2049:
3060:. 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
2214:
3206:
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
2643:-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.
3032:= 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,
2984:
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.
1728:
2293:
605:
1907:
2343:, 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
3181:
881:
2123:
1899:
1377:
1197:
1589:
1070:
2488:
117:
1285:
678:
2098:
1530:
3280:
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.
1800:
1776:
1752:
438:
3222:
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
1622:
1484:
1109:
776:
148:
2841:
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
1826:
1449:
212:
2566:
2392:
1649:
1312:
1423:
2511:
2415:
991:
2586:
2535:
2437:
2365:
2341:
2118:
1397:
1332:
1217:
1129:
1035:
1015:
968:
944:
924:
901:
823:
796:
743:
721:
482:
458:
398:
378:
350:
322:
302:
264:
236:
168:
79:
59:
2888:
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
2307:, 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
3071:
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.
1662:
684:
2222:
2044:{\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)}
532:
3052:
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
2655:. A global section of the canonical bundle is therefore the same as an everywhere-regular differential form. Classically, these were called
2209:{\displaystyle \operatorname {deg} \left({\mathcal {L}}^{-1}\right)=\chi ({\mathcal {O}}_{X})+\operatorname {length} ({\mathcal {T}})}
1132:
3102:
828:
3571:
3543:
3345:
3024:
at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a)
3478:
2964:− 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves
1831:
2656:
683:
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is
3505:
3466:
3399:
3371:
3612:
2601:
3500:
3461:
3394:
3366:
3195:
1337:
1138:
3291:
is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean
2941:
3456:
1040:
3277:
will admit a canonical model (more generally, this is true for normal complete
Gorenstein algebraic spaces).
2757:
is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the
2442:
87:
3607:
2885:
1222:
616:
3587:
3412:
3053:
2057:
1535:
3219:
3057:
2993:
2811:
2300:
1489:
239:
3495:
3389:
3361:
3296:
2651:
The best studied case is that of curves. Here, the canonical bundle is the same as the (holomorphic)
1656:
36:
3559:
3531:
3518:
3333:
3044:. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is
1781:
1757:
1733:
3308:
3266:
2997:
2940:= 5 when it is an intersection of three quadrics. There is a converse, which is a corollary to the
2921:
2807:
947:
413:
3045:
497:
329:
178:
2924:. All non-singular plane quartics arise in this way. There is explicit information for the case
1594:
3020:, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for
2635:
into a projective space of dimension one less than the dimension of the global sections of the
3567:
3539:
3341:
3313:
3284:
3235:
3191:
3037:
2514:
1457:
1082:
749:
461:
325:
39:
133:
3588:"09w5033: Complex Analysis and Complex Geometry | Banff International Research Station"
3065:
2652:
2312:
2304:
1805:
1428:
1076:
904:
267:
187:
182:
128:
2544:
2370:
1627:
1290:
3292:
3262:
3049:
2973:
2957:
2787:
2778:
2589:
799:
282:
3056:
on such curves by the differentials of the first kind; and this has consequences for the
1402:
2493:
2397:
973:
3438:
3080:
3025:
2745:
2724:
2571:
2520:
2422:
2350:
2326:
2103:
1382:
1317:
1202:
1114:
1020:
1000:
953:
929:
909:
886:
808:
781:
728:
706:
507:
467:
443:
408:
383:
363:
335:
307:
287:
249:
221:
174:
153:
124:
64:
44:
3601:
3198:
of the image of the canonical map. This can be true even when the canonical class of
3061:
3033:
2933:
2568:
defined above. In the absence of the normality hypothesis, the same result holds if
802:
278:
243:
2605:
2538:
2344:
2315:
will always admit multiple fibers and so, such a surface will not admit a section.
485:
3094:
2723:, for example, a meromorphic differential with double pole at the origin on the
1723:{\displaystyle R^{1}f_{*}{\mathcal {O}}_{X}={\mathcal {L}}\oplus {\mathcal {T}}}
82:
28:
17:
2308:
3238:
of the canonical ring. If the canonical ring is not finitely generated, then
2288:{\displaystyle \operatorname {length} ({\mathcal {T}})=0\iff a_{i}=m_{i}-1}
1532:
be the finitely many fibers that are not geometrically integral and write
1131:
are geometrically integral and all fibers are geometrically connected (by
3234:. When the canonical ring is finitely generated, the canonical model is
2785:-canonical map is a curve. The image of the 1-canonical map is called a
600:{\displaystyle \omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D)).}
2929:
3253:
admits no canonical model. One can show that if the canonical divisor
3040:). The terminology is confused, since the result is also called the
3230:
with mild singularities that could be constructed by blowing down
3519:
http://hal.archives-ouvertes.fr/docs/00/40/42/57/PDF/these-OD.pdf
687:, which allows one to deduce results about the singularities of
3176:{\displaystyle R=\bigoplus _{d=0}^{\infty }H^{0}(V,K_{V}^{d}).}
1624:
is greatest common divisor of coefficients of the expansion of
876:{\displaystyle f_{*}{\mathcal {O}}_{X}\cong {\mathcal {O}}_{B}}
2988:
More refined information is available, for larger values of
2237:
2198:
2170:
2140:
1976:
1940:
1864:
1787:
1763:
1739:
1715:
1705:
1689:
862:
845:
577:
2822:
is a polynomial of degree 6 (without repeated roots) then
518:. The adjunction formula relates the canonical bundles of
3004:: the dimension of the space of quadrics passing through
2311:. On the other hand, a minimal genus one fibration of an
3413:"Geometric Form of Riemann-Roch | Rigorous Trivialities"
2992:, but in these cases canonical curves are not generally
1894:{\displaystyle h^{0}(X_{b},{\mathcal {O}}_{X_{b}})>1}
3566:. Springer Science & Business Media. p. 123.
3538:. Springer Science & Business Media. p. 242.
3340:. Springer Science & Business Media. p. 111.
2920:= 3 the canonical curves (non-hyperelliptic case) are
2818:
a double cover of its canonical curve. For example if
3105:
2996:, and the description requires more consideration of
2574:
2547:
2523:
2496:
2445:
2425:
2400:
2373:
2353:
2329:
2225:
2126:
2106:
2060:
1910:
1834:
1808:
1784:
1760:
1736:
1665:
1630:
1597:
1538:
1492:
1460:
1431:
1405:
1385:
1340:
1320:
1293:
1225:
1205:
1141:
1117:
1085:
1043:
1023:
1003:
976:
956:
932:
912:
889:
831:
811:
784:
752:
731:
709:
619:
535:
470:
446:
416:
386:
366:
338:
324:
giving rise to the canonical bundle — it is an
310:
290:
252:
224:
214:. Equivalently, it is the line bundle of holomorphic
190:
156:
136:
90:
67:
47:
3245:
is not a variety, and so it cannot be birational to
2928:= 4, when a canonical curve is an intersection of a
2299:
For example, for the minimal genus 1 fibration of a
970:
do not contain rational curves of self-intersection
2777:has genus two or more, then the canonical class is
3175:
2695:, and the canonical class is the class of −2
2580:
2560:
2529:
2505:
2482:
2431:
2409:
2386:
2359:
2335:
2287:
2208:
2112:
2092:
2043:
1893:
1820:
1794:
1770:
1746:
1722:
1643:
1616:
1583:
1524:
1478:
1443:
1417:
1391:
1371:
1326:
1306:
1279:
1211:
1191:
1123:
1103:
1064:
1029:
1009:
985:
962:
938:
918:
895:
875:
817:
790:
770:
737:
715:
672:
599:
476:
452:
432:
392:
372:
344:
316:
296:
258:
230:
206:
162:
142:
111:
73:
53:
2908:is at least 3, the morphism is an isomorphism of
92:
3215:is any sufficiently divisible positive integer.
2394:that is referred to as the canonical divisor on
2795:always sits in a projective space of dimension
2884:This means that the canonical map is given by
2612:into projective space. This map is called the
1037:is birationally ruled, that is, birational to
1017:admits a (minimal) genus 0 fibration, then is
2972:at least 3), Riemann-Roch, and the theory of
1372:{\displaystyle m=\operatorname {gcd} (a_{i})}
1075:For a minimal genus 1 fibration (also called
8:
2541:class, which is equal to the divisor class
1651:into integral components; these are called
1192:{\displaystyle F=\sum _{i=1}^{n}a_{i}E_{i}}
2952:embedded in projective space of dimension
2620:th multiple of the canonical class is the
2255:
2251:
266:. It may equally well be considered as an
3383:
3381:
3161:
3156:
3137:
3127:
3116:
3104:
3028:and (b) non-singular plane quintics when
2707:. This follows from the calculus formula
2573:
2552:
2546:
2522:
2495:
2471:
2466:
2450:
2444:
2424:
2399:
2378:
2372:
2352:
2328:
2273:
2260:
2236:
2235:
2224:
2197:
2196:
2175:
2169:
2168:
2145:
2139:
2138:
2125:
2105:
2084:
2071:
2059:
2027:
2017:
2007:
1996:
1981:
1975:
1974:
1961:
1945:
1939:
1938:
1928:
1915:
1909:
1874:
1869:
1863:
1862:
1852:
1839:
1833:
1807:
1786:
1785:
1783:
1762:
1761:
1759:
1738:
1737:
1735:
1714:
1713:
1704:
1703:
1694:
1688:
1687:
1680:
1670:
1664:
1635:
1629:
1602:
1596:
1571:
1566:
1556:
1543:
1537:
1516:
1497:
1491:
1459:
1430:
1404:
1384:
1360:
1339:
1319:
1298:
1292:
1262:
1249:
1236:
1224:
1204:
1183:
1173:
1163:
1152:
1140:
1116:
1084:
1050:
1046:
1045:
1042:
1022:
1002:
975:
955:
931:
911:
888:
867:
861:
860:
850:
844:
843:
836:
830:
810:
783:
751:
730:
708:
661:
656:
640:
624:
618:
576:
575:
566:
553:
540:
534:
469:
445:
421:
415:
385:
365:
337:
309:
289:
251:
223:
195:
189:
155:
135:
97:
91:
89:
66:
46:
2736:and its multiples are not effective. If
2659:. The degree of the canonical class is 2
1065:{\displaystyle \mathbb {P} ^{1}\times B}
352:, and any divisor in it may be called a
3325:
2483:{\displaystyle h^{-d}(\omega _{X}^{.})}
2419:Alternately, again on a normal variety
946:is a smooth projective surface and the
112:{\displaystyle \,\!\Omega ^{n}=\omega }
1280:{\displaystyle F.E_{i}=K_{X}.E_{i}=0,}
3427:Algebraic Curves and Riemann Surfaces
2679:is a smooth algebraic curve of genus
2616:. The rational map determined by the
1454:Consider a minimal genus 1 fibration
673:{\displaystyle K_{D}=(K_{X}+D)|_{D}.}
610:In terms of canonical classes, it is
7:
2976:is rather close. Effective divisors
2639:th multiple of the canonical class.
2093:{\displaystyle 0\leq a_{i}<m_{i}}
1584:{\displaystyle F_{i}=m_{i}F_{i}^{'}}
3226:, a particular birational model of
3008:as embedded as canonical curve is (
440:. When the anticanonical bundle of
3128:
2968:(in the non-hyperelliptic case of
2912:with its image, which has degree 2
1525:{\displaystyle F_{1},\dots ,F_{r}}
137:
94:
25:
3194:, then the canonical ring is the
2900:Otherwise, for non-hyperelliptic
2513:'th cohomology of the normalized
2367:. It is this class, denoted by
1111:all but finitely many fibers of
3479:Igor Rostislavovich Shafarevich
3202:is not ample. For instance, if
2663:− 2 for a curve of genus
2657:differentials of the first kind
2537:. This sheaf corresponds to a
1133:Zariski's connectedness theorem
993:, then the fibration is called
3167:
3143:
2477:
2459:
2252:
2242:
2232:
2203:
2193:
2181:
2164:
1967:
1934:
1882:
1845:
1795:{\displaystyle {\mathcal {T}}}
1771:{\displaystyle {\mathcal {T}}}
1747:{\displaystyle {\mathcal {L}}}
1470:
1366:
1353:
1135:). In particular, for a fiber
1095:
762:
657:
652:
633:
591:
588:
582:
559:
526:. It is a natural isomorphism
360:divisor is any divisor −
1:
2791:. A canonical curve of genus
1399:is geometrically integral if
825:to a smooth curve such that
699:The canonical bundle formula
433:{\displaystyle \omega ^{-1}}
3501:Encyclopedia of Mathematics
3462:Encyclopedia of Mathematics
3395:Encyclopedia of Mathematics
3367:Encyclopedia of Mathematics
3273:is greater than zero, then
3196:homogeneous coordinate ring
2810:, the canonical curve is a
1754:is an invertible sheaf and
3629:
3457:"Noether–Enriques theorem"
3455:Iskovskih, V. A. (2001) ,
3186:If the canonical class of
3078:
2600:If the canonical class is
2301:(quasi)-bielliptic surface
1657:cohomology and base change
1617:{\displaystyle m_{i}>1}
1314:is a canonical divisor of
691:from the singularities of
495:
3000:. The field started with
3443:The Geometry of Syzygies
3388:Parshin, A. N. (2001) ,
3042:Noether–Enriques theorem
1479:{\displaystyle f:X\to B}
1104:{\displaystyle f:X\to B}
771:{\displaystyle f:X\to B}
3054:quadratic differentials
2944:: a non-singular curve
2886:homogeneous coordinates
2765:is the map to a point.
2761:-canonical map for any
2604:, then it determines a
1901:). Then, one has that
723:be a normal surface. A
685:inversion of adjunction
514:is a smooth divisor on
143:{\displaystyle \Omega }
3211:-canonical map, where
3177:
3132:
2994:complete intersections
2781:, so the image of any
2582:
2562:
2531:
2507:
2484:
2433:
2411:
2388:
2361:
2337:
2323:On a singular variety
2289:
2210:
2114:
2094:
2045:
2012:
1895:
1822:
1821:{\displaystyle b\in B}
1796:
1772:
1748:
1724:
1645:
1618:
1585:
1526:
1480:
1445:
1444:{\displaystyle m>1}
1419:
1393:
1373:
1328:
1308:
1281:
1213:
1193:
1168:
1125:
1105:
1066:
1031:
1011:
987:
964:
940:
920:
897:
877:
819:
792:
772:
739:
717:
674:
601:
492:The adjunction formula
478:
454:
434:
394:
374:
346:
318:
298:
260:
232:
208:
207:{\displaystyle T^{*}V}
164:
144:
113:
75:
55:
3220:minimal model program
3178:
3112:
3058:local Torelli theorem
3002:Max Noether's theorem
2812:rational normal curve
2631:-canonical map sends
2583:
2563:
2561:{\displaystyle K_{X}}
2532:
2508:
2485:
2434:
2412:
2389:
2387:{\displaystyle K_{X}}
2362:
2338:
2290:
2211:
2115:
2095:
2046:
1992:
1896:
1823:
1797:
1773:
1749:
1725:
1646:
1644:{\displaystyle F_{i}}
1619:
1586:
1527:
1481:
1446:
1420:
1394:
1374:
1329:
1309:
1307:{\displaystyle K_{X}}
1282:
1214:
1194:
1148:
1126:
1106:
1067:
1032:
1012:
988:
965:
941:
921:
898:
878:
820:
793:
773:
740:
718:
675:
602:
479:
455:
435:
407:is the corresponding
395:
375:
347:
319:
299:
261:
233:
209:
165:
145:
114:
76:
56:
3483:Algebraic geometry I
3297:transcendence degree
3103:
2942:Riemann–Roch theorem
2922:quartic plane curves
2916:− 2. Thus for
2572:
2545:
2521:
2494:
2443:
2423:
2398:
2371:
2351:
2327:
2223:
2124:
2104:
2058:
1908:
1832:
1806:
1782:
1778:is a torsion sheaf (
1758:
1734:
1663:
1628:
1595:
1536:
1490:
1458:
1429:
1403:
1383:
1338:
1318:
1291:
1223:
1203:
1139:
1115:
1083:
1041:
1021:
1001:
974:
954:
930:
910:
887:
829:
809:
782:
750:
729:
707:
617:
533:
468:
444:
414:
405:anticanonical bundle
384:
364:
336:
308:
288:
250:
222:
188:
154:
134:
88:
81:over a field is the
65:
45:
3613:Algebraic varieties
3309:Birational geometry
3166:
2998:commutative algebra
2808:hyperelliptic curve
2476:
2439:, one can consider
2035:
1580:
1418:{\displaystyle m=1}
1077:elliptic fibrations
181:of the holomorphic
3564:Algebraic Surfaces
3536:Algebraic Surfaces
3496:"Torelli theorems"
3338:Algebraic Surfaces
3173:
3152:
3046:normally generated
2769:Hyperelliptic case
2592:in dimension one.
2578:
2558:
2527:
2506:{\displaystyle -d}
2503:
2480:
2462:
2429:
2410:{\displaystyle X.}
2407:
2384:
2357:
2333:
2285:
2216:. One notes that
2206:
2110:
2090:
2041:
2023:
1891:
1818:
1792:
1768:
1744:
1720:
1641:
1614:
1581:
1562:
1522:
1476:
1441:
1415:
1389:
1369:
1324:
1304:
1277:
1209:
1189:
1121:
1101:
1062:
1027:
1007:
997:. For example, if
986:{\displaystyle -1}
983:
960:
936:
916:
893:
883:and all fibers of
873:
815:
788:
768:
735:
713:
670:
597:
498:Adjunction formula
474:
450:
430:
390:
370:
342:
330:linear equivalence
314:
294:
256:
228:
204:
179:determinant bundle
160:
140:
109:
71:
51:
3445:(2005), p. 181-2.
3390:"Canonical curve"
3362:"canonical class"
3314:Differential form
3285:Kodaira dimension
3267:self intersection
3249:; in particular,
3192:ample line bundle
3038:Federigo Enriques
2960:curve of degree 2
2727:. In particular,
2581:{\displaystyle X}
2530:{\displaystyle X}
2515:dualizing complex
2432:{\displaystyle X}
2360:{\displaystyle X}
2336:{\displaystyle X}
2305:Albanese morphism
2113:{\displaystyle i}
1392:{\displaystyle F}
1327:{\displaystyle X}
1212:{\displaystyle f}
1124:{\displaystyle f}
1030:{\displaystyle X}
1010:{\displaystyle X}
963:{\displaystyle f}
939:{\displaystyle X}
919:{\displaystyle g}
896:{\displaystyle f}
818:{\displaystyle f}
791:{\displaystyle X}
738:{\displaystyle g}
716:{\displaystyle X}
477:{\displaystyle V}
453:{\displaystyle V}
393:{\displaystyle K}
373:{\displaystyle K}
354:canonical divisor
345:{\displaystyle V}
326:equivalence class
317:{\displaystyle V}
297:{\displaystyle K}
259:{\displaystyle V}
231:{\displaystyle V}
163:{\displaystyle V}
74:{\displaystyle n}
54:{\displaystyle V}
40:algebraic variety
16:(Redirected from
3620:
3592:
3591:
3584:
3578:
3577:
3556:
3550:
3549:
3528:
3522:
3516:
3510:
3509:
3492:
3486:
3476:
3470:
3469:
3452:
3446:
3436:
3430:
3429:(1995), Ch. VII.
3423:
3417:
3416:
3415:. 7 August 2008.
3409:
3403:
3402:
3385:
3376:
3375:
3358:
3352:
3351:
3330:
3265:divisor and the
3244:
3182:
3180:
3179:
3174:
3165:
3160:
3142:
3141:
3131:
3126:
3066:Veronese surface
3050:symmetric powers
2974:special divisors
2879:
2878:
2861:
2860:
2801:
2703:is any point of
2653:cotangent bundle
2647:Canonical curves
2587:
2585:
2584:
2579:
2567:
2565:
2564:
2559:
2557:
2556:
2536:
2534:
2533:
2528:
2512:
2510:
2509:
2504:
2489:
2487:
2486:
2481:
2475:
2470:
2458:
2457:
2438:
2436:
2435:
2430:
2416:
2414:
2413:
2408:
2393:
2391:
2390:
2385:
2383:
2382:
2366:
2364:
2363:
2358:
2342:
2340:
2339:
2334:
2313:Enriques surface
2294:
2292:
2291:
2286:
2278:
2277:
2265:
2264:
2241:
2240:
2215:
2213:
2212:
2207:
2202:
2201:
2180:
2179:
2174:
2173:
2157:
2153:
2152:
2144:
2143:
2119:
2117:
2116:
2111:
2099:
2097:
2096:
2091:
2089:
2088:
2076:
2075:
2050:
2048:
2047:
2042:
2040:
2036:
2031:
2022:
2021:
2011:
2006:
1986:
1985:
1980:
1979:
1966:
1965:
1953:
1952:
1944:
1943:
1933:
1932:
1920:
1919:
1900:
1898:
1897:
1892:
1881:
1880:
1879:
1878:
1868:
1867:
1857:
1856:
1844:
1843:
1827:
1825:
1824:
1819:
1802:is supported on
1801:
1799:
1798:
1793:
1791:
1790:
1777:
1775:
1774:
1769:
1767:
1766:
1753:
1751:
1750:
1745:
1743:
1742:
1729:
1727:
1726:
1721:
1719:
1718:
1709:
1708:
1699:
1698:
1693:
1692:
1685:
1684:
1675:
1674:
1650:
1648:
1647:
1642:
1640:
1639:
1623:
1621:
1620:
1615:
1607:
1606:
1590:
1588:
1587:
1582:
1579:
1578:
1570:
1561:
1560:
1548:
1547:
1531:
1529:
1528:
1523:
1521:
1520:
1502:
1501:
1485:
1483:
1482:
1477:
1450:
1448:
1447:
1442:
1424:
1422:
1421:
1416:
1398:
1396:
1395:
1390:
1378:
1376:
1375:
1370:
1365:
1364:
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1303:
1302:
1286:
1284:
1283:
1278:
1267:
1266:
1254:
1253:
1241:
1240:
1218:
1216:
1215:
1210:
1198:
1196:
1195:
1190:
1188:
1187:
1178:
1177:
1167:
1162:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1071:
1069:
1068:
1063:
1055:
1054:
1049:
1036:
1034:
1033:
1028:
1016:
1014:
1013:
1008:
992:
990:
989:
984:
969:
967:
966:
961:
945:
943:
942:
937:
925:
923:
922:
917:
905:arithmetic genus
902:
900:
899:
894:
882:
880:
879:
874:
872:
871:
866:
865:
855:
854:
849:
848:
841:
840:
824:
822:
821:
816:
797:
795:
794:
789:
777:
775:
774:
769:
744:
742:
741:
736:
722:
720:
719:
714:
679:
677:
676:
671:
666:
665:
660:
645:
644:
629:
628:
606:
604:
603:
598:
581:
580:
571:
570:
558:
557:
545:
544:
483:
481:
480:
475:
459:
457:
456:
451:
439:
437:
436:
431:
429:
428:
399:
397:
396:
391:
379:
377:
376:
371:
351:
349:
348:
343:
323:
321:
320:
315:
303:
301:
300:
295:
268:invertible sheaf
265:
263:
262:
257:
240:dualising object
237:
235:
234:
229:
213:
211:
210:
205:
200:
199:
183:cotangent bundle
169:
167:
166:
161:
149:
147:
146:
141:
129:cotangent bundle
118:
116:
115:
110:
102:
101:
80:
78:
77:
72:
60:
58:
57:
52:
33:canonical bundle
21:
3628:
3627:
3623:
3622:
3621:
3619:
3618:
3617:
3598:
3597:
3596:
3595:
3586:
3585:
3581:
3574:
3560:Badescu, Lucian
3558:
3557:
3553:
3546:
3532:Badescu, Lucian
3530:
3529:
3525:
3517:
3513:
3494:
3493:
3489:
3485:(1994), p. 192.
3477:
3473:
3454:
3453:
3449:
3437:
3433:
3424:
3420:
3411:
3410:
3406:
3387:
3386:
3379:
3360:
3359:
3355:
3348:
3334:Badescu, Lucian
3332:
3331:
3327:
3322:
3305:
3293:Krull dimension
3239:
3224:canonical model
3133:
3101:
3100:
3083:
3077:
3075:Canonical rings
3026:trigonal curves
3018:Petri's theorem
2958:linearly normal
2956:− 1 as a
2898:
2869:
2867:
2851:
2849:
2796:
2788:canonical curve
2771:
2756:
2735:
2673:
2649:
2598:
2570:
2569:
2548:
2543:
2542:
2519:
2518:
2492:
2491:
2446:
2441:
2440:
2421:
2420:
2396:
2395:
2374:
2369:
2368:
2349:
2348:
2325:
2324:
2321:
2303:induced by the
2269:
2256:
2221:
2220:
2167:
2137:
2133:
2122:
2121:
2102:
2101:
2080:
2067:
2056:
2055:
2013:
1991:
1987:
1973:
1957:
1937:
1924:
1911:
1906:
1905:
1870:
1861:
1848:
1835:
1830:
1829:
1804:
1803:
1780:
1779:
1756:
1755:
1732:
1731:
1686:
1676:
1666:
1661:
1660:
1653:multiple fibers
1631:
1626:
1625:
1598:
1593:
1592:
1572:
1552:
1539:
1534:
1533:
1512:
1493:
1488:
1487:
1456:
1455:
1427:
1426:
1401:
1400:
1381:
1380:
1356:
1336:
1335:
1316:
1315:
1294:
1289:
1288:
1258:
1245:
1232:
1221:
1220:
1219:, we have that
1201:
1200:
1179:
1169:
1137:
1136:
1113:
1112:
1081:
1080:
1044:
1039:
1038:
1019:
1018:
999:
998:
972:
971:
952:
951:
928:
927:
908:
907:
885:
884:
859:
842:
832:
827:
826:
807:
806:
780:
779:
748:
747:
727:
726:
705:
704:
701:
655:
636:
620:
615:
614:
562:
549:
536:
531:
530:
500:
494:
466:
465:
442:
441:
417:
412:
411:
382:
381:
362:
361:
334:
333:
306:
305:
286:
285:
283:Cartier divisor
275:canonical class
248:
247:
220:
219:
191:
186:
185:
175:complex numbers
152:
151:
132:
131:
119:, which is the
93:
86:
85:
63:
62:
43:
42:
23:
22:
18:Canonical class
15:
12:
11:
5:
3626:
3624:
3616:
3615:
3610:
3608:Vector bundles
3600:
3599:
3594:
3593:
3579:
3572:
3551:
3544:
3523:
3511:
3487:
3471:
3447:
3439:David Eisenbud
3431:
3425:Rick Miranda,
3418:
3404:
3377:
3353:
3346:
3324:
3323:
3321:
3318:
3317:
3316:
3311:
3304:
3301:
3184:
3183:
3172:
3169:
3164:
3159:
3155:
3151:
3148:
3145:
3140:
3136:
3130:
3125:
3122:
3119:
3115:
3111:
3108:
3087:canonical ring
3081:Canonical ring
3079:Main article:
3076:
3073:
3016:− 3)/2.
2897:
2894:
2882:
2881:
2839:
2838:
2770:
2767:
2752:
2746:elliptic curve
2731:
2725:Riemann sphere
2687:is zero, then
2672:
2669:
2648:
2645:
2625:-canonical map
2597:
2596:Canonical maps
2594:
2577:
2555:
2551:
2526:
2502:
2499:
2479:
2474:
2469:
2465:
2461:
2456:
2453:
2449:
2428:
2406:
2403:
2381:
2377:
2356:
2332:
2320:
2317:
2297:
2296:
2284:
2281:
2276:
2272:
2268:
2263:
2259:
2254:
2250:
2247:
2244:
2239:
2234:
2231:
2228:
2205:
2200:
2195:
2192:
2189:
2186:
2183:
2178:
2172:
2166:
2163:
2160:
2156:
2151:
2148:
2142:
2136:
2132:
2129:
2109:
2087:
2083:
2079:
2074:
2070:
2066:
2063:
2052:
2051:
2039:
2034:
2030:
2026:
2020:
2016:
2010:
2005:
2002:
1999:
1995:
1990:
1984:
1978:
1972:
1969:
1964:
1960:
1956:
1951:
1948:
1942:
1936:
1931:
1927:
1923:
1918:
1914:
1890:
1887:
1884:
1877:
1873:
1866:
1860:
1855:
1851:
1847:
1842:
1838:
1817:
1814:
1811:
1789:
1765:
1741:
1717:
1712:
1707:
1702:
1697:
1691:
1683:
1679:
1673:
1669:
1638:
1634:
1613:
1610:
1605:
1601:
1577:
1574:
1569:
1565:
1559:
1555:
1551:
1546:
1542:
1519:
1515:
1511:
1508:
1505:
1500:
1496:
1475:
1472:
1469:
1466:
1463:
1440:
1437:
1434:
1414:
1411:
1408:
1388:
1368:
1363:
1359:
1355:
1352:
1349:
1346:
1343:
1323:
1301:
1297:
1276:
1273:
1270:
1265:
1261:
1257:
1252:
1248:
1244:
1239:
1235:
1231:
1228:
1208:
1186:
1182:
1176:
1172:
1166:
1161:
1158:
1155:
1151:
1147:
1144:
1120:
1100:
1097:
1094:
1091:
1088:
1061:
1058:
1053:
1048:
1026:
1006:
982:
979:
959:
935:
915:
892:
870:
864:
858:
853:
847:
839:
835:
814:
787:
767:
764:
761:
758:
755:
734:
712:
700:
697:
681:
680:
669:
664:
659:
654:
651:
648:
643:
639:
635:
632:
627:
623:
608:
607:
596:
593:
590:
587:
584:
579:
574:
569:
565:
561:
556:
552:
548:
543:
539:
508:smooth variety
496:Main article:
493:
490:
473:
449:
427:
424:
420:
409:inverse bundle
389:
369:
341:
313:
293:
255:
238:. This is the
227:
203:
198:
194:
159:
139:
125:exterior power
108:
105:
100:
96:
70:
50:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3625:
3614:
3611:
3609:
3606:
3605:
3603:
3589:
3583:
3580:
3575:
3573:9780387986685
3569:
3565:
3561:
3555:
3552:
3547:
3545:9780387986685
3541:
3537:
3533:
3527:
3524:
3520:
3515:
3512:
3507:
3503:
3502:
3497:
3491:
3488:
3484:
3480:
3475:
3472:
3468:
3464:
3463:
3458:
3451:
3448:
3444:
3440:
3435:
3432:
3428:
3422:
3419:
3414:
3408:
3405:
3401:
3397:
3396:
3391:
3384:
3382:
3378:
3373:
3369:
3368:
3363:
3357:
3354:
3349:
3347:9780387986685
3343:
3339:
3335:
3329:
3326:
3319:
3315:
3312:
3310:
3307:
3306:
3302:
3300:
3298:
3294:
3290:
3286:
3281:
3278:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3243:
3237:
3233:
3229:
3225:
3221:
3216:
3214:
3210:
3205:
3201:
3197:
3193:
3189:
3170:
3162:
3157:
3153:
3149:
3146:
3138:
3134:
3123:
3120:
3117:
3113:
3109:
3106:
3099:
3098:
3097:
3096:
3092:
3088:
3082:
3074:
3072:
3069:
3067:
3063:
3062:ruled surface
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3034:Oscar Chisini
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2995:
2991:
2986:
2983:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2947:
2943:
2939:
2935:
2934:cubic surface
2931:
2927:
2923:
2919:
2915:
2911:
2907:
2903:
2895:
2893:
2891:
2887:
2876:
2872:
2865:
2858:
2854:
2847:
2844:
2843:
2842:
2836:
2832:
2828:
2825:
2824:
2823:
2821:
2817:
2813:
2809:
2805:
2799:
2794:
2790:
2789:
2784:
2780:
2776:
2768:
2766:
2764:
2760:
2755:
2751:
2747:
2743:
2740:is one, then
2739:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2675:Suppose that
2670:
2668:
2666:
2662:
2658:
2654:
2646:
2644:
2642:
2638:
2634:
2630:
2626:
2624:
2619:
2615:
2614:canonical map
2611:
2607:
2603:
2595:
2593:
2591:
2575:
2553:
2549:
2540:
2524:
2516:
2500:
2497:
2472:
2467:
2463:
2454:
2451:
2447:
2426:
2417:
2404:
2401:
2379:
2375:
2354:
2346:
2330:
2319:Singular case
2318:
2316:
2314:
2310:
2306:
2302:
2282:
2279:
2274:
2270:
2266:
2261:
2257:
2248:
2245:
2229:
2226:
2219:
2218:
2217:
2190:
2187:
2184:
2176:
2161:
2158:
2154:
2149:
2146:
2134:
2130:
2127:
2107:
2085:
2081:
2077:
2072:
2068:
2064:
2061:
2037:
2032:
2028:
2024:
2018:
2014:
2008:
2003:
2000:
1997:
1993:
1988:
1982:
1970:
1962:
1958:
1954:
1949:
1946:
1929:
1925:
1921:
1916:
1912:
1904:
1903:
1902:
1888:
1885:
1875:
1871:
1858:
1853:
1849:
1840:
1836:
1815:
1812:
1809:
1710:
1700:
1695:
1681:
1677:
1671:
1667:
1659:one has that
1658:
1654:
1636:
1632:
1611:
1608:
1603:
1599:
1575:
1573:
1567:
1563:
1557:
1553:
1549:
1544:
1540:
1517:
1513:
1509:
1506:
1503:
1498:
1494:
1473:
1467:
1464:
1461:
1452:
1438:
1435:
1432:
1412:
1409:
1406:
1386:
1361:
1357:
1350:
1347:
1344:
1341:
1321:
1299:
1295:
1274:
1271:
1268:
1263:
1259:
1255:
1250:
1246:
1242:
1237:
1233:
1229:
1226:
1206:
1184:
1180:
1174:
1170:
1164:
1159:
1156:
1153:
1149:
1145:
1142:
1134:
1118:
1098:
1092:
1089:
1086:
1078:
1073:
1059:
1056:
1051:
1024:
1004:
996:
980:
977:
957:
949:
933:
913:
906:
890:
868:
856:
851:
837:
833:
812:
804:
801:
785:
765:
759:
756:
753:
746:
732:
710:
698:
696:
694:
690:
686:
667:
662:
649:
646:
641:
637:
630:
625:
621:
613:
612:
611:
594:
585:
572:
567:
563:
554:
550:
546:
541:
537:
529:
528:
527:
525:
521:
517:
513:
509:
505:
502:Suppose that
499:
491:
489:
487:
471:
463:
447:
425:
422:
418:
410:
406:
401:
387:
367:
359:
358:anticanonical
355:
339:
331:
327:
311:
291:
284:
280:
279:divisor class
276:
271:
269:
253:
245:
244:Serre duality
241:
225:
217:
201:
196:
192:
184:
180:
176:
171:
157:
130:
126:
122:
106:
103:
98:
84:
68:
61:of dimension
48:
41:
38:
34:
30:
19:
3582:
3563:
3554:
3535:
3526:
3521:, pp. 11-13.
3514:
3499:
3490:
3482:
3474:
3460:
3450:
3442:
3434:
3426:
3421:
3407:
3393:
3365:
3356:
3337:
3328:
3288:
3282:
3279:
3274:
3270:
3258:
3254:
3250:
3246:
3241:
3231:
3227:
3223:
3217:
3212:
3208:
3203:
3199:
3187:
3185:
3090:
3086:
3084:
3070:
3041:
3029:
3021:
3017:
3013:
3009:
3005:
3001:
2989:
2987:
2981:
2977:
2969:
2965:
2961:
2953:
2949:
2945:
2937:
2925:
2917:
2913:
2909:
2905:
2904:which means
2901:
2899:
2896:General case
2889:
2883:
2874:
2870:
2863:
2856:
2852:
2845:
2840:
2834:
2830:
2826:
2819:
2815:
2803:
2797:
2792:
2786:
2782:
2774:
2772:
2762:
2758:
2753:
2749:
2741:
2737:
2732:
2728:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2674:
2664:
2660:
2650:
2640:
2636:
2632:
2628:
2622:
2621:
2617:
2613:
2609:
2606:rational map
2599:
2539:Weil divisor
2418:
2345:Weil divisor
2322:
2298:
2053:
1652:
1453:
1074:
994:
724:
702:
692:
688:
682:
609:
523:
519:
515:
511:
503:
501:
486:Fano variety
484:is called a
404:
402:
357:
353:
274:
272:
215:
177:, it is the
172:
120:
37:non-singular
32:
26:
3095:graded ring
3012:− 2)(
2715:) = −
1451:otherwise.
400:canonical.
83:line bundle
29:mathematics
3602:Categories
2936:; and for
2590:Gorenstein
2588:is S2 and
2309:K3 surface
1828:such that
218:-forms on
3506:EMS Press
3467:EMS Press
3400:EMS Press
3372:EMS Press
3129:∞
3114:⨁
2948:of genus
2862:,
2800:− 1
2671:Low genus
2602:effective
2498:−
2464:ω
2452:−
2347:class on
2280:−
2253:⟺
2230:
2191:
2162:χ
2147:−
2131:
2100:for each
2065:≤
1994:∑
1971:⊗
1959:ω
1955:⊗
1947:−
1930:∗
1922:≅
1913:ω
1813:∈
1711:⊕
1682:∗
1507:…
1471:→
1351:
1334:; so for
1150:∑
1096:→
1057:×
978:−
857:≅
838:∗
805:morphism
763:→
745:fibration
573:⊗
564:ω
555:∗
538:ω
510:and that
423:−
419:ω
197:∗
173:Over the
138:Ω
107:ω
95:Ω
3562:(2001).
3534:(2001).
3336:(2001).
3303:See also
2699:, where
2033:′
1576:′
3508:, 2001
3374:, 2001
3093:is the
2930:quadric
2868:√
2850:√
2802:. When
995:minimal
277:is the
127:of the
3570:
3542:
3344:
3190:is an
3064:and a
3048:: the
2932:and a
2814:, and
2748:, and
2744:is an
2627:. The
2490:, the
2227:length
2188:length
2054:where
1730:where
1591:where
1486:. Let
1287:where
948:fibers
800:proper
725:genus
31:, the
3320:Notes
3261:is a
3240:Proj
2806:is a
2683:. If
2608:from
1655:. By
1379:, if
926:. If
903:have
798:is a
506:is a
462:ample
380:with
356:. An
281:of a
35:of a
3568:ISBN
3540:ISBN
3342:ISBN
3283:The
3236:Proj
3218:The
3085:The
3036:and
2864:x dx
2120:and
2078:<
1886:>
1609:>
1436:>
1425:and
803:flat
703:Let
522:and
403:The
328:for
273:The
242:for
3295:or
3287:of
3269:of
3263:nef
3257:of
3089:of
2980:on
2779:big
2773:If
2711:(1/
2691:is
2517:of
2128:deg
1348:gcd
1199:of
950:of
778:of
460:is
332:on
304:on
246:on
150:on
123:th
27:In
3604::
3504:,
3498:,
3481:,
3465:,
3459:,
3441:,
3398:,
3392:,
3380:^
3370:,
3364:,
3299:.
3068:.
2892:.
2846:dx
2829:=
2717:dt
2667:.
1079:)
1072:.
695:.
488:.
464:,
270:.
170:.
3590:.
3576:.
3548:.
3350:.
3289:V
3275:V
3271:K
3259:V
3255:K
3251:V
3247:V
3242:R
3232:V
3228:V
3213:k
3209:k
3204:V
3200:V
3188:V
3171:.
3168:)
3163:d
3158:V
3154:K
3150:,
3147:V
3144:(
3139:0
3135:H
3124:0
3121:=
3118:d
3110:=
3107:R
3091:V
3030:g
3022:g
3014:g
3010:g
3006:C
2990:g
2982:C
2978:D
2970:g
2966:C
2962:g
2954:g
2950:g
2946:C
2938:g
2926:g
2918:g
2914:g
2910:C
2906:g
2902:C
2890:x
2880:.
2877:)
2875:x
2873:(
2871:P
2866:/
2859:)
2857:x
2855:(
2853:P
2848:/
2837:)
2835:x
2833:(
2831:P
2827:y
2820:P
2816:C
2804:C
2798:g
2793:g
2783:n
2775:C
2763:n
2759:n
2754:C
2750:K
2742:C
2738:g
2733:C
2729:K
2721:t
2719:/
2713:t
2709:d
2705:C
2701:P
2697:P
2693:P
2689:C
2685:g
2681:g
2677:C
2665:g
2661:g
2641:n
2637:n
2633:V
2629:n
2623:n
2618:n
2610:V
2576:X
2554:X
2550:K
2525:X
2501:d
2478:)
2473:.
2468:X
2460:(
2455:d
2448:h
2427:X
2405:.
2402:X
2380:X
2376:K
2355:X
2331:X
2295:.
2283:1
2275:i
2271:m
2267:=
2262:i
2258:a
2249:0
2246:=
2243:)
2238:T
2233:(
2204:)
2199:T
2194:(
2185:+
2182:)
2177:X
2171:O
2165:(
2159:=
2155:)
2150:1
2141:L
2135:(
2108:i
2086:i
2082:m
2073:i
2069:a
2062:0
2038:)
2029:i
2025:F
2019:i
2015:a
2009:r
2004:1
2001:=
1998:i
1989:(
1983:X
1977:O
1968:)
1963:B
1950:1
1941:L
1935:(
1926:f
1917:X
1889:1
1883:)
1876:b
1872:X
1865:O
1859:,
1854:b
1850:X
1846:(
1841:0
1837:h
1816:B
1810:b
1788:T
1764:T
1740:L
1716:T
1706:L
1701:=
1696:X
1690:O
1678:f
1672:1
1668:R
1637:i
1633:F
1612:1
1604:i
1600:m
1568:i
1564:F
1558:i
1554:m
1550:=
1545:i
1541:F
1518:r
1514:F
1510:,
1504:,
1499:1
1495:F
1474:B
1468:X
1465::
1462:f
1439:1
1433:m
1413:1
1410:=
1407:m
1387:F
1367:)
1362:i
1358:a
1354:(
1345:=
1342:m
1322:X
1300:X
1296:K
1275:,
1272:0
1269:=
1264:i
1260:E
1256:.
1251:X
1247:K
1243:=
1238:i
1234:E
1230:.
1227:F
1207:f
1185:i
1181:E
1175:i
1171:a
1165:n
1160:1
1157:=
1154:i
1146:=
1143:F
1119:f
1099:B
1093:X
1090::
1087:f
1060:B
1052:1
1047:P
1025:X
1005:X
981:1
958:f
934:X
914:g
891:f
869:B
863:O
852:X
846:O
834:f
813:f
786:X
766:B
760:X
757::
754:f
733:g
711:X
693:D
689:X
668:.
663:D
658:|
653:)
650:D
647:+
642:X
638:K
634:(
631:=
626:D
622:K
595:.
592:)
589:)
586:D
583:(
578:O
568:X
560:(
551:i
547:=
542:D
524:D
520:X
516:X
512:D
504:X
472:V
448:V
426:1
388:K
368:K
340:V
312:V
292:K
254:V
226:V
216:n
202:V
193:T
158:V
121:n
104:=
99:n
69:n
49:V
20:)
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