2038:
3049:. Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a
2203:
3195:
is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a
2632:-canonical maps may have base points, meaning that they are not defined everywhere (i.e., they may not be a morphism of varieties). They may have positive dimensional fibers, and even if they have zero-dimensional fibers, they need not be local analytic isomorphisms.
3021:= 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof,
2973:
consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.
1717:
2282:
594:
1896:
2332:, there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique
3170:
870:
2112:
1888:
1366:
1186:
1578:
1059:
2477:
106:
1274:
667:
2087:
1519:
3269:
A fundamental theorem of Birkar–Cascini–Hacon–McKernan from 2006 is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
1789:
1765:
1741:
427:
3211:
proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated. In particular, this was known to imply the existence of a
1611:
1473:
1098:
765:
137:
2830:
is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by
1815:
1438:
201:
2555:
2381:
1638:
1301:
1412:
2500:
2404:
980:
2575:
2524:
2426:
2354:
2330:
2107:
1386:
1321:
1206:
1118:
1024:
1004:
957:
933:
913:
890:
812:
785:
732:
710:
471:
447:
387:
367:
339:
311:
291:
253:
225:
157:
68:
48:
2877:
as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in
673:
2296:, the canonical bundle formula gives that this fibration has no multiple fibers. A similar deduction can be made for any minimal genus 1 fibration of a
3060:
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.
1651:
3548:
3520:
3322:
2211:
2033:{\displaystyle \omega _{X}\cong f^{*}({\mathcal {L}}^{-1}\otimes \omega _{B})\otimes {\mathcal {O}}_{X}\left(\sum _{i=1}^{r}a_{i}F_{i}'\right)}
521:
3041:
of the space of sections of the canonical bundle map onto the sections of its tensor powers. This implies for instance the generation of the
2644:. A global section of the canonical bundle is therefore the same as an everywhere-regular differential form. Classically, these were called
2198:{\displaystyle \operatorname {deg} \left({\mathcal {L}}^{-1}\right)=\chi ({\mathcal {O}}_{X})+\operatorname {length} ({\mathcal {T}})}
3091:
817:
3013:
at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a)
2953:− 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves
1820:
672:
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is
1121:
3560:
3532:
3334:
1326:
3467:
1127:
3280:
is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean
2645:
3445:
1029:
3494:
3455:
3388:
3360:
3266:
will admit a canonical model (more generally, this is true for normal complete
Gorenstein algebraic spaces).
2746:
is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the
2431:
76:
1211:
3601:
2590:
605:
3576:
3401:
2046:
1524:
3489:
3450:
3383:
3355:
3184:
1478:
3484:
3378:
3350:
2640:
The best studied case is that of curves. Here, the canonical bundle is the same as the (holomorphic)
3507:
3033:. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is
1770:
1746:
1722:
2930:
2929:= 5 when it is an intersection of three quadrics. There is a converse, which is a corollary to the
402:
3596:
2874:
2913:. All non-singular plane quartics arise in this way. There is explicit information for the case
1583:
3042:
3009:, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for
2624:
into a projective space of dimension one less than the dimension of the global sections of the
1446:
1071:
738:
3208:
3046:
2982:
2800:
2289:
228:
122:
3577:"09w5033: Complex Analysis and Complex Geometry | Banff International Research Station"
1794:
1417:
176:
3285:
2533:
2359:
1645:
1616:
1279:
25:
8:
3297:
3255:
3045:
on such curves by the differentials of the first kind; and this has consequences for the
2986:
2910:
2796:
1391:
936:
2482:
2386:
962:
3034:
2560:
2509:
2411:
2339:
2315:
2092:
1371:
1306:
1191:
1103:
1009:
989:
942:
918:
898:
875:
797:
770:
717:
695:
486:
456:
432:
372:
352:
324:
318:
296:
276:
238:
210:
167:
142:
53:
33:
3556:
3528:
3330:
3302:
3273:
3224:
3187:
of the image of the canonical map. This can be true even when the canonical class of
3180:
3026:
2557:
defined above. In the absence of the normality hypothesis, the same result holds if
2503:
450:
314:
28:
3054:
2641:
2304:
will always admit multiple fibers and so, such a surface will not admit a section.
2301:
2293:
1065:
893:
256:
171:
117:
3281:
3251:
3038:
2962:
2946:
2776:
2767:
2578:
788:
271:
2712:, for example, a meromorphic differential with double pole at the origin on the
1712:{\displaystyle R^{1}f_{*}{\mathcal {O}}_{X}={\mathcal {L}}\oplus {\mathcal {T}}}
3427:
3069:
3014:
2734:
2713:
496:
397:
163:
113:
3590:
3227:
of the canonical ring. If the canonical ring is not finitely generated, then
3050:
3022:
2922:
791:
267:
232:
2594:
2527:
2333:
2277:{\displaystyle \operatorname {length} ({\mathcal {T}})=0\iff a_{i}=m_{i}-1}
474:
1521:
be the finitely many fibers that are not geometrically integral and write
1120:
are geometrically integral and all fibers are geometrically connected (by
3083:
71:
17:
3223:. When the canonical ring is finitely generated, the canonical model is
2774:-canonical map is a curve. The image of the 1-canonical map is called a
589:{\displaystyle \omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D)).}
2297:
3242:
admits no canonical model. One can show that if the canonical divisor
3029:). The terminology is confused, since the result is also called the
3219:
with mild singularities that could be constructed by blowing down
3508:
http://hal.archives-ouvertes.fr/docs/00/40/42/57/PDF/these-OD.pdf
2918:
676:, which allows one to deduce results about the singularities of
3165:{\displaystyle R=\bigoplus _{d=0}^{\infty }H^{0}(V,K_{V}^{d}).}
1613:
is greatest common divisor of coefficients of the expansion of
865:{\displaystyle f_{*}{\mathcal {O}}_{X}\cong {\mathcal {O}}_{B}}
2977:
More refined information is available, for larger values of
2811:
is a polynomial of degree 6 (without repeated roots) then
507:. The adjunction formula relates the canonical bundles of
2993:: the dimension of the space of quadrics passing through
2300:. On the other hand, a minimal genus one fibration of an
3402:"Geometric Form of Riemann-Roch | Rigorous Trivialities"
2981:, but in these cases canonical curves are not generally
1883:{\displaystyle h^{0}(X_{b},{\mathcal {O}}_{X_{b}})>1}
3555:. Springer Science & Business Media. p. 123.
3527:. Springer Science & Business Media. p. 242.
3329:. Springer Science & Business Media. p. 111.
2909:= 3 the canonical curves (non-hyperelliptic case) are
2807:
a double cover of its canonical curve. For example if
3094:
2985:, and the description requires more consideration of
2563:
2536:
2512:
2485:
2434:
2414:
2389:
2362:
2342:
2318:
2214:
2115:
2095:
2049:
1899:
1823:
1797:
1773:
1749:
1725:
1654:
1619:
1586:
1527:
1481:
1449:
1420:
1394:
1374:
1329:
1309:
1282:
1214:
1194:
1130:
1106:
1074:
1032:
1012:
992:
965:
945:
921:
901:
878:
820:
800:
773:
741:
720:
698:
608:
524:
459:
435:
405:
375:
355:
327:
313:
giving rise to the canonical bundle — it is an
299:
279:
241:
213:
203:. Equivalently, it is the line bundle of holomorphic
179:
145:
125:
79:
56:
36:
3234:
is not a variety, and so it cannot be birational to
2917:= 4, when a canonical curve is an intersection of a
2288:
For example, for the minimal genus 1 fibration of a
959:
do not contain rational curves of self-intersection
2766:has genus two or more, then the canonical class is
3164:
2684:, and the canonical class is the class of −2
2569:
2549:
2518:
2494:
2471:
2420:
2398:
2375:
2348:
2324:
2276:
2197:
2101:
2081:
2032:
1882:
1809:
1783:
1759:
1735:
1711:
1632:
1605:
1572:
1513:
1467:
1432:
1406:
1380:
1360:
1315:
1295:
1268:
1200:
1180:
1112:
1092:
1053:
1018:
998:
974:
951:
927:
907:
884:
864:
806:
779:
759:
726:
704:
661:
588:
465:
441:
421:
381:
361:
333:
305:
285:
247:
219:
195:
151:
131:
100:
62:
42:
2897:is at least 3, the morphism is an isomorphism of
81:
3588:
3204:is any sufficiently divisible positive integer.
2383:that is referred to as the canonical divisor on
2784:always sits in a projective space of dimension
687:
2873:This means that the canonical map is given by
2601:into projective space. This map is called the
1026:is birationally ruled, that is, birational to
1006:admits a (minimal) genus 0 fibration, then is
2961:at least 3), Riemann-Roch, and the theory of
1361:{\displaystyle m=\operatorname {gcd} (a_{i})}
1064:For a minimal genus 1 fibration (also called
2530:class, which is equal to the divisor class
1640:into integral components; these are called
1181:{\displaystyle F=\sum _{i=1}^{n}a_{i}E_{i}}
2941:embedded in projective space of dimension
2609:th multiple of the canonical class is the
2244:
2240:
255:. It may equally well be considered as an
3443:
3372:
3370:
3017:and (b) non-singular plane quintics when
2696:. This follows from the calculus formula
1035:
480:
80:
2725:and its multiples are not effective. If
2648:. The degree of the canonical class is 2
1054:{\displaystyle \mathbb {P} ^{1}\times B}
341:, and any divisor in it may be called a
3547:
3519:
3376:
3321:
2472:{\displaystyle h^{-d}(\omega _{X}^{.})}
2408:Alternately, again on a normal variety
935:is a smooth projective surface and the
101:{\displaystyle \,\!\Omega ^{n}=\omega }
3589:
3367:
1269:{\displaystyle F.E_{i}=K_{X}.E_{i}=0,}
3416:Algebraic Curves and Riemann Surfaces
2757:
2668:is a smooth algebraic curve of genus
2605:. The rational map determined by the
1443:Consider a minimal genus 1 fibration
662:{\displaystyle K_{D}=(K_{X}+D)|_{D}.}
599:In terms of canonical classes, it is
2965:is rather close. Effective divisors
2628:th multiple of the canonical class.
2082:{\displaystyle 0\leq a_{i}<m_{i}}
1573:{\displaystyle F_{i}=m_{i}F_{i}^{'}}
3215:, a particular birational model of
2997:as embedded as canonical curve is (
2635:
429:. When the anticanonical bundle of
13:
3117:
3063:
2957:(in the non-hyperelliptic case of
2901:with its image, which has degree 2
2226:
2187:
2159:
2129:
1965:
1929:
1853:
1776:
1752:
1728:
1704:
1694:
1678:
1514:{\displaystyle F_{1},\dots ,F_{r}}
851:
834:
566:
126:
83:
14:
3613:
3183:, then the canonical ring is the
2889:Otherwise, for non-hyperelliptic
2584:
2502:'th cohomology of the normalized
2356:. It is this class, denoted by
2307:
1100:all but finitely many fibers of
3569:
3541:
3513:
3501:
3468:Igor Rostislavovich Shafarevich
3191:is not ample. For instance, if
2884:
2652:− 2 for a curve of genus
2646:differentials of the first kind
2526:. This sheaf corresponds to a
1122:Zariski's connectedness theorem
982:, then the fibration is called
3477:
3461:
3437:
3421:
3408:
3394:
3343:
3315:
3156:
3132:
2466:
2448:
2241:
2231:
2221:
2192:
2182:
2170:
2153:
1956:
1923:
1871:
1834:
1784:{\displaystyle {\mathcal {T}}}
1760:{\displaystyle {\mathcal {T}}}
1736:{\displaystyle {\mathcal {L}}}
1459:
1355:
1342:
1124:). In particular, for a fiber
1084:
751:
646:
641:
622:
580:
577:
571:
548:
515:. It is a natural isomorphism
349:divisor is any divisor −
1:
2780:. A canonical curve of genus
1388:is geometrically integral if
2659:
814:to a smooth curve such that
688:The canonical bundle formula
422:{\displaystyle \omega ^{-1}}
7:
3490:Encyclopedia of Mathematics
3451:Encyclopedia of Mathematics
3384:Encyclopedia of Mathematics
3356:Encyclopedia of Mathematics
3291:
3262:is greater than zero, then
3185:homogeneous coordinate ring
2799:, the canonical curve is a
1743:is an invertible sheaf and
10:
3618:
3446:"Noether–Enriques theorem"
3444:Iskovskih, V. A. (2001) ,
3175:If the canonical class of
3067:
2589:If the canonical class is
2290:(quasi)-bielliptic surface
1646:cohomology and base change
1606:{\displaystyle m_{i}>1}
1303:is a canonical divisor of
680:from the singularities of
484:
2989:. The field started with
3432:The Geometry of Syzygies
3377:Parshin, A. N. (2001) ,
3308:
3031:Noether–Enriques theorem
1468:{\displaystyle f:X\to B}
1093:{\displaystyle f:X\to B}
760:{\displaystyle f:X\to B}
3043:quadratic differentials
2933:: a non-singular curve
2875:homogeneous coordinates
2754:is the map to a point.
2750:-canonical map for any
2593:, then it determines a
1890:). Then, one has that
712:be a normal surface. A
674:inversion of adjunction
503:is a smooth divisor on
132:{\displaystyle \Omega }
3200:-canonical map, where
3166:
3121:
2983:complete intersections
2770:, so the image of any
2571:
2551:
2520:
2496:
2473:
2422:
2400:
2377:
2350:
2326:
2312:On a singular variety
2278:
2199:
2103:
2083:
2034:
2001:
1884:
1811:
1810:{\displaystyle b\in B}
1785:
1761:
1737:
1713:
1634:
1607:
1574:
1515:
1469:
1434:
1433:{\displaystyle m>1}
1408:
1382:
1362:
1317:
1297:
1270:
1202:
1182:
1157:
1114:
1094:
1055:
1020:
1000:
976:
953:
929:
909:
886:
866:
808:
781:
761:
728:
706:
663:
590:
481:The adjunction formula
467:
443:
423:
383:
363:
335:
307:
287:
249:
221:
197:
196:{\displaystyle T^{*}V}
153:
133:
102:
64:
44:
3209:minimal model program
3167:
3101:
3047:local Torelli theorem
2991:Max Noether's theorem
2801:rational normal curve
2620:-canonical map sends
2572:
2552:
2550:{\displaystyle K_{X}}
2521:
2497:
2474:
2423:
2401:
2378:
2376:{\displaystyle K_{X}}
2351:
2327:
2279:
2200:
2104:
2084:
2035:
1981:
1885:
1812:
1786:
1762:
1738:
1714:
1635:
1633:{\displaystyle F_{i}}
1608:
1575:
1516:
1470:
1435:
1409:
1383:
1363:
1318:
1298:
1296:{\displaystyle K_{X}}
1271:
1203:
1183:
1137:
1115:
1095:
1056:
1021:
1001:
977:
954:
930:
910:
887:
867:
809:
782:
762:
729:
707:
664:
591:
468:
444:
424:
396:is the corresponding
384:
364:
336:
308:
288:
250:
222:
198:
154:
134:
103:
65:
45:
3472:Algebraic geometry I
3286:transcendence degree
3092:
2931:Riemann–Roch theorem
2911:quartic plane curves
2905:− 2. Thus for
2561:
2534:
2510:
2483:
2432:
2412:
2387:
2360:
2340:
2316:
2212:
2113:
2093:
2047:
1897:
1821:
1795:
1771:
1767:is a torsion sheaf (
1747:
1723:
1652:
1617:
1584:
1525:
1479:
1447:
1418:
1392:
1372:
1327:
1307:
1280:
1212:
1192:
1128:
1104:
1072:
1030:
1010:
990:
963:
943:
919:
899:
876:
818:
798:
771:
739:
718:
696:
606:
522:
457:
433:
403:
394:anticanonical bundle
373:
353:
325:
297:
277:
239:
211:
177:
143:
123:
77:
70:over a field is the
54:
34:
3602:Algebraic varieties
3298:Birational geometry
3155:
2987:commutative algebra
2797:hyperelliptic curve
2465:
2428:, one can consider
2024:
1569:
1407:{\displaystyle m=1}
1066:elliptic fibrations
170:of the holomorphic
3553:Algebraic Surfaces
3525:Algebraic Surfaces
3485:"Torelli theorems"
3327:Algebraic Surfaces
3162:
3141:
3035:normally generated
2758:Hyperelliptic case
2581:in dimension one.
2567:
2547:
2516:
2495:{\displaystyle -d}
2492:
2469:
2451:
2418:
2399:{\displaystyle X.}
2396:
2373:
2346:
2322:
2274:
2205:. One notes that
2195:
2099:
2079:
2030:
2012:
1880:
1807:
1781:
1757:
1733:
1709:
1630:
1603:
1570:
1551:
1511:
1465:
1430:
1404:
1378:
1358:
1313:
1293:
1266:
1198:
1178:
1110:
1090:
1051:
1016:
996:
986:. For example, if
975:{\displaystyle -1}
972:
949:
925:
905:
882:
872:and all fibers of
862:
804:
777:
757:
724:
702:
659:
586:
487:Adjunction formula
463:
439:
419:
379:
359:
331:
319:linear equivalence
303:
283:
245:
217:
193:
168:determinant bundle
149:
129:
98:
60:
40:
3434:(2005), p. 181-2.
3379:"Canonical curve"
3351:"canonical class"
3303:Differential form
3274:Kodaira dimension
3256:self intersection
3238:; in particular,
3181:ample line bundle
3027:Federigo Enriques
2949:curve of degree 2
2716:. In particular,
2570:{\displaystyle X}
2519:{\displaystyle X}
2504:dualizing complex
2421:{\displaystyle X}
2349:{\displaystyle X}
2325:{\displaystyle X}
2294:Albanese morphism
2102:{\displaystyle i}
1381:{\displaystyle F}
1316:{\displaystyle X}
1201:{\displaystyle f}
1113:{\displaystyle f}
1019:{\displaystyle X}
999:{\displaystyle X}
952:{\displaystyle f}
928:{\displaystyle X}
908:{\displaystyle g}
885:{\displaystyle f}
807:{\displaystyle f}
780:{\displaystyle X}
727:{\displaystyle g}
705:{\displaystyle X}
466:{\displaystyle V}
442:{\displaystyle V}
382:{\displaystyle K}
362:{\displaystyle K}
343:canonical divisor
334:{\displaystyle V}
315:equivalence class
306:{\displaystyle V}
286:{\displaystyle K}
248:{\displaystyle V}
220:{\displaystyle V}
152:{\displaystyle V}
63:{\displaystyle n}
43:{\displaystyle V}
29:algebraic variety
3609:
3581:
3580:
3573:
3567:
3566:
3545:
3539:
3538:
3517:
3511:
3505:
3499:
3498:
3481:
3475:
3465:
3459:
3458:
3441:
3435:
3425:
3419:
3418:(1995), Ch. VII.
3412:
3406:
3405:
3404:. 7 August 2008.
3398:
3392:
3391:
3374:
3365:
3364:
3347:
3341:
3340:
3319:
3254:divisor and the
3233:
3171:
3169:
3168:
3163:
3154:
3149:
3131:
3130:
3120:
3115:
3055:Veronese surface
3039:symmetric powers
2963:special divisors
2868:
2867:
2850:
2849:
2790:
2692:is any point of
2642:cotangent bundle
2636:Canonical curves
2576:
2574:
2573:
2568:
2556:
2554:
2553:
2548:
2546:
2545:
2525:
2523:
2522:
2517:
2501:
2499:
2498:
2493:
2478:
2476:
2475:
2470:
2464:
2459:
2447:
2446:
2427:
2425:
2424:
2419:
2405:
2403:
2402:
2397:
2382:
2380:
2379:
2374:
2372:
2371:
2355:
2353:
2352:
2347:
2331:
2329:
2328:
2323:
2302:Enriques surface
2283:
2281:
2280:
2275:
2267:
2266:
2254:
2253:
2230:
2229:
2204:
2202:
2201:
2196:
2191:
2190:
2169:
2168:
2163:
2162:
2146:
2142:
2141:
2133:
2132:
2108:
2106:
2105:
2100:
2088:
2086:
2085:
2080:
2078:
2077:
2065:
2064:
2039:
2037:
2036:
2031:
2029:
2025:
2020:
2011:
2010:
2000:
1995:
1975:
1974:
1969:
1968:
1955:
1954:
1942:
1941:
1933:
1932:
1922:
1921:
1909:
1908:
1889:
1887:
1886:
1881:
1870:
1869:
1868:
1867:
1857:
1856:
1846:
1845:
1833:
1832:
1816:
1814:
1813:
1808:
1791:is supported on
1790:
1788:
1787:
1782:
1780:
1779:
1766:
1764:
1763:
1758:
1756:
1755:
1742:
1740:
1739:
1734:
1732:
1731:
1718:
1716:
1715:
1710:
1708:
1707:
1698:
1697:
1688:
1687:
1682:
1681:
1674:
1673:
1664:
1663:
1639:
1637:
1636:
1631:
1629:
1628:
1612:
1610:
1609:
1604:
1596:
1595:
1579:
1577:
1576:
1571:
1568:
1567:
1559:
1550:
1549:
1537:
1536:
1520:
1518:
1517:
1512:
1510:
1509:
1491:
1490:
1474:
1472:
1471:
1466:
1439:
1437:
1436:
1431:
1413:
1411:
1410:
1405:
1387:
1385:
1384:
1379:
1367:
1365:
1364:
1359:
1354:
1353:
1322:
1320:
1319:
1314:
1302:
1300:
1299:
1294:
1292:
1291:
1275:
1273:
1272:
1267:
1256:
1255:
1243:
1242:
1230:
1229:
1207:
1205:
1204:
1199:
1187:
1185:
1184:
1179:
1177:
1176:
1167:
1166:
1156:
1151:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1060:
1058:
1057:
1052:
1044:
1043:
1038:
1025:
1023:
1022:
1017:
1005:
1003:
1002:
997:
981:
979:
978:
973:
958:
956:
955:
950:
934:
932:
931:
926:
914:
912:
911:
906:
894:arithmetic genus
891:
889:
888:
883:
871:
869:
868:
863:
861:
860:
855:
854:
844:
843:
838:
837:
830:
829:
813:
811:
810:
805:
786:
784:
783:
778:
766:
764:
763:
758:
733:
731:
730:
725:
711:
709:
708:
703:
668:
666:
665:
660:
655:
654:
649:
634:
633:
618:
617:
595:
593:
592:
587:
570:
569:
560:
559:
547:
546:
534:
533:
472:
470:
469:
464:
448:
446:
445:
440:
428:
426:
425:
420:
418:
417:
388:
386:
385:
380:
368:
366:
365:
360:
340:
338:
337:
332:
312:
310:
309:
304:
292:
290:
289:
284:
257:invertible sheaf
254:
252:
251:
246:
229:dualising object
226:
224:
223:
218:
202:
200:
199:
194:
189:
188:
172:cotangent bundle
158:
156:
155:
150:
138:
136:
135:
130:
118:cotangent bundle
107:
105:
104:
99:
91:
90:
69:
67:
66:
61:
49:
47:
46:
41:
22:canonical bundle
3617:
3616:
3612:
3611:
3610:
3608:
3607:
3606:
3587:
3586:
3585:
3584:
3575:
3574:
3570:
3563:
3549:Badescu, Lucian
3546:
3542:
3535:
3521:Badescu, Lucian
3518:
3514:
3506:
3502:
3483:
3482:
3478:
3474:(1994), p. 192.
3466:
3462:
3442:
3438:
3426:
3422:
3413:
3409:
3400:
3399:
3395:
3375:
3368:
3349:
3348:
3344:
3337:
3323:Badescu, Lucian
3320:
3316:
3311:
3294:
3282:Krull dimension
3228:
3213:canonical model
3150:
3145:
3126:
3122:
3116:
3105:
3093:
3090:
3089:
3072:
3066:
3064:Canonical rings
3015:trigonal curves
3007:Petri's theorem
2947:linearly normal
2945:− 1 as a
2887:
2858:
2856:
2840:
2838:
2785:
2777:canonical curve
2760:
2745:
2724:
2662:
2638:
2587:
2562:
2559:
2558:
2541:
2537:
2535:
2532:
2531:
2511:
2508:
2507:
2484:
2481:
2480:
2460:
2455:
2439:
2435:
2433:
2430:
2429:
2413:
2410:
2409:
2388:
2385:
2384:
2367:
2363:
2361:
2358:
2357:
2341:
2338:
2337:
2317:
2314:
2313:
2310:
2292:induced by the
2262:
2258:
2249:
2245:
2225:
2224:
2213:
2210:
2209:
2186:
2185:
2164:
2158:
2157:
2156:
2134:
2128:
2127:
2126:
2122:
2114:
2111:
2110:
2094:
2091:
2090:
2073:
2069:
2060:
2056:
2048:
2045:
2044:
2016:
2006:
2002:
1996:
1985:
1980:
1976:
1970:
1964:
1963:
1962:
1950:
1946:
1934:
1928:
1927:
1926:
1917:
1913:
1904:
1900:
1898:
1895:
1894:
1863:
1859:
1858:
1852:
1851:
1850:
1841:
1837:
1828:
1824:
1822:
1819:
1818:
1796:
1793:
1792:
1775:
1774:
1772:
1769:
1768:
1751:
1750:
1748:
1745:
1744:
1727:
1726:
1724:
1721:
1720:
1703:
1702:
1693:
1692:
1683:
1677:
1676:
1675:
1669:
1665:
1659:
1655:
1653:
1650:
1649:
1642:multiple fibers
1624:
1620:
1618:
1615:
1614:
1591:
1587:
1585:
1582:
1581:
1561:
1560:
1555:
1545:
1541:
1532:
1528:
1526:
1523:
1522:
1505:
1501:
1486:
1482:
1480:
1477:
1476:
1448:
1445:
1444:
1419:
1416:
1415:
1393:
1390:
1389:
1373:
1370:
1369:
1349:
1345:
1328:
1325:
1324:
1308:
1305:
1304:
1287:
1283:
1281:
1278:
1277:
1251:
1247:
1238:
1234:
1225:
1221:
1213:
1210:
1209:
1208:, we have that
1193:
1190:
1189:
1172:
1168:
1162:
1158:
1152:
1141:
1129:
1126:
1125:
1105:
1102:
1101:
1073:
1070:
1069:
1039:
1034:
1033:
1031:
1028:
1027:
1011:
1008:
1007:
991:
988:
987:
964:
961:
960:
944:
941:
940:
920:
917:
916:
900:
897:
896:
877:
874:
873:
856:
850:
849:
848:
839:
833:
832:
831:
825:
821:
819:
816:
815:
799:
796:
795:
772:
769:
768:
740:
737:
736:
719:
716:
715:
697:
694:
693:
690:
650:
645:
644:
629:
625:
613:
609:
607:
604:
603:
565:
564:
555:
551:
542:
538:
529:
525:
523:
520:
519:
489:
483:
458:
455:
454:
434:
431:
430:
410:
406:
404:
401:
400:
374:
371:
370:
354:
351:
350:
326:
323:
322:
298:
295:
294:
278:
275:
274:
272:Cartier divisor
264:canonical class
240:
237:
236:
212:
209:
208:
184:
180:
178:
175:
174:
164:complex numbers
144:
141:
140:
124:
121:
120:
108:, which is the
86:
82:
78:
75:
74:
55:
52:
51:
35:
32:
31:
12:
11:
5:
3615:
3605:
3604:
3599:
3597:Vector bundles
3583:
3582:
3568:
3561:
3540:
3533:
3512:
3500:
3476:
3460:
3436:
3428:David Eisenbud
3420:
3414:Rick Miranda,
3407:
3393:
3366:
3342:
3335:
3313:
3312:
3310:
3307:
3306:
3305:
3300:
3293:
3290:
3173:
3172:
3161:
3158:
3153:
3148:
3144:
3140:
3137:
3134:
3129:
3125:
3119:
3114:
3111:
3108:
3104:
3100:
3097:
3076:canonical ring
3070:Canonical ring
3068:Main article:
3065:
3062:
3005:− 3)/2.
2886:
2883:
2871:
2870:
2828:
2827:
2759:
2756:
2741:
2735:elliptic curve
2720:
2714:Riemann sphere
2676:is zero, then
2661:
2658:
2637:
2634:
2614:-canonical map
2586:
2585:Canonical maps
2583:
2566:
2544:
2540:
2515:
2491:
2488:
2468:
2463:
2458:
2454:
2450:
2445:
2442:
2438:
2417:
2395:
2392:
2370:
2366:
2345:
2321:
2309:
2306:
2286:
2285:
2273:
2270:
2265:
2261:
2257:
2252:
2248:
2243:
2239:
2236:
2233:
2228:
2223:
2220:
2217:
2194:
2189:
2184:
2181:
2178:
2175:
2172:
2167:
2161:
2155:
2152:
2149:
2145:
2140:
2137:
2131:
2125:
2121:
2118:
2098:
2076:
2072:
2068:
2063:
2059:
2055:
2052:
2041:
2040:
2028:
2023:
2019:
2015:
2009:
2005:
1999:
1994:
1991:
1988:
1984:
1979:
1973:
1967:
1961:
1958:
1953:
1949:
1945:
1940:
1937:
1931:
1925:
1920:
1916:
1912:
1907:
1903:
1879:
1876:
1873:
1866:
1862:
1855:
1849:
1844:
1840:
1836:
1831:
1827:
1806:
1803:
1800:
1778:
1754:
1730:
1706:
1701:
1696:
1691:
1686:
1680:
1672:
1668:
1662:
1658:
1627:
1623:
1602:
1599:
1594:
1590:
1566:
1563:
1558:
1554:
1548:
1544:
1540:
1535:
1531:
1508:
1504:
1500:
1497:
1494:
1489:
1485:
1464:
1461:
1458:
1455:
1452:
1429:
1426:
1423:
1403:
1400:
1397:
1377:
1357:
1352:
1348:
1344:
1341:
1338:
1335:
1332:
1312:
1290:
1286:
1265:
1262:
1259:
1254:
1250:
1246:
1241:
1237:
1233:
1228:
1224:
1220:
1217:
1197:
1175:
1171:
1165:
1161:
1155:
1150:
1147:
1144:
1140:
1136:
1133:
1109:
1089:
1086:
1083:
1080:
1077:
1050:
1047:
1042:
1037:
1015:
995:
971:
968:
948:
924:
904:
881:
859:
853:
847:
842:
836:
828:
824:
803:
776:
756:
753:
750:
747:
744:
723:
701:
689:
686:
670:
669:
658:
653:
648:
643:
640:
637:
632:
628:
624:
621:
616:
612:
597:
596:
585:
582:
579:
576:
573:
568:
563:
558:
554:
550:
545:
541:
537:
532:
528:
497:smooth variety
485:Main article:
482:
479:
462:
438:
416:
413:
409:
398:inverse bundle
378:
358:
330:
302:
282:
244:
227:. This is the
216:
192:
187:
183:
148:
128:
114:exterior power
97:
94:
89:
85:
59:
39:
9:
6:
4:
3:
2:
3614:
3603:
3600:
3598:
3595:
3594:
3592:
3578:
3572:
3564:
3562:9780387986685
3558:
3554:
3550:
3544:
3536:
3534:9780387986685
3530:
3526:
3522:
3516:
3509:
3504:
3496:
3492:
3491:
3486:
3480:
3473:
3469:
3464:
3457:
3453:
3452:
3447:
3440:
3433:
3429:
3424:
3417:
3411:
3403:
3397:
3390:
3386:
3385:
3380:
3373:
3371:
3362:
3358:
3357:
3352:
3346:
3338:
3336:9780387986685
3332:
3328:
3324:
3318:
3314:
3304:
3301:
3299:
3296:
3295:
3289:
3287:
3283:
3279:
3275:
3270:
3267:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3232:
3226:
3222:
3218:
3214:
3210:
3205:
3203:
3199:
3194:
3190:
3186:
3182:
3178:
3159:
3151:
3146:
3142:
3138:
3135:
3127:
3123:
3112:
3109:
3106:
3102:
3098:
3095:
3088:
3087:
3086:
3085:
3081:
3077:
3071:
3061:
3058:
3056:
3052:
3051:ruled surface
3048:
3044:
3040:
3036:
3032:
3028:
3024:
3023:Oscar Chisini
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2984:
2980:
2975:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2944:
2940:
2936:
2932:
2928:
2924:
2923:cubic surface
2920:
2916:
2912:
2908:
2904:
2900:
2896:
2892:
2882:
2880:
2876:
2865:
2861:
2854:
2847:
2843:
2836:
2833:
2832:
2831:
2825:
2821:
2817:
2814:
2813:
2812:
2810:
2806:
2802:
2798:
2794:
2788:
2783:
2779:
2778:
2773:
2769:
2765:
2755:
2753:
2749:
2744:
2740:
2736:
2732:
2729:is one, then
2728:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2664:Suppose that
2657:
2655:
2651:
2647:
2643:
2633:
2631:
2627:
2623:
2619:
2615:
2613:
2608:
2604:
2603:canonical map
2600:
2596:
2592:
2582:
2580:
2564:
2542:
2538:
2529:
2513:
2505:
2489:
2486:
2461:
2456:
2452:
2443:
2440:
2436:
2415:
2406:
2393:
2390:
2368:
2364:
2343:
2335:
2319:
2308:Singular case
2305:
2303:
2299:
2295:
2291:
2271:
2268:
2263:
2259:
2255:
2250:
2246:
2237:
2234:
2218:
2215:
2208:
2207:
2206:
2179:
2176:
2173:
2165:
2150:
2147:
2143:
2138:
2135:
2123:
2119:
2116:
2096:
2074:
2070:
2066:
2061:
2057:
2053:
2050:
2026:
2021:
2017:
2013:
2007:
2003:
1997:
1992:
1989:
1986:
1982:
1977:
1971:
1959:
1951:
1947:
1943:
1938:
1935:
1918:
1914:
1910:
1905:
1901:
1893:
1892:
1891:
1877:
1874:
1864:
1860:
1847:
1842:
1838:
1829:
1825:
1804:
1801:
1798:
1699:
1689:
1684:
1670:
1666:
1660:
1656:
1648:one has that
1647:
1643:
1625:
1621:
1600:
1597:
1592:
1588:
1564:
1562:
1556:
1552:
1546:
1542:
1538:
1533:
1529:
1506:
1502:
1498:
1495:
1492:
1487:
1483:
1462:
1456:
1453:
1450:
1441:
1427:
1424:
1421:
1401:
1398:
1395:
1375:
1350:
1346:
1339:
1336:
1333:
1330:
1310:
1288:
1284:
1263:
1260:
1257:
1252:
1248:
1244:
1239:
1235:
1231:
1226:
1222:
1218:
1215:
1195:
1173:
1169:
1163:
1159:
1153:
1148:
1145:
1142:
1138:
1134:
1131:
1123:
1107:
1087:
1081:
1078:
1075:
1067:
1062:
1048:
1045:
1040:
1013:
993:
985:
969:
966:
946:
938:
922:
902:
895:
879:
857:
845:
840:
826:
822:
801:
793:
790:
774:
754:
748:
745:
742:
735:
721:
699:
685:
683:
679:
675:
656:
651:
638:
635:
630:
626:
619:
614:
610:
602:
601:
600:
583:
574:
561:
556:
552:
543:
539:
535:
530:
526:
518:
517:
516:
514:
510:
506:
502:
498:
494:
491:Suppose that
488:
478:
476:
460:
452:
436:
414:
411:
407:
399:
395:
390:
376:
356:
348:
347:anticanonical
344:
328:
320:
316:
300:
280:
273:
269:
268:divisor class
265:
260:
258:
242:
234:
233:Serre duality
230:
214:
206:
190:
185:
181:
173:
169:
165:
160:
146:
119:
115:
111:
95:
92:
87:
73:
57:
50:of dimension
37:
30:
27:
23:
19:
3571:
3552:
3543:
3524:
3515:
3510:, pp. 11-13.
3503:
3488:
3479:
3471:
3463:
3449:
3439:
3431:
3423:
3415:
3410:
3396:
3382:
3354:
3345:
3326:
3317:
3277:
3271:
3268:
3263:
3259:
3247:
3243:
3239:
3235:
3230:
3220:
3216:
3212:
3206:
3201:
3197:
3192:
3188:
3176:
3174:
3079:
3075:
3073:
3059:
3030:
3018:
3010:
3006:
3002:
2998:
2994:
2990:
2978:
2976:
2970:
2966:
2958:
2954:
2950:
2942:
2938:
2934:
2926:
2914:
2906:
2902:
2898:
2894:
2893:which means
2890:
2888:
2885:General case
2878:
2872:
2863:
2859:
2852:
2845:
2841:
2834:
2829:
2823:
2819:
2815:
2808:
2804:
2792:
2786:
2781:
2775:
2771:
2763:
2761:
2751:
2747:
2742:
2738:
2730:
2726:
2721:
2717:
2709:
2705:
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2663:
2653:
2649:
2639:
2629:
2625:
2621:
2617:
2611:
2610:
2606:
2602:
2598:
2595:rational map
2588:
2528:Weil divisor
2407:
2334:Weil divisor
2311:
2287:
2042:
1641:
1442:
1063:
983:
713:
691:
681:
677:
671:
598:
512:
508:
504:
500:
492:
490:
475:Fano variety
473:is called a
393:
391:
346:
342:
263:
261:
204:
166:, it is the
161:
109:
26:non-singular
21:
15:
3084:graded ring
3001:− 2)(
2704:) = −
1440:otherwise.
389:canonical.
72:line bundle
18:mathematics
3591:Categories
2925:; and for
2579:Gorenstein
2577:is S2 and
2298:K3 surface
1817:such that
207:-forms on
3495:EMS Press
3456:EMS Press
3389:EMS Press
3361:EMS Press
3118:∞
3103:⨁
2937:of genus
2851:,
2789:− 1
2660:Low genus
2591:effective
2487:−
2453:ω
2441:−
2336:class on
2269:−
2242:⟺
2219:
2180:
2151:χ
2136:−
2120:
2089:for each
2054:≤
1983:∑
1960:⊗
1948:ω
1944:⊗
1936:−
1919:∗
1911:≅
1902:ω
1802:∈
1700:⊕
1671:∗
1496:…
1460:→
1340:
1323:; so for
1139:∑
1085:→
1046:×
967:−
846:≅
827:∗
794:morphism
752:→
734:fibration
562:⊗
553:ω
544:∗
527:ω
499:and that
412:−
408:ω
186:∗
162:Over the
127:Ω
96:ω
84:Ω
3551:(2001).
3523:(2001).
3325:(2001).
3292:See also
2688:, where
2022:′
1565:′
3497:, 2001
3363:, 2001
3082:is the
2919:quadric
2857:√
2839:√
2791:. When
984:minimal
266:is the
116:of the
3559:
3531:
3333:
3179:is an
3053:and a
3037:: the
2921:and a
2803:, and
2737:, and
2733:is an
2616:. The
2479:, the
2216:length
2177:length
2043:where
1719:where
1580:where
1475:. Let
1276:where
937:fibers
789:proper
714:genus
20:, the
3309:Notes
3250:is a
3229:Proj
2795:is a
2672:. If
2597:from
1644:. By
1368:, if
915:. If
892:have
787:is a
495:is a
451:ample
369:with
345:. An
270:of a
24:of a
3557:ISBN
3529:ISBN
3331:ISBN
3272:The
3225:Proj
3207:The
3074:The
3025:and
2853:x dx
2109:and
2067:<
1875:>
1598:>
1425:>
1414:and
792:flat
692:Let
511:and
392:The
317:for
262:The
231:for
3284:or
3276:of
3258:of
3252:nef
3246:of
3078:of
2969:on
2768:big
2762:If
2700:(1/
2680:is
2506:of
2117:deg
1337:gcd
1188:of
939:of
767:of
449:is
321:on
293:on
235:on
139:on
112:th
16:In
3593::
3493:,
3487:,
3470:,
3454:,
3448:,
3430:,
3387:,
3381:,
3369:^
3359:,
3353:,
3288:.
3057:.
2881:.
2835:dx
2818:=
2706:dt
2656:.
1068:)
1061:.
684:.
477:.
453:,
259:.
159:.
3579:.
3565:.
3537:.
3339:.
3278:V
3264:V
3260:K
3248:V
3244:K
3240:V
3236:V
3231:R
3221:V
3217:V
3202:k
3198:k
3193:V
3189:V
3177:V
3160:.
3157:)
3152:d
3147:V
3143:K
3139:,
3136:V
3133:(
3128:0
3124:H
3113:0
3110:=
3107:d
3099:=
3096:R
3080:V
3019:g
3011:g
3003:g
2999:g
2995:C
2979:g
2971:C
2967:D
2959:g
2955:C
2951:g
2943:g
2939:g
2935:C
2927:g
2915:g
2907:g
2903:g
2899:C
2895:g
2891:C
2879:x
2869:.
2866:)
2864:x
2862:(
2860:P
2855:/
2848:)
2846:x
2844:(
2842:P
2837:/
2826:)
2824:x
2822:(
2820:P
2816:y
2809:P
2805:C
2793:C
2787:g
2782:g
2772:n
2764:C
2752:n
2748:n
2743:C
2739:K
2731:C
2727:g
2722:C
2718:K
2710:t
2708:/
2702:t
2698:d
2694:C
2690:P
2686:P
2682:P
2678:C
2674:g
2670:g
2666:C
2654:g
2650:g
2630:n
2626:n
2622:V
2618:n
2612:n
2607:n
2599:V
2565:X
2543:X
2539:K
2514:X
2490:d
2467:)
2462:.
2457:X
2449:(
2444:d
2437:h
2416:X
2394:.
2391:X
2369:X
2365:K
2344:X
2320:X
2284:.
2272:1
2264:i
2260:m
2256:=
2251:i
2247:a
2238:0
2235:=
2232:)
2227:T
2222:(
2193:)
2188:T
2183:(
2174:+
2171:)
2166:X
2160:O
2154:(
2148:=
2144:)
2139:1
2130:L
2124:(
2097:i
2075:i
2071:m
2062:i
2058:a
2051:0
2027:)
2018:i
2014:F
2008:i
2004:a
1998:r
1993:1
1990:=
1987:i
1978:(
1972:X
1966:O
1957:)
1952:B
1939:1
1930:L
1924:(
1915:f
1906:X
1878:1
1872:)
1865:b
1861:X
1854:O
1848:,
1843:b
1839:X
1835:(
1830:0
1826:h
1805:B
1799:b
1777:T
1753:T
1729:L
1705:T
1695:L
1690:=
1685:X
1679:O
1667:f
1661:1
1657:R
1626:i
1622:F
1601:1
1593:i
1589:m
1557:i
1553:F
1547:i
1543:m
1539:=
1534:i
1530:F
1507:r
1503:F
1499:,
1493:,
1488:1
1484:F
1463:B
1457:X
1454::
1451:f
1428:1
1422:m
1402:1
1399:=
1396:m
1376:F
1356:)
1351:i
1347:a
1343:(
1334:=
1331:m
1311:X
1289:X
1285:K
1264:,
1261:0
1258:=
1253:i
1249:E
1245:.
1240:X
1236:K
1232:=
1227:i
1223:E
1219:.
1216:F
1196:f
1174:i
1170:E
1164:i
1160:a
1154:n
1149:1
1146:=
1143:i
1135:=
1132:F
1108:f
1088:B
1082:X
1079::
1076:f
1049:B
1041:1
1036:P
1014:X
994:X
970:1
947:f
923:X
903:g
880:f
858:B
852:O
841:X
835:O
823:f
802:f
775:X
755:B
749:X
746::
743:f
722:g
700:X
682:D
678:X
657:.
652:D
647:|
642:)
639:D
636:+
631:X
627:K
623:(
620:=
615:D
611:K
584:.
581:)
578:)
575:D
572:(
567:O
557:X
549:(
540:i
536:=
531:D
513:D
509:X
505:X
501:D
493:X
461:V
437:V
415:1
377:K
357:K
329:V
301:V
281:K
243:V
215:V
205:n
191:V
182:T
147:V
110:n
93:=
88:n
58:n
38:V
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