184:,O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable open cover. On triple overlaps, any lift of these transition functions satisfies the cocycle condition up to an invertible function. The collection of these functions forms a 2-cocycle which vanishes in
1177:
1497:
2089:
1596:
1352:
1055:
849:
753:
501:
1028:
are generically elliptic curves. Because every elliptic curve is a genus 1 curve with a distinguished point, there exists a global section of the fibration. Because of this global section, there exists a model of
1920:
105:
1731:
1392:
961:
1026:
1628:
890:
144:
252:
1387:
991:
1648:
1047:
917:
893:
1967:
1505:
1191:
1172:{\displaystyle X\to \mathbb {P} ({\mathcal {O}}_{\mathbb {P} ^{1}}(4)\oplus {\mathcal {O}}_{\mathbb {P} ^{1}}(6)\oplus {\mathcal {O}}_{\mathbb {P} ^{1}})}
761:
691:
2256:
2305:
418:
1826:
169:
gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an
57:
2352:
2317:
1492:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{1}}(4),{\mathcal {O}}_{\mathbb {P} ^{1}}(6),{\mathcal {O}}_{\mathbb {P} ^{1}}}
1653:
925:
390:
2344:
996:
2383:
2378:
1604:
866:
2339:
2161:
2150:
118:
1630:. Note this equation is well-defined because each term in the Weierstrass equation has total degree
342:
192:,O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if
1950:
Chern classes as the coefficients in the relation; this is the approach taken by
Grothendieck.
221:
2362:
2327:
2285:
1360:
1183:
969:
8:
1786:
1633:
2289:
2223:
2166:
2129:
1032:
902:
170:
2084:{\displaystyle A_{k}(\mathbf {P} (E))=\bigoplus _{i=0}^{r-1}\zeta ^{i}A_{k-r+1+i}(X).}
1591:{\displaystyle a_{i}\in H^{0}(\mathbb {P} ^{1},{\mathcal {O}}_{\mathbb {P} ^{1}}(2i))}
2348:
2313:
2293:
2273:
2145:
1650:(meaning the degree of the coefficient plus the degree of the monomial. For example,
599:. Moreover, this natural map vanishes at a point exactly when the point is a line in
163:
1347:{\displaystyle y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}}
2334:
2265:
863:
Many non-trivial examples of projective bundles can be found using fibrations over
358:
216:
174:
32:
2254:
Elencwajg, G.; Narasimhan, M. S. (1983), "Projective bundles on a complex torus",
1961:
is smooth). In particular, for Chow groups, there is the direct sum decomposition
2358:
2323:
2309:
2281:
1953:
Over fields other than the complex field, the same description remains true with
1774:
844:{\displaystyle g^{*}({\mathcal {O}}(-1))\simeq {\mathcal {O}}(-1)\otimes p^{*}L.}
748:{\displaystyle g:\mathbf {P} (E){\overset {\sim }{\to }}\mathbf {P} (E\otimes L)}
197:
412:), there is a natural exact sequence (called the tautological exact sequence):
112:
2372:
2277:
2155:
496:{\displaystyle 0\to {\mathcal {O}}_{\mathbf {P} (E)}(-1)\to p^{*}E\to Q\to 0}
159:
2269:
662:⊕ 1) is referred to as the projective completion (or "compactification") of
28:
1798:
20:
897:
1954:
574:
855:
by the universal property applied to the line bundle on the right.)
2228:
1915:{\displaystyle \zeta ^{r}+c_{1}(E)\zeta ^{r-1}+\cdots +c_{r}(E)=0}
2222:
Propp, Oron Y. (2019-05-22). "Constructing explicit K3 spectra".
642:⊕ 1), called the hyperplane at infinity, and the complement of
107:
and transition automorphisms are linear. Over a regular scheme
1946:. One interesting feature of this description is that one can
626:
and the trivial line bundle (i.e., the structure sheaf). Then
614:
A particularly useful instance of this construction is when
2308:. 3. Folge., vol. 2 (2nd ed.), Berlin, New York:
2094:
As it turned out, this decomposition remains valid even if
521:
be vector bundles (locally free sheaves of finite rank) on
153:
100:{\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}}
273:
is characterized by the universal property that says:
1970:
1829:
1656:
1636:
1607:
1508:
1395:
1363:
1194:
1058:
1035:
999:
972:
928:
905:
869:
764:
694:
421:
224:
121:
60:
603:; in other words, the zero-locus of this section is
2253:
1726:{\displaystyle {\text{deg}}(a_{1}xyz)=2+(4+6+0)=12}
2083:
1914:
1725:
1642:
1622:
1590:
1491:
1381:
1346:
1171:
1041:
1020:
985:
955:
911:
884:
843:
747:
495:
246:
138:
99:
1736:
681:by a line bundle; precisely, given a line bundle
2370:
2306:Ergebnisse der Mathematik und ihrer Grenzgebiete
2257:Journal für die reine und angewandte Mathematik
16:Fiber bundle whose fibers are projective spaces
1957:in place of cohomology ring (still assuming
510:is called the tautological quotient-bundle.
146:for some vector bundle (locally free sheaf)
2347:, vol. 52, New York: Springer-Verlag,
2098:is not smooth nor projective. In contrast,
1745:be a complex smooth projective variety and
2333:
2209:
2197:
2185:
1049:giving a morphism to the projective bundle
956:{\displaystyle \pi :X\to \mathbb {P} ^{1}}
208:,O*)=0, and so this obstruction vanishes.
2227:
1610:
1561:
1537:
1477:
1442:
1407:
1154:
1119:
1084:
1066:
1008:
943:
872:
211:The projective bundle of a vector bundle
123:
115:, every projective bundle is of the form
82:
553:be the projection. Then the natural map
154:The projective bundle of a vector bundle
2136:, the vector spaces, are contractible.
2371:
2299:
2242:
50:-bundle if it is locally a projective
2221:
2132:, morally because that the fibers of
1021:{\displaystyle p\in \mathbb {P} ^{1}}
361:in the sense that when a line bundle
1499:, respectively, and the coefficients
685:, there is the natural isomorphism:
397:for a more explicit construction of
2158:(an example of a projective bundle)
1389:represent the local coordinates of
13:
1553:
1469:
1434:
1399:
1146:
1111:
1076:
805:
780:
431:
311:is to specify a line subbundle of
14:
2395:
1985:
1749:a complex vector bundle of rank
1623:{\displaystyle \mathbb {P} ^{1}}
919:is a K3 surface with a fibration
885:{\displaystyle \mathbb {P} ^{1}}
726:
702:
438:
139:{\displaystyle \mathbb {P} (E)}
2236:
2215:
2203:
2191:
2179:
2075:
2069:
1998:
1995:
1989:
1981:
1903:
1897:
1859:
1853:
1737:Cohomology ring and Chow group
1714:
1696:
1684:
1662:
1585:
1582:
1573:
1532:
1460:
1454:
1425:
1419:
1166:
1137:
1131:
1102:
1096:
1070:
1062:
938:
819:
810:
797:
794:
785:
775:
742:
730:
717:
712:
706:
487:
481:
465:
462:
453:
448:
442:
425:
330:, one gets the line subbundle
241:
235:
133:
127:
1:
2345:Graduate Texts in Mathematics
2188:, Ch. II, Exercise 7.10. (c).
2172:
1769:be the projective bundle of
7:
2200:, Ch. II, Proposition 7.12.
2139:
1601:are sections of sheaves on
896:. For example, an elliptic
858:
677:) is stable under twisting
573:is a global section of the
293:through the projection map
10:
2400:
2151:cone (algebraic geometry)
650:) can be identified with
215:is the same thing as the
42:over a Noetherian scheme
2300:Fulton, William (1998),
343:tautological line bundle
247:{\displaystyle G_{1}(E)}
38:By definition, a scheme
2270:10.1515/crll.1983.340.1
1793:) through the pullback
2085:
2030:
1916:
1727:
1644:
1624:
1599:
1592:
1493:
1383:
1355:
1348:
1180:
1173:
1043:
1022:
987:
964:
957:
913:
886:
845:
749:
669:The projective bundle
497:
365:gives a factorization
261:The projective bundle
248:
140:
101:
2162:Severi–Brauer variety
2086:
2004:
1917:
1820:)) with the relation
1728:
1645:
1625:
1593:
1501:
1494:
1384:
1382:{\displaystyle x,y,z}
1349:
1187:
1174:
1051:
1044:
1023:
988:
986:{\displaystyle E_{p}}
966:such that the fibers
958:
921:
914:
887:
846:
750:
634:) is a hyperplane in
498:
269:) of a vector bundle
249:
141:
102:
2212:, Ch. II, Lemma 7.9.
1968:
1827:
1654:
1634:
1605:
1506:
1393:
1361:
1192:
1184:Weierstrass equation
1056:
1033:
997:
970:
926:
903:
894:Lefschetz fibrations
867:
762:
692:
419:
322:For example, taking
222:
119:
58:
2302:Intersection theory
1942:-th Chern class of
851:(In fact, one gets
381:is the pullback of
96:
2384:Algebraic geometry
2379:Algebraic topology
2340:Algebraic Geometry
2167:Hirzebruch surface
2130:Gysin homomorphism
2081:
1912:
1723:
1643:{\displaystyle 12}
1640:
1620:
1588:
1489:
1379:
1344:
1169:
1039:
1018:
983:
953:
909:
882:
841:
745:
618:is the direct sum
493:
353:). Moreover, this
244:
136:
97:
80:
2354:978-0-387-90244-9
2335:Hartshorne, Robin
2319:978-3-540-62046-4
2146:Proj construction
1812:(1)) generates H(
1797:. Then the first
1660:
1042:{\displaystyle X}
912:{\displaystyle X}
723:
277:Given a morphism
33:projective spaces
31:whose fibers are
25:projective bundle
2391:
2365:
2330:
2296:
2246:
2240:
2234:
2233:
2231:
2219:
2213:
2207:
2201:
2195:
2189:
2183:
2090:
2088:
2087:
2082:
2068:
2067:
2040:
2039:
2029:
2018:
1988:
1980:
1979:
1921:
1919:
1918:
1913:
1896:
1895:
1877:
1876:
1852:
1851:
1839:
1838:
1732:
1730:
1729:
1724:
1674:
1673:
1661:
1658:
1649:
1647:
1646:
1641:
1629:
1627:
1626:
1621:
1619:
1618:
1613:
1597:
1595:
1594:
1589:
1572:
1571:
1570:
1569:
1564:
1557:
1556:
1546:
1545:
1540:
1531:
1530:
1518:
1517:
1498:
1496:
1495:
1490:
1488:
1487:
1486:
1485:
1480:
1473:
1472:
1453:
1452:
1451:
1450:
1445:
1438:
1437:
1418:
1417:
1416:
1415:
1410:
1403:
1402:
1388:
1386:
1385:
1380:
1353:
1351:
1350:
1345:
1343:
1342:
1333:
1332:
1320:
1319:
1307:
1306:
1291:
1290:
1281:
1280:
1268:
1267:
1255:
1254:
1242:
1241:
1220:
1219:
1204:
1203:
1178:
1176:
1175:
1170:
1165:
1164:
1163:
1162:
1157:
1150:
1149:
1130:
1129:
1128:
1127:
1122:
1115:
1114:
1095:
1094:
1093:
1092:
1087:
1080:
1079:
1069:
1048:
1046:
1045:
1040:
1027:
1025:
1024:
1019:
1017:
1016:
1011:
992:
990:
989:
984:
982:
981:
962:
960:
959:
954:
952:
951:
946:
918:
916:
915:
910:
891:
889:
888:
883:
881:
880:
875:
850:
848:
847:
842:
834:
833:
809:
808:
784:
783:
774:
773:
754:
752:
751:
746:
729:
724:
716:
705:
598:
572:
502:
500:
499:
494:
477:
476:
452:
451:
441:
435:
434:
359:universal bundle
310:
253:
251:
250:
245:
234:
233:
217:Grassmann bundle
175:cohomology group
145:
143:
142:
137:
126:
106:
104:
103:
98:
95:
90:
85:
73:
72:
2399:
2398:
2394:
2393:
2392:
2390:
2389:
2388:
2369:
2368:
2355:
2320:
2310:Springer-Verlag
2250:
2249:
2241:
2237:
2220:
2216:
2210:Hartshorne 1977
2208:
2204:
2198:Hartshorne 1977
2196:
2192:
2186:Hartshorne 1977
2184:
2180:
2175:
2142:
2123:
2106:
2045:
2041:
2035:
2031:
2019:
2008:
1984:
1975:
1971:
1969:
1966:
1965:
1933:
1891:
1887:
1866:
1862:
1847:
1843:
1834:
1830:
1828:
1825:
1824:
1807:
1775:cohomology ring
1739:
1669:
1665:
1657:
1655:
1652:
1651:
1635:
1632:
1631:
1614:
1609:
1608:
1606:
1603:
1602:
1565:
1560:
1559:
1558:
1552:
1551:
1550:
1541:
1536:
1535:
1526:
1522:
1513:
1509:
1507:
1504:
1503:
1481:
1476:
1475:
1474:
1468:
1467:
1466:
1446:
1441:
1440:
1439:
1433:
1432:
1431:
1411:
1406:
1405:
1404:
1398:
1397:
1396:
1394:
1391:
1390:
1362:
1359:
1358:
1338:
1334:
1328:
1324:
1315:
1311:
1302:
1298:
1286:
1282:
1276:
1272:
1263:
1259:
1250:
1246:
1237:
1233:
1215:
1211:
1199:
1195:
1193:
1190:
1189:
1182:defined by the
1158:
1153:
1152:
1151:
1145:
1144:
1143:
1123:
1118:
1117:
1116:
1110:
1109:
1108:
1088:
1083:
1082:
1081:
1075:
1074:
1073:
1065:
1057:
1054:
1053:
1034:
1031:
1030:
1012:
1007:
1006:
998:
995:
994:
977:
973:
971:
968:
967:
947:
942:
941:
927:
924:
923:
904:
901:
900:
876:
871:
870:
868:
865:
864:
861:
829:
825:
804:
803:
779:
778:
769:
765:
763:
760:
759:
725:
715:
701:
693:
690:
689:
654:. In this way,
577:
554:
472:
468:
437:
436:
430:
429:
428:
420:
417:
416:
294:
289:, to factorize
254:of 1-planes in
229:
225:
223:
220:
219:
198:Riemann surface
156:
122:
120:
117:
116:
91:
86:
81:
68:
64:
59:
56:
55:
17:
12:
11:
5:
2397:
2387:
2386:
2381:
2367:
2366:
2353:
2331:
2318:
2297:
2248:
2247:
2245:, Theorem 3.3.
2235:
2214:
2202:
2190:
2177:
2176:
2174:
2171:
2170:
2169:
2164:
2159:
2153:
2148:
2141:
2138:
2115:
2102:
2092:
2091:
2080:
2077:
2074:
2071:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2044:
2038:
2034:
2028:
2025:
2022:
2017:
2014:
2011:
2007:
2003:
2000:
1997:
1994:
1991:
1987:
1983:
1978:
1974:
1929:
1923:
1922:
1911:
1908:
1905:
1902:
1899:
1894:
1890:
1886:
1883:
1880:
1875:
1872:
1869:
1865:
1861:
1858:
1855:
1850:
1846:
1842:
1837:
1833:
1805:
1738:
1735:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1672:
1668:
1664:
1639:
1617:
1612:
1587:
1584:
1581:
1578:
1575:
1568:
1563:
1555:
1549:
1544:
1539:
1534:
1529:
1525:
1521:
1516:
1512:
1484:
1479:
1471:
1465:
1462:
1459:
1456:
1449:
1444:
1436:
1430:
1427:
1424:
1421:
1414:
1409:
1401:
1378:
1375:
1372:
1369:
1366:
1341:
1337:
1331:
1327:
1323:
1318:
1314:
1310:
1305:
1301:
1297:
1294:
1289:
1285:
1279:
1275:
1271:
1266:
1262:
1258:
1253:
1249:
1245:
1240:
1236:
1232:
1229:
1226:
1223:
1218:
1214:
1210:
1207:
1202:
1198:
1168:
1161:
1156:
1148:
1142:
1139:
1136:
1133:
1126:
1121:
1113:
1107:
1104:
1101:
1098:
1091:
1086:
1078:
1072:
1068:
1064:
1061:
1038:
1015:
1010:
1005:
1002:
980:
976:
950:
945:
940:
937:
934:
931:
908:
879:
874:
860:
857:
840:
837:
832:
828:
824:
821:
818:
815:
812:
807:
802:
799:
796:
793:
790:
787:
782:
777:
772:
768:
756:
755:
744:
741:
738:
735:
732:
728:
722:
719:
714:
711:
708:
704:
700:
697:
504:
503:
492:
489:
486:
483:
480:
475:
471:
467:
464:
461:
458:
455:
450:
447:
444:
440:
433:
427:
424:
320:
319:
243:
240:
237:
232:
228:
155:
152:
135:
132:
129:
125:
113:smooth variety
94:
89:
84:
79:
76:
71:
67:
63:
54:-space; i.e.,
15:
9:
6:
4:
3:
2:
2396:
2385:
2382:
2380:
2377:
2376:
2374:
2364:
2360:
2356:
2350:
2346:
2342:
2341:
2336:
2332:
2329:
2325:
2321:
2315:
2311:
2307:
2303:
2298:
2295:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2259:
2258:
2252:
2251:
2244:
2239:
2230:
2225:
2218:
2211:
2206:
2199:
2194:
2187:
2182:
2178:
2168:
2165:
2163:
2160:
2157:
2156:ruled surface
2154:
2152:
2149:
2147:
2144:
2143:
2137:
2135:
2131:
2127:
2122:
2118:
2114:
2110:
2105:
2101:
2097:
2078:
2072:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2042:
2036:
2032:
2026:
2023:
2020:
2015:
2012:
2009:
2005:
2001:
1992:
1976:
1972:
1964:
1963:
1962:
1960:
1956:
1951:
1949:
1945:
1941:
1937:
1932:
1928:
1909:
1906:
1900:
1892:
1888:
1884:
1881:
1878:
1873:
1870:
1867:
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2243:Fulton 1998
2128:), via the
1799:Chern class
1773:. Then the
1753:on it. Let
389:. See also
385:(-1) along
171:obstruction
21:mathematics
2373:Categories
2229:1810.08953
2173:References
898:K3 surface
758:such that
357:(-1) is a
111:such as a
2294:122557310
2278:0075-4102
2050:−
2033:ζ
2024:−
2006:⨁
1955:Chow ring
1938:) is the
1882:⋯
1871:−
1864:ζ
1832:ζ
1785:)) is an
1520:∈
1141:⊕
1106:⊕
1063:→
1004:∈
939:→
930:π
831:∗
823:⊗
814:−
801:≃
789:−
771:∗
737:⊗
721:∼
718:→
575:sheaf hom
488:→
482:→
474:∗
466:→
457:−
426:→
78:≃
66:×
2337:(1977),
2140:See also
892:such as
859:Examples
334:(-1) of
2363:0463157
2328:1644323
2286:0691957
622:⊕ 1 of
558:(-1) →
173:in the
164:variety
162:over a
2361:
2351:
2326:
2316:
2292:
2284:
2276:
1948:define
1925:where
1357:where
582:(-1),
537:. Let
506:where
401:(-1).
326:to be
158:Every
2290:S2CID
2224:arXiv
586:G) =
391:Cone#
200:then
46:is a
27:is a
2349:ISBN
2314:ISBN
2274:ISSN
2262:1983
2111:) =
1801:ζ =
1765:) →
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993:for
578:Hom(
549:) →
525:and
513:Let
306:) →
23:, a
2266:doi
1733:).
1659:deg
611:).
597:(1)
404:On
395:(1)
345:on
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2324:MR
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593:⊗
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517:⊂
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373:∘
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