
1762:
1726:
1722:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1606:
1604:
1599:
1595:
1587:
1583:
1579:
1575:
1571:
1563:
1559:
1551:
1497:
1493:
1457:
1453:
1443:
1441:
1430:
1338:
1291:
1287:
1280:
1269:
1265:
1257:
1250:
1246:
1238:
1234:
1233:is nef, but
1230:
1226:
1132:
1128:
1120:
1116:
1112:
1075:
1071:
1067:
1030:
1026:
1022:
967:
963:
959:
955:
953:
948:
944:
940:
936:
873:
812:
808:
804:
796:. Following
762:
760:
750:
746:
742:
738:
734:
694:
690:
689:-divisor on
686:
682:
678:
638:
634:
630:
624:
501:
499:
494:
490:
450:
446:
442:
438:
393:vector space
391:form a real
384:
372:
370:
367:The nef cone
360:
356:
352:
308:
305:Picard group
300:
298:
293:
289:
256:
252:
251:) is nef on
248:
244:
237:
233:
228:
194:tensor power
189:
185:
181:
177:
173:
164:which has a
161:
157:
153:
151:
147:Zariski 1962
140:
131:
127:
123:
119:
115:
111:
105:
101:
93:
89:
82:
72:
70:
60:
55:(built from
40:
26:
2566:Reid, Miles
2471:: 295–345,
1996:at a point
1725:, the face
1644:) with the
1578:) < dim(
1445:contraction
841:there is a
100:) curve in
98:irreducible
61:nef divisor
57:codimension
49:convex cone
33:line bundle
2447:References
2037:, and let
1847:Proj
1500:such that
1290:such that
1070:is nef on
880:satisfies
775:(1,1)-form
190:semi-ample
188:is called
143:Miles Reid
67:Definition
2196:→
2153:→
2058:→
2052::
2049:π
1891:⊗
1859:≥
1852:⨁
1746:→
1740::
1690:⊂
1660:∗
1572:fibration
1556:connected
1513:∗
1477:→
1471::
1409:⊗
1188:−
1152:→
1146::
1143:π
1051:≥
997:≥
979:Θ
954:Even for
923:ω
920:ϵ
917:−
914:≥
898:ϵ
889:Θ
878:curvature
858:ϵ
823:ϵ
784:ω
620:dual cone
273:⋅
208:⊗
2647:Category
2531:(2004),
2490:(1998),
1646:pullback
1586:) = dim(
1262:Jacobian
1076:distinct
1016:Examples
753:> 0.
737:nef and
491:nef cone
435:tensored
53:divisors
2637:0141668
2629:1970376
2598:0715649
2559:2095471
2522:1658959
2477:1257325
2082:. Then
1825:by the
1574:if dim(
1243:torsion
1125:blow-up
1123:is the
1111:, then
1066:, then
935:. When
741:ample,
104:. (The
85:over a
2635:
2627:
2596:
2586:
2557:
2547:
2520:
2510:
2475:
2177:, and
2000:. Let
1450:normal
1249:. For
876:whose
843:smooth
489:, the
445:.) An
107:degree
80:scheme
77:proper
2625:JSTOR
2611:, 2,
2461:(PDF)
2294:Notes
1492:with
1448:of a
1255:genus
765:be a
627:ample
176:over
160:over
118:over
87:field
75:on a
45:curve
35:on a
2584:ISBN
2545:ISBN
2508:ISBN
2126:and
1607:face
1566:has
1554:has
1167:has
1096:<
826:>
761:Let
616:dual
614:are
578:and
500:The
493:Nef(
381:real
31:, a
2617:doi
2574:doi
2537:doi
2500:doi
1925:in
1765:on
1264:of
1253:of
1241:is
1210:or
1021:If
951:).
872:on
813:nef
807:on
497:).
359:of
292:in
130:of
94:nef
41:nef
39:is
27:In
2649::
2633:MR
2631:,
2623:,
2613:76
2594:MR
2592:,
2582:,
2555:MR
2553:,
2543:,
2518:MR
2516:,
2506:,
2498:,
2494:,
2486:;
2473:MR
2467:,
2463:,
2181:−
2138::
2130:−
1941:.
1829::
1605:A
1442:A
747:cA
745:+
363:.
296:.
229:A
138:.
63:.
2619::
2576::
2539::
2502::
2469:3
2288:X
2284:X
2280:p
2264:2
2259:P
2235:1
2230:P
2206:1
2201:P
2193:X
2183:E
2179:H
2163:2
2158:P
2150:X
2140:H
2136:X
2132:E
2128:H
2124:H
2110:)
2107:X
2104:(
2099:1
2095:N
2084:X
2068:2
2063:P
2055:X
2039:E
2023:2
2018:P
2006:X
2002:H
1998:p
1982:2
1977:P
1965:X
1950:X
1946:X
1939:C
1935:L
1931:Y
1927:X
1923:C
1919:Y
1902:.
1899:)
1894:a
1887:L
1883:,
1880:X
1877:(
1872:0
1868:H
1862:0
1856:a
1843:=
1840:Y
1823:Y
1819:Y
1815:X
1811:L
1807:F
1793:)
1790:X
1787:(
1782:1
1778:N
1767:X
1763:L
1749:Y
1743:X
1737:f
1727:F
1723:X
1709:)
1706:X
1703:(
1698:1
1694:N
1687:)
1684:)
1681:Y
1678:(
1673:1
1669:N
1665:(
1656:f
1642:X
1638:X
1634:F
1630:X
1626:F
1622:F
1618:N
1614:N
1610:F
1600:Y
1596:X
1588:X
1584:Y
1580:X
1576:Y
1564:k
1560:f
1552:f
1536:Y
1532:O
1528:=
1523:X
1519:O
1509:f
1498:k
1494:Y
1480:Y
1474:X
1468:f
1458:k
1454:X
1433:.
1431:a
1417:)
1412:a
1405:L
1401:,
1398:X
1395:(
1390:0
1386:H
1365:)
1362:L
1359:(
1354:1
1350:c
1339:L
1323:2
1319:)
1315:L
1312:(
1307:1
1303:c
1292:L
1288:X
1281:L
1272:.
1270:g
1266:X
1258:g
1251:X
1247:X
1239:L
1235:L
1231:X
1227:L
1222:.
1203:.
1191:1
1185:=
1180:2
1176:E
1155:Y
1149:X
1133:E
1129:Y
1121:X
1117:X
1113:C
1099:0
1091:2
1087:C
1072:X
1068:C
1054:0
1046:2
1042:C
1031:X
1027:C
1023:X
1000:0
994:)
991:L
988:(
983:h
968:h
964:L
960:C
956:X
949:X
945:L
941:C
937:X
911:)
908:L
905:(
894:h
874:L
854:h
829:0
809:X
805:L
763:X
751:c
743:D
739:A
735:D
721:)
718:X
715:(
710:1
706:N
695:R
691:X
687:R
683:X
679:R
665:)
662:X
659:(
654:1
650:N
639:R
635:X
602:)
599:X
596:(
591:1
587:N
566:)
563:X
560:(
555:1
551:N
530:)
527:X
524:(
519:1
515:N
495:X
477:)
474:X
471:(
466:1
462:N
451:R
447:R
443:X
439:R
418:)
415:X
412:(
407:1
403:N
385:R
373:R
361:L
357:s
353:s
339:)
336:L
333:(
328:1
324:c
309:X
294:X
290:C
276:C
270:D
257:D
253:X
249:D
247:(
245:O
238:X
234:D
211:a
204:L
186:L
182:X
178:k
174:X
162:k
158:C
154:L
132:L
128:s
124:s
120:k
116:C
112:L
102:X
90:k
83:X
73:L
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.