1875:
2323:
715:
1977:
393:
192:
562:
The Proj construction in fact gives more than a mere scheme: a sheaf in graded modules over the structure sheaf is defined in the process. The homogeneous components of this graded sheaf are denoted
1269:
1042:
840:
2034:
1397:
659:, which shows that any Weil divisor is linearly equivalent to some power of a hyperplane divisor. This consideration proves that the Picard group of a projective space is free of rank 1. That is
1580:
1145:
1528:
1459:
944:
1709:
1621:
772:
1752:
1317:
305:
117:) is non-trivial (i.e. the common zero locus contains more than the single element {0}), and, more precisely, the ideal of polynomials that vanish on that locus coincides with the
878:
1760:
2093:
1484:
1212:
977:
638:
593:
542:
900:
445:
2193:
497:
It can be noted that the ring of global sections of this scheme is a field, which implies that the scheme is not affine. Any two open sets intersect non-trivially:
1898:
662:
1909:
332:
138:
1175:
1223:
997:
792:
494:
the projective schemes can be thought of as being obtained by the gluing via projectivization of isomorphic affine schemes.
1995:
1344:
1533:
2343:
2356:
1497:
1070:
2348:
1487:
to a projective space. A line bundle whose base can be embedded in a projective space by such a morphism is called
1154:
states that on the projective line, any vector bundle splits in a unique way as a direct sum of the line bundles.
1409:
912:
2390:
1673:
1585:
1151:
739:
1870:{\displaystyle j^{*}(\Gamma (\mathbf {P} ^{n},{\mathcal {O}}(1)))\subset \Gamma (X,j^{*}{\mathcal {O}}(1)).}
1714:
1274:
270:
2385:
2380:
2105:
for an application of the
Veronese embedding to the calculation of cohomology groups of smooth projective
849:
2375:
644:
1181:
2039:
949:
610:
565:
1901:
1638:
552:
516:
883:
2318:{\displaystyle {\frac {P(X_{0},\ldots ,X_{n})}{X_{0}^{\deg(P)}}}\mapsto P(1,X_{1},\ldots ,X_{n})}
455:. Its elements are therefore the rational functions with homogeneous numerator and some power of
401:
2171:
1660:
1488:
448:
502:
1334:, equivalently, their anticanonical line bundle is ample (in fact very ample). Their index (
2139:
652:
596:
510:
232:
43:
8:
2166:
1167:
1163:
787:
779:
225:
118:
46:
2144:
1883:
726:
218:
54:
27:
23:
1330:
The negativity of the canonical line bundle makes projective spaces prime examples of
2352:
2149:
1900:
is not contained in a hyperplane divisor, then the pull-back is an injection and the
903:
99:
69:
65:
2122:
128:
This last assertion is best summarized by the formula : for any relevant ideal
2338:
2161:
1470:
1323:
This fact derives from a fundamental geometric statement on projective spaces: the
1215:
734:
545:
106:
18:
1476:
Any choice of a finite system of nonsimultaneously vanishing global sections of a
656:
604:
311:
95:
76:
1324:
88:
2369:
2118:
471:
105:
that does not contain all polynomials of a certain degree (referred to as an
710:{\displaystyle \mathrm {Pic} \ \mathbf {P} _{\mathbf {k} }^{n}=\mathbb {Z} }
2106:
2102:
1338:
1331:
843:
221:
57:
1643:
1479:
648:
600:
463:
326:
varies over the set of homogeneous polynomials, by setting the sections
2121:. The intersection theory of curves in the projective plane yields the
1171:
487:
1659:
projective transformations is likewise equivalent to the choice of a
2098:
1972:{\displaystyle j^{*}(\Gamma (\mathbf {P} ^{n},{\mathcal {O}}(1)))}
1399:, and, by a theorem of Kobayashi-Ochiai, projective spaces are
388:{\displaystyle \Gamma (D(P),{\mathcal {O}}_{\mathbb {P} (V)})}
187:{\displaystyle {\mathcal {I}}({\mathcal {V}}(I))={\sqrt {I}}.}
26:. This article aims to define the notion in terms of abstract
1469:
As affine spaces can be embedded in projective spaces, all
240:
717:, and the isomorphism is given by the degree of divisors.
466:φ. The restriction of the structure sheaf to the open set
208:
1264:{\displaystyle {\mathcal {K}}(\mathbb {P} _{k}^{n}),\,}
1037:{\displaystyle \Gamma (\mathbb {P} ,{\mathcal {O}}(m))}
459:
as the denominator, with same degree as the numerator.
2042:
1987:
1073:
835:{\displaystyle {\mathcal {O}}(m),\ m\in \mathbb {Z} ,}
197:
In particular, maximal homogeneous relevant ideals of
30:
and to describe some basic uses of projective spaces.
2196:
1998:
1912:
1886:
1763:
1717:
1676:
1588:
1536:
1500:
1412:
1347:
1277:
1226:
1184:
1000:
952:
915:
886:
852:
795:
742:
665:
613:
568:
519:
404:
335:
273:
141:
2029:{\displaystyle \mathbb {P} ^{n}\to \mathbb {P} ^{N}}
1392:{\displaystyle \mathrm {Ind} (\mathbb {P} ^{n})=n+1}
447:, the zero degree component of the ring obtained by
1530:is the group of projectivized linear automorphisms
720:
2317:
2087:
2028:
1971:
1892:
1869:
1746:
1703:
1615:
1575:{\displaystyle \mathrm {PGL} _{n+1}(\mathbf {k} )}
1574:
1522:
1464:
1453:
1391:
1311:
1263:
1206:
1139:
1036:
971:
938:
894:
872:
834:
766:
709:
632:
587:
536:
439:
387:
299:
186:
1582:. The choice of a morphism to a projective space
2367:
2337:
1494:The group of symmetries of the projective space
1140:{\textstyle {\binom {m+n}{m}}={\binom {m+n}{n}}}
557:
201:are one-to-one with lines through the origin of
33:
2190:In coordinates this correspondence is given by
1523:{\displaystyle \mathbb {P} _{\mathbf {k} }^{n}}
991:. In particular, the space of global sections
462:The situation is most clear at a non-vanishing
109:), the common zero locus of all polynomials in
2155:
2112:
2133:
2117:As Fano varieties, the projective spaces are
2073:
2052:
1131:
1110:
1098:
1077:
470:(φ) is then canonically identified with the
1711:defines a globally generated line bundle by
1059:=0 and of homogeneous polynomials of degree
267:is the projectivization of the vector space
1454:{\displaystyle \mathrm {Ind} (X)=\dim X+1.}
909:The space of local sections on an open set
607:and line bundles, the first twisting sheaf
310:The definition of the sheaf is done on the
1651:. The choice of a projective embedding of
1473:can be embedded in projective spaces too.
939:{\displaystyle U\subseteq \mathbb {P} (V)}
2016:
2001:
1503:
1364:
1260:
1239:
1157:
1008:
923:
888:
855:
825:
763:
745:
703:
521:
367:
276:
1403:amongst Fano varieties by the property
1992:The Veronese embeddings are embeddings
1704:{\displaystyle j:X\to \mathbf {P} ^{n}}
1616:{\displaystyle j:X\to \mathbf {P} ^{n}}
767:{\displaystyle \mathbb {P} _{k}^{n},\,}
2368:
1747:{\displaystyle j^{*}{\mathcal {O}}(1)}
1312:{\displaystyle {\mathcal {O}}(-(n+1))}
1051:< 0, and consists of constants in
640:is equivalent to hyperplane divisors.
300:{\displaystyle \mathbb {A} _{k}^{n+1}}
209:Construction of projectivized schemes
1626:the action of this group is in fact
1166:, which appears for instance as the
873:{\displaystyle \mathbb {P} _{k}^{n}}
1988:An example: the Veronese embeddings
979:is the space of homogeneous degree
643:Since the ring of polynomials is a
599:. All of these sheaves are in fact
13:
2088:{\textstyle N={\binom {n+d}{d}}-1}
2056:
1949:
1926:
1847:
1823:
1800:
1777:
1730:
1545:
1542:
1539:
1420:
1417:
1414:
1355:
1352:
1349:
1280:
1229:
1207:{\displaystyle {\mathcal {O}}(-1)}
1187:
1114:
1081:
1017:
1001:
955:
902:. The isomorphism is given by the
798:
673:
670:
667:
616:
571:
360:
336:
154:
144:
14:
2402:
1670:A morphism to a projective space
983:regular functions on the cone in
972:{\displaystyle {\mathcal {O}}(k)}
633:{\displaystyle {\mathcal {O}}(1)}
588:{\displaystyle {\mathcal {O}}(i)}
1979:is a linear system of dimension
1934:
1785:
1691:
1603:
1565:
1509:
721:Classification of vector bundles
688:
682:
603:. By the correspondence between
548:, that furthermore is complete.
1465:Morphisms to projective schemes
537:{\displaystyle \mathbb {P} (V)}
2312:
2274:
2268:
2261:
2255:
2235:
2203:
2184:
2011:
1966:
1963:
1960:
1954:
1929:
1923:
1861:
1858:
1852:
1826:
1817:
1814:
1811:
1805:
1780:
1774:
1741:
1735:
1686:
1598:
1569:
1561:
1430:
1424:
1374:
1359:
1306:
1303:
1291:
1285:
1254:
1234:
1201:
1192:
1031:
1028:
1022:
1004:
966:
960:
933:
927:
809:
803:
627:
621:
582:
576:
531:
525:
428:
418:
411:
405:
382:
377:
371:
351:
345:
339:
168:
165:
159:
149:
91:by the degree of polynomials.
1:
2331:
1880:If the range of the morphism
1152:Birkhoff-Grothendieck theorem
558:Divisors and twisting sheaves
34:Homogeneous polynomial ideals
895:{\displaystyle \mathbb {Z} }
314:of principal open sets
7:
2156:General projective geometry
2128:
2113:Curves in projective spaces
645:unique factorization domain
440:{\displaystyle (k_{P})_{0}}
10:
2407:
2134:General algebraic geometry
1902:linear system of divisors
1639:linear system of divisors
553:Glossary of scheme theory
2177:
22:plays a central role in
1067:. (Hence has dimension
2319:
2172:Homogeneous polynomial
2089:
2030:
1973:
1894:
1871:
1748:
1705:
1661:very ample line bundle
1617:
1576:
1524:
1455:
1393:
1313:
1265:
1208:
1158:Important line bundles
1141:
1038:
973:
940:
896:
874:
836:
768:
711:
634:
597:Serre twisting sheaves
589:
538:
441:
389:
301:
188:
2320:
2090:
2031:
1974:
1895:
1872:
1754:and a linear system
1749:
1706:
1618:
1577:
1525:
1456:
1394:
1314:
1266:
1209:
1142:
1039:
974:
941:
897:
875:
837:
769:
712:
635:
590:
539:
442:
390:
302:
189:
98:states that, for any
2391:Geometry of divisors
2194:
2140:Scheme (mathematics)
2040:
1996:
1910:
1884:
1761:
1715:
1674:
1586:
1534:
1498:
1410:
1345:
1275:
1224:
1182:
1071:
998:
950:
913:
884:
850:
793:
740:
663:
611:
566:
517:
511:algebraically closed
402:
333:
271:
139:
87:. It is a naturally
44:algebraically closed
2386:Algebraic varieties
2381:Projective geometry
2265:
2167:Projective geometry
2109:(smooth divisors).
1632:globally generating
1630:to the choice of a
1519:
1478:globally generated
1253:
1168:exceptional divisor
1164:tautological bundle
946:of the line bundle
869:
759:
698:
296:
2376:Algebraic geometry
2344:Algebraic Geometry
2315:
2239:
2145:Projective variety
2085:
2026:
1969:
1890:
1867:
1744:
1701:
1613:
1572:
1520:
1501:
1451:
1389:
1309:
1261:
1237:
1204:
1137:
1034:
969:
936:
892:
870:
853:
832:
764:
743:
727:invertible sheaves
707:
680:
630:
585:
534:
437:
385:
297:
274:
219:finite-dimensional
184:
55:finite-dimensional
28:algebraic geometry
24:algebraic geometry
2266:
2150:Proj construction
2071:
1893:{\displaystyle j}
1129:
1096:
904:first Chern class
880:is isomorphic to
817:
679:
505:. When the field
312:base of open sets
179:
100:homogeneous ideal
70:dual vector space
66:symmetric algebra
16:The concept of a
2398:
2362:
2339:Robin Hartshorne
2325:
2324:
2322:
2321:
2316:
2311:
2310:
2292:
2291:
2267:
2264:
2247:
2238:
2234:
2233:
2215:
2214:
2198:
2188:
2162:Projective space
2094:
2092:
2091:
2086:
2078:
2077:
2076:
2067:
2055:
2035:
2033:
2032:
2027:
2025:
2024:
2019:
2010:
2009:
2004:
1978:
1976:
1975:
1970:
1953:
1952:
1943:
1942:
1937:
1922:
1921:
1899:
1897:
1896:
1891:
1876:
1874:
1873:
1868:
1851:
1850:
1844:
1843:
1804:
1803:
1794:
1793:
1788:
1773:
1772:
1753:
1751:
1750:
1745:
1734:
1733:
1727:
1726:
1710:
1708:
1707:
1702:
1700:
1699:
1694:
1622:
1620:
1619:
1614:
1612:
1611:
1606:
1581:
1579:
1578:
1573:
1568:
1560:
1559:
1548:
1529:
1527:
1526:
1521:
1518:
1513:
1512:
1506:
1471:affine varieties
1460:
1458:
1457:
1452:
1423:
1398:
1396:
1395:
1390:
1373:
1372:
1367:
1358:
1318:
1316:
1315:
1310:
1284:
1283:
1270:
1268:
1267:
1262:
1252:
1247:
1242:
1233:
1232:
1216:canonical bundle
1213:
1211:
1210:
1205:
1191:
1190:
1146:
1144:
1143:
1138:
1136:
1135:
1134:
1125:
1113:
1103:
1102:
1101:
1092:
1080:
1043:
1041:
1040:
1035:
1021:
1020:
1011:
978:
976:
975:
970:
959:
958:
945:
943:
942:
937:
926:
901:
899:
898:
893:
891:
879:
877:
876:
871:
868:
863:
858:
841:
839:
838:
833:
828:
815:
802:
801:
773:
771:
770:
765:
758:
753:
748:
735:projective space
716:
714:
713:
708:
706:
697:
692:
691:
685:
677:
676:
639:
637:
636:
631:
620:
619:
605:Cartier divisors
594:
592:
591:
586:
575:
574:
546:abstract variety
543:
541:
540:
535:
524:
446:
444:
443:
438:
436:
435:
426:
425:
394:
392:
391:
386:
381:
380:
370:
364:
363:
306:
304:
303:
298:
295:
284:
279:
249:projectivization
193:
191:
190:
185:
180:
175:
158:
157:
148:
147:
107:irrelevant ideal
19:Projective space
2406:
2405:
2401:
2400:
2399:
2397:
2396:
2395:
2366:
2365:
2359:
2349:Springer-Verlag
2334:
2329:
2328:
2306:
2302:
2287:
2283:
2248:
2243:
2229:
2225:
2210:
2206:
2199:
2197:
2195:
2192:
2191:
2189:
2185:
2180:
2158:
2136:
2131:
2119:ruled varieties
2115:
2072:
2057:
2051:
2050:
2049:
2041:
2038:
2037:
2020:
2015:
2014:
2005:
2000:
1999:
1997:
1994:
1993:
1990:
1948:
1947:
1938:
1933:
1932:
1917:
1913:
1911:
1908:
1907:
1885:
1882:
1881:
1846:
1845:
1839:
1835:
1799:
1798:
1789:
1784:
1783:
1768:
1764:
1762:
1759:
1758:
1729:
1728:
1722:
1718:
1716:
1713:
1712:
1695:
1690:
1689:
1675:
1672:
1671:
1607:
1602:
1601:
1587:
1584:
1583:
1564:
1549:
1538:
1537:
1535:
1532:
1531:
1514:
1508:
1507:
1502:
1499:
1496:
1495:
1467:
1413:
1411:
1408:
1407:
1368:
1363:
1362:
1348:
1346:
1343:
1342:
1279:
1278:
1276:
1273:
1272:
1248:
1243:
1238:
1228:
1227:
1225:
1222:
1221:
1186:
1185:
1183:
1180:
1179:
1160:
1130:
1115:
1109:
1108:
1107:
1097:
1082:
1076:
1075:
1074:
1072:
1069:
1068:
1016:
1015:
1007:
999:
996:
995:
954:
953:
951:
948:
947:
922:
914:
911:
910:
887:
885:
882:
881:
864:
859:
854:
851:
848:
847:
824:
797:
796:
794:
791:
790:
754:
749:
744:
741:
738:
737:
723:
702:
693:
687:
686:
681:
666:
664:
661:
660:
615:
614:
612:
609:
608:
570:
569:
567:
564:
563:
560:
520:
518:
515:
514:
431:
427:
421:
417:
403:
400:
399:
398:to be the ring
366:
365:
359:
358:
357:
334:
331:
330:
285:
280:
275:
272:
269:
268:
211:
174:
153:
152:
143:
142:
140:
137:
136:
96:Nullstellensatz
94:The projective
83:and denoted by
77:polynomial ring
36:
12:
11:
5:
2404:
2394:
2393:
2388:
2383:
2378:
2364:
2363:
2357:
2333:
2330:
2327:
2326:
2314:
2309:
2305:
2301:
2298:
2295:
2290:
2286:
2282:
2279:
2276:
2273:
2270:
2263:
2260:
2257:
2254:
2251:
2246:
2242:
2237:
2232:
2228:
2224:
2221:
2218:
2213:
2209:
2205:
2202:
2182:
2181:
2179:
2176:
2175:
2174:
2169:
2164:
2157:
2154:
2153:
2152:
2147:
2142:
2135:
2132:
2130:
2127:
2123:Bézout theorem
2114:
2111:
2084:
2081:
2075:
2070:
2066:
2063:
2060:
2054:
2048:
2045:
2023:
2018:
2013:
2008:
2003:
1989:
1986:
1985:
1984:
1968:
1965:
1962:
1959:
1956:
1951:
1946:
1941:
1936:
1931:
1928:
1925:
1920:
1916:
1889:
1878:
1877:
1866:
1863:
1860:
1857:
1854:
1849:
1842:
1838:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1802:
1797:
1792:
1787:
1782:
1779:
1776:
1771:
1767:
1743:
1740:
1737:
1732:
1725:
1721:
1698:
1693:
1688:
1685:
1682:
1679:
1610:
1605:
1600:
1597:
1594:
1591:
1571:
1567:
1563:
1558:
1555:
1552:
1547:
1544:
1541:
1517:
1511:
1505:
1466:
1463:
1462:
1461:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1422:
1419:
1416:
1388:
1385:
1382:
1379:
1376:
1371:
1366:
1361:
1357:
1354:
1351:
1341:) is given by
1339:Fano varieties
1332:Fano varieties
1325:Euler sequence
1321:
1320:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1282:
1259:
1256:
1251:
1246:
1241:
1236:
1231:
1203:
1200:
1197:
1194:
1189:
1159:
1156:
1133:
1128:
1124:
1121:
1118:
1112:
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987:associated to
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623:
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584:
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544:is in fact an
533:
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501:the scheme is
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89:graded algebra
75:is called the
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2358:0-387-90244-9
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2107:hypersurfaces
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1178:is the sheaf
1177:
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786:the twisting
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493:
489:
485:
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478:). Since the
477:
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472:affine scheme
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121:of the ideal
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2103:MathOverflow
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1047:vanishes if
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731:line bundles
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601:line bundles
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449:localization
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256:
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244:
236:
228:
222:vector space
214:
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58:vector space
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1480:line bundle
649:prime ideal
503:irreducible
464:linear form
257:projective
239:defined by
2370:Categories
2332:References
1628:equivalent
1489:very ample
1483:defines a
1172:blowing up
733:, on the
488:open cover
486:) form an
115:Nullstelle
2297:…
2269:↦
2253:
2220:…
2080:−
2012:→
1927:Γ
1919:∗
1841:∗
1824:Γ
1821:⊂
1778:Γ
1770:∗
1724:∗
1687:→
1599:→
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1289:−
1196:−
1002:Γ
920:⊆
822:∈
657:principal
337:Γ
322:), where
2341:(1977).
2129:See also
2097:See the
1485:morphism
1065:m > 0
1170:of the
842:so the
788:sheaves
784:exactly
224:over a
119:radical
68:of the
2355:
2099:answer
1657:modulo
1624:modulo
1214:. The
816:
782:, are
678:
653:height
647:, any
595:, the
261:-space
255:. The
233:scheme
231:. The
64:. The
49:, and
42:be an
2178:Notes
1641:on a
1174:of a
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655:1 is
474:spec(
235:over
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217:be a
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53:be a
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1063:for
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550:cf.
241:Proj
213:Let
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