31:
2170:, says that any birational map between two smooth complex projective varieties can be decomposed into finitely many blow-ups or blow-downs of smooth subvarieties. This is important to know, but it can still be very hard to determine whether two smooth projective varieties are birational.
2255:
Minimal models are not unique in dimensions at least 3, but any two minimal varieties which are birational are very close. For example, they are isomorphic outside subsets of codimension at least 2, and more precisely they are related by a sequence of
2286:
if it is covered by rational curves. A uniruled variety does not have a minimal model, but there is a good substitute: Birkar, Cascini, Hacon, and McKernan showed that every uniruled variety over a field of characteristic zero is birational to a
2263:
The conjecture was proved in dimension 3 by Mori. There has been great progress in higher dimensions, although the general problem remains open. In particular, Birkar, Cascini, Hacon, and McKernan (2010) proved that every variety of
2620:
showed that the birational automorphism group of a smooth quartic 3-fold is equal to its automorphism group, which is finite. In this sense, quartic 3-folds are far from being rational, since the birational automorphism group of a
2659:, in the development of explicit invariants of Fano varieties to test K-stability by computing on birational models, and in the construction of moduli spaces of Fano varieties. Important results in birational geometry such as
651:
1241:
953:
2025:
1481:
1949:. This is a measure of the complexity of a variety, with projective space having Kodaira dimension −∞. The most complicated varieties are those with Kodaira dimension equal to their dimension
870:
1682:
At first, it is not clear how to show that there are any algebraic varieties which are not rational. In order to prove this, some birational invariants of algebraic varieties are needed. A
2128:
2210:
of dimension 2, it is enough to consider smooth varieties in this definition. In dimensions at least 3, minimal varieties must be allowed to have certain mild singularities, for which
2358:. Nonetheless, the problem of determining exactly which Fano varieties are rational is far from solved. For example, it is not known whether there is any smooth cubic hypersurface in
1642:
753:
1408:
2514:
2565:
2391:
2597:
2439:
1510:
1437:
1132:
1023:
691:
525:
491:
1345:
183:
2333:
834:
985:
2777:, Corollary 1.3.3), implies that every uniruled variety in characteristic zero is birational to a Fano fiber space, using the easier result that a uniruled variety
1550:
construction. By blowing up, every smooth projective variety of dimension at least 2 is birational to infinitely many "bigger" varieties, for example with bigger
1044:
890:
795:
439:
247:
227:
203:
141:
121:
2412:
is extremely rigid, in the sense that its birational automorphism group is finite. At the other extreme, the birational automorphism group of projective space
2346:) over an algebraically closed field is rational. A major discovery in the 1970s was that starting in dimension 3, there are many Fano varieties which are not
3212:
2868:
2809:
1528:). So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting.
538:
1187:
1540:
348:
3190:
3065:
3035:
3001:
1598:
249:, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.
1602:
1546:
In dimension 1, if two smooth projective curves are birational, then they are isomorphic. But that fails in dimension at least 2, by the
379:
is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of
2633:
Birational geometry has found applications in other areas of geometry, but especially in traditional problems in algebraic geometry.
3102:
2807:
Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Włodarczyk, Jarosław (2002), "Torification and factorization of birational maps",
895:
1971:
1561:: is there a unique simplest variety in each birational equivalence class? The modern definition is that a projective variety
2291:. This leads to the problem of the birational classification of Fano fiber spaces and (as the most interesting special case)
1442:
2260:. So the minimal model conjecture would give strong information about the birational classification of algebraic varieties.
2656:
2652:
1597:
This notion works perfectly for algebraic surfaces (varieties of dimension 2). In modern terms, one central result of the
3311:
1055:
1686:
is any kind of number, ring, etc which is the same, or isomorphic, for all varieties that are birationally equivalent.
1536:
839:
3174:
3023:
1883:; then the plurigenera are birational invariants for smooth projective varieties. In particular, if any plurigenus
1543:
projective variety. Given that, it is enough to classify smooth projective varieties up to birational equivalence.
2625:
is enormous. This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces.
2065:
2645:
1067:
43:
1612:
699:
1154:. This is a birational equivalence but not an isomorphism of varieties, because it fails to be defined where
413:
minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.
3126:
2218:
1353:
2940:
2478:
2527:
2931:
2361:
2288:
2225:
2179:
1661:
1558:
100:
2570:
2339:. Fano varieties can be considered the algebraic varieties which are most similar to projective space.
2415:
1486:
1413:
1108:
999:
667:
501:
3135:
2887:
2672:
1677:
444:
2945:
2663:
proof of boundedness of Fano varieties have been used to prove existence results for moduli spaces.
2408:
Algebraic varieties differ widely in how many birational automorphisms they have. Every variety of
1539:: over a field of characteristic 0 (such as the complex numbers), every variety is birational to a
63:
1249:
3289:
3269:
3237:
3088:
2966:
2911:
2877:
2844:
2818:
2257:
1521:
771:
156:
75:
59:
3162:
2641:
162:
2306:
800:
3229:
3186:
3098:
3061:
3031:
3019:
2997:
2958:
2926:
2859:
2604:
2409:
2343:
2336:
2265:
2150:
1954:
1918:
1912:
1695:
958:
71:
3279:
3221:
3178:
3143:
3108:
3053:
2950:
2895:
2828:
2622:
2347:
2193:
1737:
1699:
1570:
1532:
406:
401:
206:
96:
3249:
3200:
3155:
3075:
3045:
3011:
2978:
2907:
2840:
3245:
3196:
3151:
3112:
3071:
3041:
3007:
2993:
2985:
2974:
2922:
2903:
2836:
2203:
1591:
1181:
767:
1525:
17:
3139:
2891:
3207:
3166:
2863:
2855:
2660:
2233:
1733:
1083:
1029:
875:
780:
424:
232:
212:
188:
126:
106:
2166:
The "Weak factorization theorem", proved by
Abramovich, Karu, Matsuki, and Włodarczyk
3305:
3293:
2446:
2277:
3147:
2848:
2915:
2637:
2292:
2057:
1551:
646:{\displaystyle f(t)=\left({\frac {2t}{1+t^{2}}},{\frac {1-t^{2}}{1+t^{2}}}\right),}
2899:
2832:
3124:(1971), "Three-dimensional quartics and counterexamples to the Lüroth problem",
2600:
1711:
79:
51:
3121:
2651:
Birational geometry has recently found important applications in the study of
2056:
is a birational invariant for smooth projective varieties. In particular, the
1547:
67:
3233:
3182:
2962:
39:
30:
1934:
goes to infinity. The
Kodaira dimension divides all varieties of dimension
1236:{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}\to \mathbb {P} ^{3}}
409:) of some dimension. Rationality is a very natural property: it means that
82:; the map may fail to be defined where the rational functions have poles.
2866:(2010), "Existence of minimal models for varieties of log general type",
493:
in the affine plane is a rational curve, because there is a rational map
297:. A birational map induces an isomorphism from a nonempty open subset of
3210:(1988), "Flip theorem and the existence of minimal models for 3-folds",
2882:
1439:. That gives another proof that this quadric surface is rational, since
3241:
2970:
3284:
3257:
2823:
2163:) is a birational invariant for smooth complex projective varieties.
35:
3225:
2954:
3274:
3093:
1093:
is algebraically closed.) To define stereographic projection, let
305:, and vice versa: an isomorphism between nonempty open subsets of
29:
1594:. It is easy to check that blown-up varieties are never minimal.
2614:
is very much a mystery: no explicit set of generators is known.
1164:(and the inverse map fails to be defined at those lines through
3258:"K-stability of Fano varieties: An algebro-geometric approach"
3083:
Kollár, János (2013). "Moduli of varieties of general type".
2929:(1972), "The intermediate Jacobian of the cubic threefold",
70:
outside lower-dimensional subsets. This amounts to studying
1800:
between smooth projective varieties induces an isomorphism
1483:
is obviously rational, having an open subset isomorphic to
2774:
2727:
2350:. In particular, smooth cubic 3-folds are not rational by
948:{\displaystyle U=\mathbb {A} _{\mathbb {C} }^{1}-\{i,-i\}}
2636:
Famously the minimal model program was used to construct
2268:
over a field of characteristic zero has a minimal model.
2020:{\displaystyle E(\Omega ^{1})=\bigotimes ^{k}\Omega ^{1}}
405:
if it is birational to affine space (or equivalently, to
2788:
has negative degree. A reference for the latter fact is
2141:
are not birational invariants, as shown by blowing up.)
1476:{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}}
2573:
2530:
2481:
2418:
2364:
2309:
2068:
1974:
1615:
1489:
1445:
1416:
1356:
1252:
1190:
1111:
1032:
1002:
961:
898:
878:
842:
803:
783:
702:
670:
541:
504:
447:
427:
235:
215:
191:
165:
129:
109:
2461:), is large (in a sense, infinite-dimensional) for
375:, meaning a morphism which is birational. That is,
209:used in algebraic geometry, a nonempty open subset
2591:
2559:
2508:
2433:
2385:
2327:
2122:
2019:
1786:has the remarkable property that a birational map
1636:
1504:
1475:
1431:
1402:
1339:
1235:
1126:
1038:
1017:
979:
947:
884:
864:
828:
789:
747:
685:
645:
519:
485:
433:
241:
221:
197:
177:
135:
115:
2354:, and smooth quartic 3-folds are not rational by
343:. In algebraic terms, two varieties over a field
2351:
2145:Fundamental group of smooth projective varieties
1945:types, with Kodaira dimension −∞, 0, 1, ..., or
1694:One useful set of birational invariants are the
2607:. By contrast, the Cremona group in dimensions
2516:is generated by the "quadratic transformation"
2342:In dimension 2, every Fano variety (known as a
2034:th tensor power of the cotangent bundle Ω with
1921:, which measures the growth of the plurigenera
1050:Birational equivalence of smooth quadrics and P
865:{\displaystyle \mathbb {A} _{\mathbb {C} }^{1}}
2617:
2355:
1516:Minimal models and resolution of singularities
8:
3213:Journal of the American Mathematical Society
2869:Journal of the American Mathematical Society
2810:Journal of the American Mathematical Society
2123:{\displaystyle h^{p,0}=H^{0}(X,\Omega ^{p})}
1652:. The two cases are mutually exclusive, and
942:
927:
1520:Every algebraic variety is birational to a
62:in which the goal is to determine when two
3171:Birational Geometry of Algebraic Varieties
2781:is covered by a family of curves on which
2703:
2691:
1917:A fundamental birational invariant is the
3283:
3273:
3092:
2944:
2881:
2822:
2580:
2576:
2575:
2572:
2550:
2549:
2529:
2499:
2498:
2489:
2480:
2425:
2421:
2420:
2417:
2371:
2367:
2366:
2363:
2319:
2314:
2308:
2111:
2092:
2073:
2067:
2011:
2001:
1985:
1973:
1622:
1618:
1617:
1614:
1579:has nonnegative degree on every curve in
1496:
1492:
1491:
1488:
1467:
1463:
1462:
1452:
1448:
1447:
1444:
1423:
1419:
1418:
1415:
1394:
1384:
1371:
1361:
1355:
1251:
1227:
1223:
1222:
1212:
1208:
1207:
1197:
1193:
1192:
1189:
1176:Birational equivalence of quadric surface
1118:
1114:
1113:
1110:
1031:
1009:
1005:
1004:
1001:
960:
918:
913:
912:
911:
907:
906:
897:
877:
856:
851:
850:
849:
845:
844:
841:
814:
802:
782:
724:
701:
677:
673:
672:
669:
626:
608:
595:
583:
562:
540:
511:
507:
506:
503:
465:
452:
446:
426:
234:
214:
190:
164:
128:
108:
2217:is still well-behaved; these are called
1965:More generally, for any natural summand
1637:{\displaystyle \mathbb {P} ^{1}\times C}
748:{\displaystyle g(x,y)={\frac {1-y}{x}}.}
2789:
2767:
2684:
417:Birational equivalence of a plane conic
2739:
2655:through general existence results for
2295:. By definition, a projective variety
2041:, the vector space of global sections
1766:, the vector space of global sections
1026:is not defined at the point (0,−1) in
391:Birational equivalence and rationality
351:are isomorphic as extension fields of
2990:Higher-Dimensional Algebraic Geometry
2792:, Corollary 4.11) and Example 4.7(1).
1961:Summands of ⊗Ω and some Hodge numbers
1863:as the dimension of the vector space
1403:{\displaystyle x_{0}x_{3}=x_{1}x_{2}}
313:by definition gives a birational map
42:. One birational map between them is
7:
3262:EMS Surveys in Mathematical Sciences
2715:
2509:{\displaystyle Cr_{2}(\mathbb {C} )}
1599:Italian school of algebraic geometry
347:are birational if and only if their
2560:{\displaystyle PGL(3,\mathbb {C} )}
2236:or birational to a minimal variety
2174:Minimal models in higher dimensions
770:gives a systematic construction of
99:from one variety (understood to be
2751:
2648:, now known as KSB moduli spaces.
2386:{\displaystyle \mathbb {P} ^{n+1}}
2108:
2008:
1982:
1609:is birational either to a product
797:is not defined on the locus where
283:such that there is a rational map
38:is birationally equivalent to the
25:
3087:. Vol. 2. pp. 131–157.
2592:{\displaystyle \mathbb {P} ^{2},}
1350:The image is the quadric surface
892:is a morphism on the open subset
836:. So, on the complex affine line
3028:Principles of Algebraic Geometry
2640:of varieties of general type by
2434:{\displaystyle \mathbb {P} ^{n}}
1505:{\displaystyle \mathbb {A} ^{2}}
1432:{\displaystyle \mathbb {P} ^{3}}
1127:{\displaystyle \mathbb {P} ^{n}}
1018:{\displaystyle \mathbb {A} ^{1}}
686:{\displaystyle \mathbb {A} ^{1}}
520:{\displaystyle \mathbb {A} ^{1}}
3148:10.1070/SM1971v015n01ABEH001536
2228:would imply that every variety
486:{\displaystyle x^{2}+y^{2}-1=0}
2554:
2540:
2503:
2495:
2404:Birational automorphism groups
2117:
2098:
1991:
1978:
1331:
1295:
1292:
1289:
1286:
1274:
1268:
1256:
1253:
1218:
971:
718:
706:
551:
545:
1:
2900:10.1090/S0894-0347-09-00649-3
2833:10.1090/S0894-0347-02-00396-X
2653:K-stability of Fano varieties
2133:are birational invariants of
1656:is unique if it exists. When
1101:. Then a birational map from
987:. Likewise, the rational map
656:which has a rational inverse
301:to a nonempty open subset of
2475:, the complex Cremona group
2303:if the anticanonical bundle
2167:
2137:. (Most other Hodge numbers
1759:is again a line bundle. For
1601:from 1890–1910, part of the
1340:{\displaystyle (,)\mapsto .}
1138:is given by sending a point
159:from a nonempty open subset
143:, written as a dashed arrow
1537:resolution of singularities
27:Field of algebraic geometry
3328:
3175:Cambridge University Press
2275:
2177:
1910:
1675:
1603:classification of surfaces
1557:This leads to the idea of
1082:must be assumed to have a
178:{\displaystyle U\subset X}
3030:. John Wiley & Sons.
2328:{\displaystyle K_{X}^{*}}
1660:exists, it is called the
1054:More generally, a smooth
829:{\displaystyle 1+t^{2}=0}
18:Birational classification
3183:10.1017/CBO9780511662560
2646:Nicholas Shepherd-Barron
2524:together with the group
2352:Clemens–Griffiths (1972)
2226:minimal model conjecture
1648:or to a minimal surface
1605:, is that every surface
1105:to the projective space
1068:stereographic projection
1058:(degree 2) hypersurface
980:{\displaystyle f:U\to X}
421:For example, the circle
44:stereographic projection
3127:Matematicheskii Sbornik
2657:Kähler–Einstein metrics
2618:Iskovskikh–Manin (1971)
2400:which is not rational.
2356:Iskovskikh–Manin (1971)
1168:which are contained in
1089:; this is automatic if
1074:a quadric over a field
341:birationally equivalent
205:. By definition of the
2692:Kollár & Mori 1998
2593:
2561:
2510:
2435:
2387:
2329:
2219:terminal singularities
2124:
2021:
1953:, called varieties of
1638:
1506:
1477:
1433:
1404:
1341:
1237:
1128:
1040:
1019:
981:
949:
886:
866:
830:
791:
749:
687:
647:
521:
487:
435:
243:
223:
199:
179:
137:
117:
47:
3256:Xu, Chenyang (2021).
2932:Annals of Mathematics
2927:Griffiths, Phillip A.
2860:Hacon, Christopher D.
2594:
2562:
2511:
2436:
2388:
2330:
2232:is either covered by
2224:That being said, the
2184:A projective variety
2180:Minimal model program
2125:
2022:
1672:Birational invariants
1639:
1571:canonical line bundle
1507:
1478:
1434:
1405:
1342:
1238:
1129:
1041:
1020:
982:
950:
887:
867:
831:
792:
750:
688:
648:
522:
488:
436:
244:
224:
200:
180:
138:
118:
33:
2673:Abundance conjecture
2571:
2567:of automorphisms of
2528:
2479:
2416:
2362:
2307:
2282:A variety is called
2066:
1972:
1702:of a smooth variety
1684:birational invariant
1678:Birational invariant
1613:
1487:
1443:
1414:
1354:
1250:
1188:
1146:to the line through
1109:
1030:
1000:
959:
896:
876:
840:
801:
781:
700:
668:
539:
502:
445:
425:
358:A special case is a
233:
213:
189:
163:
127:
107:
3312:Birational geometry
3140:1971SbMat..15..141I
3060:. Springer-Verlag.
2923:Clemens, C. Herbert
2892:2010JAMS...23..405B
2775:Birkar et al. (2010
2706:, Exercise II.8.8..
2324:
1752:th tensor power of
1535:'s 1964 theorem on
1184:gives an embedding
923:
861:
772:Pythagorean triples
360:birational morphism
229:is always dense in
123:to another variety
64:algebraic varieties
56:birational geometry
3120:Iskovskih, V. A.;
3085:Handbook of moduli
3058:Algebraic Geometry
3020:Griffiths, Phillip
2858:; Cascini, Paolo;
2728:Birkar et al. 2010
2589:
2557:
2506:
2431:
2383:
2325:
2310:
2272:Uniruled varieties
2240:. When it exists,
2120:
2017:
1899:is not zero, then
1634:
1583:; in other words,
1522:projective variety
1502:
1473:
1429:
1400:
1337:
1233:
1124:
1036:
1015:
977:
945:
905:
882:
862:
843:
826:
787:
745:
683:
643:
517:
483:
431:
269:is a rational map
239:
219:
195:
175:
155:, is defined as a
133:
113:
76:rational functions
74:that are given by
60:algebraic geometry
48:
3192:978-0-521-63277-5
3130:, Novaya Seriya,
3067:978-0-387-90244-9
3054:Hartshorne, Robin
3037:978-0-471-32792-9
3003:978-0-387-95227-7
2935:, Second Series,
2344:Del Pezzo surface
2151:fundamental group
2006:
1919:Kodaira dimension
1913:Kodaira dimension
1907:Kodaira dimension
1903:is not rational.
1744:. For an integer
1134:of lines through
1062:of any dimension
1039:{\displaystyle X}
885:{\displaystyle f}
790:{\displaystyle f}
777:The rational map
758:Applying the map
740:
633:
590:
434:{\displaystyle X}
242:{\displaystyle X}
222:{\displaystyle U}
198:{\displaystyle Y}
136:{\displaystyle Y}
116:{\displaystyle X}
16:(Redirected from
3319:
3297:
3287:
3277:
3252:
3203:
3158:
3116:
3096:
3079:
3049:
3015:
2986:Debarre, Olivier
2981:
2948:
2918:
2885:
2851:
2826:
2793:
2772:
2755:
2749:
2743:
2737:
2731:
2725:
2719:
2713:
2707:
2701:
2695:
2694:, Theorem 1.29..
2689:
2623:rational variety
2613:
2598:
2596:
2595:
2590:
2585:
2584:
2579:
2566:
2564:
2563:
2558:
2553:
2515:
2513:
2512:
2507:
2502:
2494:
2493:
2474:
2467:
2440:
2438:
2437:
2432:
2430:
2429:
2424:
2399:
2392:
2390:
2389:
2384:
2382:
2381:
2370:
2334:
2332:
2331:
2326:
2323:
2318:
2289:Fano fiber space
2194:canonical bundle
2129:
2127:
2126:
2121:
2116:
2115:
2097:
2096:
2084:
2083:
2055:
2040:
2026:
2024:
2023:
2018:
2016:
2015:
2005:
1997:
1990:
1989:
1944:
1898:
1882:
1846:
1836:
1799:
1785:
1765:
1738:cotangent bundle
1727:
1700:canonical bundle
1643:
1641:
1640:
1635:
1627:
1626:
1621:
1511:
1509:
1508:
1503:
1501:
1500:
1495:
1482:
1480:
1479:
1474:
1472:
1471:
1466:
1457:
1456:
1451:
1438:
1436:
1435:
1430:
1428:
1427:
1422:
1409:
1407:
1406:
1401:
1399:
1398:
1389:
1388:
1376:
1375:
1366:
1365:
1346:
1344:
1343:
1338:
1242:
1240:
1239:
1234:
1232:
1231:
1226:
1217:
1216:
1211:
1202:
1201:
1196:
1163:
1133:
1131:
1130:
1125:
1123:
1122:
1117:
1066:is rational, by
1045:
1043:
1042:
1037:
1025:
1024:
1022:
1021:
1016:
1014:
1013:
1008:
986:
984:
983:
978:
954:
952:
951:
946:
922:
917:
916:
910:
891:
889:
888:
883:
871:
869:
868:
863:
860:
855:
854:
848:
835:
833:
832:
827:
819:
818:
796:
794:
793:
788:
754:
752:
751:
746:
741:
736:
725:
692:
690:
689:
684:
682:
681:
676:
652:
650:
649:
644:
639:
635:
634:
632:
631:
630:
614:
613:
612:
596:
591:
589:
588:
587:
571:
563:
531:
526:
524:
523:
518:
516:
515:
510:
492:
490:
489:
484:
470:
469:
457:
456:
440:
438:
437:
432:
407:projective space
374:
327:. In this case,
326:
292:
282:
248:
246:
245:
240:
228:
226:
225:
220:
207:Zariski topology
204:
202:
201:
196:
184:
182:
181:
176:
154:
150:
142:
140:
139:
134:
122:
120:
119:
114:
46:, pictured here.
21:
3327:
3326:
3322:
3321:
3320:
3318:
3317:
3316:
3302:
3301:
3300:
3285:10.4171/EMSS/51
3255:
3226:10.2307/1990969
3208:Mori, Shigefumi
3206:
3193:
3167:Mori, Shigefumi
3161:
3119:
3105:
3082:
3068:
3052:
3038:
3018:
3004:
2994:Springer-Verlag
2984:
2955:10.2307/1970801
2946:10.1.1.401.4550
2921:
2883:math.AG/0610203
2864:McKernan, James
2856:Birkar, Caucher
2854:
2806:
2802:
2797:
2796:
2786:
2773:
2769:
2764:
2759:
2758:
2750:
2746:
2738:
2734:
2726:
2722:
2714:
2710:
2704:Hartshorne 1977
2702:
2698:
2690:
2686:
2681:
2669:
2631:
2608:
2574:
2569:
2568:
2526:
2525:
2485:
2477:
2476:
2469:
2462:
2456:
2445:, known as the
2419:
2414:
2413:
2406:
2394:
2365:
2360:
2359:
2305:
2304:
2280:
2274:
2234:rational curves
2215:
2200:
2182:
2176:
2158:
2147:
2107:
2088:
2069:
2064:
2063:
2042:
2035:
2007:
1981:
1970:
1969:
1963:
1939:
1929:
1915:
1909:
1893:
1891:
1880:
1864:
1862:
1841:
1834:
1817:
1801:
1787:
1783:
1767:
1760:
1757:
1728:, which is the
1724:
1719:
1692:
1680:
1674:
1644:for some curve
1616:
1611:
1610:
1588:
1577:
1531:Much deeper is
1518:
1490:
1485:
1484:
1461:
1446:
1441:
1440:
1417:
1412:
1411:
1390:
1380:
1367:
1357:
1352:
1351:
1248:
1247:
1221:
1206:
1191:
1186:
1185:
1182:Segre embedding
1178:
1155:
1112:
1107:
1106:
1087:-rational point
1052:
1028:
1027:
1003:
998:
997:
988:
957:
956:
894:
893:
874:
873:
838:
837:
810:
799:
798:
779:
778:
768:rational number
726:
698:
697:
671:
666:
665:
622:
615:
604:
597:
579:
572:
564:
561:
557:
537:
536:
505:
500:
499:
494:
461:
448:
443:
442:
423:
422:
419:
393:
362:
349:function fields
335:are said to be
314:
284:
270:
255:
253:Birational maps
231:
230:
211:
210:
187:
186:
161:
160:
148:
144:
125:
124:
105:
104:
93:
88:
86:Birational maps
28:
23:
22:
15:
12:
11:
5:
3325:
3323:
3315:
3314:
3304:
3303:
3299:
3298:
3253:
3220:(1): 117–253,
3204:
3191:
3159:
3134:(1): 140–166,
3117:
3103:
3080:
3066:
3050:
3036:
3024:Harris, Joseph
3016:
3002:
2982:
2939:(2): 281–356,
2919:
2876:(2): 405–468,
2852:
2817:(3): 531–572,
2803:
2801:
2798:
2795:
2794:
2784:
2766:
2765:
2763:
2760:
2757:
2756:
2744:
2732:
2720:
2708:
2696:
2683:
2682:
2680:
2677:
2676:
2675:
2668:
2665:
2630:
2627:
2588:
2583:
2578:
2556:
2552:
2548:
2545:
2542:
2539:
2536:
2533:
2522:
2521:
2505:
2501:
2497:
2492:
2488:
2484:
2452:
2428:
2423:
2405:
2402:
2380:
2377:
2374:
2369:
2322:
2317:
2313:
2293:Fano varieties
2276:Main article:
2273:
2270:
2213:
2198:
2178:Main article:
2175:
2172:
2156:
2146:
2143:
2131:
2130:
2119:
2114:
2110:
2106:
2103:
2100:
2095:
2091:
2087:
2082:
2079:
2076:
2072:
2028:
2027:
2014:
2010:
2004:
2000:
1996:
1993:
1988:
1984:
1980:
1977:
1962:
1959:
1925:
1911:Main article:
1908:
1905:
1887:
1876:
1858:
1830:
1813:
1779:
1755:
1734:exterior power
1722:
1691:
1688:
1676:Main article:
1673:
1670:
1633:
1630:
1625:
1620:
1586:
1575:
1559:minimal models
1517:
1514:
1499:
1494:
1470:
1465:
1460:
1455:
1450:
1426:
1421:
1397:
1393:
1387:
1383:
1379:
1374:
1370:
1364:
1360:
1348:
1347:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1230:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1177:
1174:
1121:
1116:
1097:be a point in
1051:
1048:
1035:
1012:
1007:
976:
973:
970:
967:
964:
944:
941:
938:
935:
932:
929:
926:
921:
915:
909:
904:
901:
881:
859:
853:
847:
825:
822:
817:
813:
809:
806:
786:
756:
755:
744:
739:
735:
732:
729:
723:
720:
717:
714:
711:
708:
705:
680:
675:
654:
653:
642:
638:
629:
625:
621:
618:
611:
607:
603:
600:
594:
586:
582:
578:
575:
570:
567:
560:
556:
553:
550:
547:
544:
514:
509:
482:
479:
476:
473:
468:
464:
460:
455:
451:
441:with equation
430:
418:
415:
399:is said to be
392:
389:
259:birational map
254:
251:
238:
218:
194:
174:
171:
168:
132:
112:
92:
89:
87:
84:
58:is a field of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3324:
3313:
3310:
3309:
3307:
3295:
3291:
3286:
3281:
3276:
3271:
3267:
3263:
3259:
3254:
3251:
3247:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3215:
3214:
3209:
3205:
3202:
3198:
3194:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3163:Kollár, János
3160:
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3128:
3123:
3122:Manin, Ju. I.
3118:
3114:
3110:
3106:
3104:9781571462589
3100:
3095:
3090:
3086:
3081:
3077:
3073:
3069:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3005:
2999:
2995:
2991:
2987:
2983:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2947:
2942:
2938:
2934:
2933:
2928:
2924:
2920:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2884:
2879:
2875:
2871:
2870:
2865:
2861:
2857:
2853:
2850:
2846:
2842:
2838:
2834:
2830:
2825:
2820:
2816:
2812:
2811:
2805:
2804:
2799:
2791:
2790:Debarre (2001
2787:
2780:
2776:
2771:
2768:
2761:
2753:
2748:
2745:
2741:
2736:
2733:
2729:
2724:
2721:
2717:
2712:
2709:
2705:
2700:
2697:
2693:
2688:
2685:
2678:
2674:
2671:
2670:
2666:
2664:
2662:
2658:
2654:
2649:
2647:
2643:
2639:
2638:moduli spaces
2634:
2628:
2626:
2624:
2619:
2615:
2611:
2606:
2602:
2586:
2581:
2546:
2543:
2537:
2534:
2531:
2519:
2518:
2517:
2490:
2486:
2482:
2472:
2465:
2460:
2455:
2451:
2448:
2447:Cremona group
2444:
2441:over a field
2426:
2411:
2403:
2401:
2397:
2378:
2375:
2372:
2357:
2353:
2349:
2345:
2340:
2338:
2320:
2315:
2311:
2302:
2298:
2294:
2290:
2285:
2279:
2278:Ruled variety
2271:
2269:
2267:
2261:
2259:
2253:
2251:
2247:
2246:minimal model
2243:
2239:
2235:
2231:
2227:
2222:
2220:
2216:
2209:
2205:
2201:
2195:
2191:
2187:
2181:
2173:
2171:
2169:
2164:
2162:
2155:
2152:
2144:
2142:
2140:
2136:
2112:
2104:
2101:
2093:
2089:
2085:
2080:
2077:
2074:
2070:
2062:
2061:
2060:
2059:
2058:Hodge numbers
2053:
2049:
2045:
2038:
2033:
2012:
2002:
1998:
1994:
1986:
1975:
1968:
1967:
1966:
1960:
1958:
1956:
1952:
1948:
1942:
1937:
1933:
1928:
1924:
1920:
1914:
1906:
1904:
1902:
1896:
1890:
1886:
1879:
1875:
1871:
1867:
1861:
1857:
1854:
1850:
1847:, define the
1844:
1838:
1833:
1829:
1825:
1821:
1816:
1812:
1808:
1804:
1798:
1794:
1790:
1782:
1778:
1774:
1770:
1763:
1758:
1751:
1747:
1743:
1739:
1735:
1731:
1725:
1717:
1713:
1709:
1706:of dimension
1705:
1701:
1697:
1689:
1687:
1685:
1679:
1671:
1669:
1667:
1663:
1662:minimal model
1659:
1655:
1651:
1647:
1631:
1628:
1623:
1608:
1604:
1600:
1595:
1593:
1589:
1582:
1578:
1572:
1568:
1564:
1560:
1555:
1553:
1552:Betti numbers
1549:
1544:
1542:
1538:
1534:
1529:
1527:
1523:
1515:
1513:
1497:
1468:
1458:
1453:
1424:
1395:
1391:
1385:
1381:
1377:
1372:
1368:
1362:
1358:
1334:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1283:
1280:
1277:
1271:
1265:
1262:
1259:
1246:
1245:
1244:
1228:
1213:
1203:
1198:
1183:
1175:
1173:
1171:
1167:
1162:
1158:
1153:
1149:
1145:
1141:
1137:
1119:
1104:
1100:
1096:
1092:
1088:
1086:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1049:
1047:
1033:
1010:
995:
991:
974:
968:
965:
962:
939:
936:
933:
930:
924:
919:
902:
899:
879:
857:
823:
820:
815:
811:
807:
804:
784:
775:
773:
769:
765:
761:
742:
737:
733:
730:
727:
721:
715:
712:
709:
703:
696:
695:
694:
678:
663:
659:
640:
636:
627:
623:
619:
616:
609:
605:
601:
598:
592:
584:
580:
576:
573:
568:
565:
558:
554:
548:
542:
535:
534:
533:
530:
512:
497:
480:
477:
474:
471:
466:
462:
458:
453:
449:
428:
416:
414:
412:
408:
404:
403:
398:
390:
388:
386:
383:to points in
382:
378:
373:
369:
365:
361:
356:
354:
350:
346:
342:
338:
334:
330:
325:
321:
317:
312:
308:
304:
300:
296:
291:
287:
281:
277:
273:
268:
264:
260:
252:
250:
236:
216:
208:
192:
172:
169:
166:
158:
153:
147:
130:
110:
102:
98:
91:Rational maps
90:
85:
83:
81:
77:
73:
69:
65:
61:
57:
53:
45:
41:
37:
32:
19:
3265:
3261:
3217:
3211:
3170:
3131:
3125:
3084:
3057:
3027:
2989:
2936:
2930:
2873:
2867:
2824:math/9904135
2814:
2808:
2782:
2778:
2770:
2747:
2735:
2723:
2711:
2699:
2687:
2650:
2642:János Kollár
2635:
2632:
2629:Applications
2616:
2609:
2523:
2470:
2463:
2458:
2453:
2449:
2442:
2410:general type
2407:
2395:
2341:
2300:
2296:
2283:
2281:
2266:general type
2262:
2254:
2249:
2245:
2244:is called a
2241:
2237:
2229:
2223:
2211:
2207:
2196:
2189:
2185:
2183:
2165:
2160:
2153:
2148:
2138:
2134:
2132:
2051:
2047:
2043:
2036:
2031:
2029:
1964:
1955:general type
1950:
1946:
1940:
1935:
1931:
1926:
1922:
1916:
1900:
1894:
1888:
1884:
1877:
1873:
1869:
1865:
1859:
1855:
1852:
1848:
1842:
1839:
1831:
1827:
1823:
1819:
1814:
1810:
1806:
1802:
1796:
1792:
1788:
1780:
1776:
1772:
1768:
1761:
1753:
1749:
1745:
1741:
1729:
1720:
1715:
1707:
1703:
1693:
1683:
1681:
1665:
1657:
1653:
1649:
1645:
1606:
1596:
1584:
1580:
1573:
1566:
1562:
1556:
1545:
1530:
1526:Chow's lemma
1519:
1349:
1179:
1169:
1165:
1160:
1156:
1151:
1147:
1143:
1139:
1135:
1102:
1098:
1094:
1090:
1084:
1079:
1075:
1071:
1063:
1059:
1053:
993:
989:
776:
763:
759:
757:
661:
657:
655:
528:
495:
420:
410:
400:
396:
394:
384:
380:
376:
371:
367:
363:
359:
357:
352:
344:
340:
336:
332:
328:
323:
319:
315:
310:
306:
302:
298:
294:
289:
285:
279:
275:
271:
266:
262:
258:
256:
151:
145:
97:rational map
94:
78:rather than
55:
49:
3268:: 265–354.
2740:Kollár 2013
2605:Castelnuovo
2601:Max Noether
1712:line bundle
1696:plurigenera
1690:Plurigenera
293:inverse to
101:irreducible
80:polynomials
52:mathematics
3275:2011.10477
3113:1322.14006
2800:References
2188:is called
1853:plurigenus
1710:means the
1548:blowing up
395:A variety
337:birational
68:isomorphic
3294:204829174
3234:0894-0347
3094:1008.0621
2963:0003-486X
2941:CiteSeerX
2716:Mori 1988
2679:Citations
2321:∗
2109:Ω
2009:Ω
1999:⨂
1983:Ω
1629:×
1459:×
1293:↦
1243:given by
1219:→
1204:×
972:→
937:−
925:−
731:−
693:given by
602:−
532:given by
472:−
170:⊂
3306:Category
3169:(1998),
3056:(1977).
3026:(1978).
2988:(2001).
2849:18211120
2667:See also
2661:Birkar's
2348:rational
2284:uniruled
1791: :
1664:of
1533:Hironaka
992: :
498: :
402:rational
366: :
318: :
274: :
157:morphism
72:mappings
3250:0924704
3242:1990969
3201:1658959
3156:0291172
3136:Bibcode
3076:0463157
3046:0507725
3012:1841091
2979:0302652
2971:1970801
2916:3342362
2908:2601039
2888:Bibcode
2841:1896232
2752:Xu 2021
2192:if the
2190:minimal
2030:of the
1736:of the
1718:-forms
1569:if the
1567:minimal
1070:. (For
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3292:
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2206:. For
2168:(2002)
1897:> 0
1748:, the
1698:. The
1541:smooth
36:circle
3290:S2CID
3270:arXiv
3238:JSTOR
3089:arXiv
2967:JSTOR
2912:S2CID
2878:arXiv
2845:S2CID
2819:arXiv
2762:Notes
2393:with
2337:ample
2258:flops
1938:into
1892:with
762:with
339:, or
261:from
3230:ISSN
3187:ISBN
3099:ISBN
3062:ISBN
3032:ISBN
2998:ISBN
2959:ISSN
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2603:and
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2149:The
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1840:For
1818:) ≅
1180:The
1150:and
331:and
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34:The
3280:doi
3222:doi
3179:doi
3144:doi
3109:Zbl
2951:doi
2896:doi
2829:doi
2612:≥ 3
2599:by
2473:= 2
2466:≥ 2
2398:≥ 4
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2202:is
2039:≥ 0
1943:+ 2
1930:as
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