1032:
164:
should contain a dense subset parameterizing those curves with the minimum in the way of special divisors. One goal of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to
1224:
374:
538:
1177:
1083:
429:
930:
898:
851:
798:
758:
712:
655:
593:
65:
that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected"
1502:
1459:
1925:
1219:
Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Philip A.; Harris, Joe (1985). "The Basic
Results of the Brill-Noether Theory".
2017:
1654:
1614:
1495:
1368:
1251:
1414:
284:
1206:
2094:
1705:
1604:
437:
2084:
1342:
1240:
1299:
Griffiths, Phillip; Harris, Joseph (1980). "On the variety of special linear systems on a general algebraic curve".
1783:
1488:
1090:
17:
1930:
1851:
1841:
1778:
666:
1528:
55:
1748:
1644:
2124:
2007:
1971:
1670:
1583:
1981:
1619:
1356:
77:
1059:
2119:
2027:
1301:
66:
1940:
1856:
1773:
1675:
1634:
1027:{\displaystyle H^{0}({\mathcal {O}}_{\mathbb {P} ^{r}}(n))\rightarrow H^{0}({\mathcal {O}}_{C}(n))}
243:
130:
113:
386:
1831:
1639:
1624:
871:
824:
771:
731:
685:
628:
566:
1738:
2002:
1700:
1649:
1538:
1256:
2089:
1950:
1609:
1330:
1322:
1861:
1277:
8:
1915:
1793:
1758:
1715:
1695:
39:
2056:
1836:
1816:
1629:
1439:
1393:
1281:
166:
31:
1788:
1945:
1892:
1763:
1578:
1573:
1364:
1352:
1338:
1285:
1236:
670:
1438:
Larson, Eric; Vogt, Isabel (2022-05-05). "Interpolation for Brill--Noether curves".
1935:
1821:
1798:
1310:
1273:
1265:
1228:
101:
88:
1314:
2061:
1866:
1808:
1710:
1533:
1512:
1318:
59:
1232:
2035:
1733:
1558:
1543:
1520:
812:
185:
73:
2113:
2076:
1846:
1826:
1753:
1548:
1480:
208:
126:
2012:
1986:
1976:
1966:
1768:
1588:
613:
153:
1887:
1725:
855:
is smooth. By the connectedness result this implies it is irreducible if
105:
43:
1252:"Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie"
1882:
1269:
97:
1743:
267:
122:
cohomology or space of holomorphic sections is larger than expected.
1444:
1398:
383:. The formula can be memorized via the mnemonic (using our desired
1460:"Old Problem About Algebraic Curves Falls to Young Mathematicians"
2066:
2051:
1218:
2046:
1034:
are of maximal rank, also known as the maximal rank conjecture.
1351:
866:
Other more recent results not necessarily in terms of space
1363:. Wiley Classics Library. Wiley Interscience. p. 245.
72:
Throughout, we consider a projective smooth curve over the
1392:
Larson, Eric (2018-09-18). "The
Maximal Rank Conjecture".
659:
is not empty, and every component has dimension at least
143:
184:
The basic statement can be formulated in terms of the
1093:
1062:
933:
874:
827:
774:
762:
is reduced and all components have dimension exactly
734:
688:
631:
569:
440:
389:
287:
1329:
369:{\displaystyle \dim(d,r,g)\geq \rho =g-(r+1)(g-d+r)}
1171:
1077:
1039:Eric Larson and Isabel Vogt (2022) proved that if
1026:
892:
845:
792:
752:
706:
649:
587:
532:
423:
368:
1415:"Tinkertoy Models Produce New Geometric Insights"
533:{\displaystyle g-(r+1)(g-d+r)=g-h^{0}(D)h^{1}(D)}
2111:
1249:
47:
1298:
1214:(Master's thesis). Radboud University Nijmegen.
719:
1510:
1496:
1225:Grundlehren der Mathematischen Wissenschaften
1172:{\displaystyle (r-1)n\leq (r+1)d-(r-3)(g-1),}
1335:Algebraic Curves, the Brill and Noether way
1250:von Brill, Alexander; Noether, Max (1874).
1503:
1489:
1437:
1443:
1397:
1065:
958:
1412:
1193:) ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.
14:
2112:
1926:Clifford's theorem on special divisors
1391:
1204:
83:The condition to be a special divisor
1484:
181:be present on a curve of that genus.
144:Main theorems of Brill–Noether theory
1227:267. Vol. I. pp. 203–224.
129:, the condition is that there exist
91:terms, as the non-vanishing of the
24:
2095:Vector bundles on algebraic curves
2018:Weber's theorem (Algebraic curves)
1615:Hasse's theorem on elliptic curves
1605:Counting points on elliptic curves
1001:
950:
906:Eric Larson (2017) proved that if
561:the basic results about the space
25:
2136:
1361:Principles of Algebraic Geometry
1078:{\displaystyle \mathbb {P} ^{r}}
100:of the sheaf of sections of the
1706:Hurwitz's automorphisms theorem
1179:except in 4 exceptional cases:
1931:Gonality of an algebraic curve
1842:Differential of the first kind
1452:
1431:
1413:Hartnett, Kevin (2018-09-05).
1406:
1385:
1208:Algebraic Brill–Noether Theory
1163:
1151:
1148:
1136:
1127:
1115:
1106:
1094:
1021:
1018:
1012:
995:
982:
979:
976:
970:
944:
527:
521:
508:
502:
480:
462:
459:
447:
406:
400:
363:
345:
342:
330:
312:
294:
13:
1:
2085:Birkhoff–Grothendieck theorem
1784:Nagata's conjecture on curves
1655:Schoof–Elkies–Atkin algorithm
1529:Five points determine a conic
1315:10.1215/s0012-7094-80-04717-1
1198:
720:Griffiths & Harris (1980)
1645:Supersingular elliptic curve
1221:Geometry of Algebraic Curves
424:{\displaystyle h^{0}(D)=r+1}
7:
1852:Riemann's existence theorem
1779:Hilbert's sixteenth problem
1671:Elliptic curve cryptography
1584:Fundamental pair of periods
1233:10.1007/978-1-4757-5323-3_5
27:Field of algebraic geometry
10:
2141:
1982:Moduli of algebraic curves
1337:. Universitext. Springer.
112:. This means that, by the
78:algebraically closed field
2075:
2026:
1995:
1959:
1908:
1901:
1875:
1807:
1724:
1688:
1663:
1597:
1566:
1557:
1519:
1302:Duke Mathematical Journal
893:{\displaystyle G_{d}^{r}}
846:{\displaystyle G_{d}^{r}}
793:{\displaystyle G_{d}^{r}}
753:{\displaystyle G_{d}^{r}}
707:{\displaystyle G_{d}^{r}}
650:{\displaystyle G_{d}^{r}}
588:{\displaystyle G_{d}^{r}}
246:. There is a lower bound
131:holomorphic differentials
67:linear system of divisors
1749:Cayley–Bacharach theorem
1676:Elliptic curve primality
1378:
2008:Riemann–Hurwitz formula
1972:Gromov–Witten invariant
1832:Compact Riemann surface
1620:Mazur's torsion theorem
1205:Barbon, Andrea (2014).
927:, the restriction maps
902:of linear systems are:
242:in the notation of the
40:Alexander von Brill
1625:Modular elliptic curve
1173:
1079:
1050:interpolating through
1046:then there is a curve
1028:
894:
847:
794:
754:
708:
651:
589:
534:
425:
370:
1539:Rational normal curve
1257:Mathematische Annalen
1174:
1080:
1029:
895:
848:
795:
755:
709:
652:
597:of linear systems on
590:
535:
426:
371:
87:can be formulated in
2090:Stable vector bundle
1951:Weil reciprocity law
1941:Riemann–Roch theorem
1921:Brill–Noether theory
1857:Riemann–Roch theorem
1774:Genus–degree formula
1635:Mordell–Weil theorem
1610:Division polynomials
1331:Eduardo Casas-Alvero
1091:
1060:
931:
872:
825:
772:
732:
686:
629:
567:
438:
387:
381:Brill–Noether number
285:
244:Riemann–Roch theorem
215:, with given values
199:, and the subset of
169:) of a given degree
114:Riemann–Roch theorem
76:(or over some other
36:Brill–Noether theory
1902:Structure of curves
1794:Quartic plane curve
1716:Hyperelliptic curve
1696:De Franchis theorem
1640:Nagell–Lutz theorem
1353:Philip A. Griffiths
889:
842:
789:
749:
703:
646:
584:
431:and Riemann-Roch)
173:, as a function of
50:), is the study of
2125:Algebraic surfaces
1909:Divisors on curves
1701:Faltings's theorem
1650:Schoof's algorithm
1630:Modularity theorem
1270:10.1007/BF02104804
1169:
1075:
1054:general points in
1024:
890:
875:
843:
828:
790:
775:
766:(so in particular
750:
735:
704:
689:
647:
632:
585:
570:
543:For smooth curves
530:
421:
366:
250:for the dimension
195:of a smooth curve
167:linear equivalence
148:For a given genus
125:Alternatively, by
32:algebraic geometry
2107:
2106:
2103:
2102:
2003:Hasse–Witt matrix
1946:Weierstrass point
1893:Smooth completion
1862:Teichmüller space
1764:Cubic plane curve
1684:
1683:
1598:Arithmetic theory
1579:Elliptic integral
1574:Elliptic function
1370:978-0-471-05059-9
671:Robert Lazarsfeld
207:corresponding to
16:(Redirected from
2132:
2120:Algebraic curves
1936:Jacobian variety
1906:
1905:
1809:Riemann surfaces
1799:Real plane curve
1759:Cramer's paradox
1739:Bézout's theorem
1564:
1563:
1513:algebraic curves
1505:
1498:
1491:
1482:
1481:
1475:
1474:
1472:
1471:
1456:
1450:
1449:
1447:
1435:
1429:
1428:
1426:
1425:
1410:
1404:
1403:
1401:
1389:
1374:
1348:
1326:
1295:
1293:
1292:
1246:
1215:
1213:
1194:
1178:
1176:
1175:
1170:
1086:
1084:
1082:
1081:
1076:
1074:
1073:
1068:
1053:
1049:
1045:
1033:
1031:
1030:
1025:
1011:
1010:
1005:
1004:
994:
993:
969:
968:
967:
966:
961:
954:
953:
943:
942:
926:
919:
912:
901:
899:
897:
896:
891:
888:
883:
861:
854:
852:
850:
849:
844:
841:
836:
819:is generic then
818:
808:
801:
799:
797:
796:
791:
788:
783:
765:
761:
759:
757:
756:
751:
748:
743:
726:is generic then
725:
722:showed that if
715:
713:
711:
710:
705:
702:
697:
679:
662:
658:
656:
654:
653:
648:
645:
640:
622:
609:are as follows.
608:
604:
600:
596:
594:
592:
591:
586:
583:
578:
560:
553:
546:
539:
537:
536:
531:
520:
519:
501:
500:
430:
428:
427:
422:
399:
398:
375:
373:
372:
367:
277:
265:
249:
241:
230:
226:
218:
214:
206:
198:
194:
176:
172:
163:
159:
151:
139:
121:
111:
102:invertible sheaf
96:
89:sheaf cohomology
86:
64:
52:special divisors
38:, introduced by
21:
2140:
2139:
2135:
2134:
2133:
2131:
2130:
2129:
2110:
2109:
2108:
2099:
2071:
2062:Delta invariant
2040:
2022:
1991:
1955:
1916:Abel–Jacobi map
1897:
1871:
1867:Torelli theorem
1837:Dessin d'enfant
1817:Belyi's theorem
1803:
1789:Plücker formula
1720:
1711:Hurwitz surface
1680:
1659:
1593:
1567:Analytic theory
1559:Elliptic curves
1553:
1534:Projective line
1521:Rational curves
1515:
1509:
1479:
1478:
1469:
1467:
1464:Quanta Magazine
1458:
1457:
1453:
1436:
1432:
1423:
1421:
1419:Quanta Magazine
1411:
1407:
1390:
1386:
1381:
1371:
1345:
1290:
1288:
1243:
1211:
1201:
1180:
1092:
1089:
1088:
1087:if and only if
1069:
1064:
1063:
1061:
1058:
1057:
1055:
1051:
1047:
1040:
1006:
1000:
999:
998:
989:
985:
962:
957:
956:
955:
949:
948:
947:
938:
934:
932:
929:
928:
921:
914:
907:
884:
879:
873:
870:
869:
867:
856:
837:
832:
826:
823:
822:
820:
816:
815:proved that if
803:
784:
779:
773:
770:
769:
767:
763:
744:
739:
733:
730:
729:
727:
723:
698:
693:
687:
684:
683:
681:
674:
673:proved that if
660:
641:
636:
630:
627:
626:
624:
617:
616:proved that if
606:
602:
598:
579:
574:
568:
565:
564:
562:
555:
548:
544:
515:
511:
496:
492:
439:
436:
435:
394:
390:
388:
385:
384:
286:
283:
282:
271:
251:
247:
232:
228:
220:
216:
212:
209:divisor classes
200:
196:
188:
174:
170:
161:
157:
149:
146:
134:
117:
109:
92:
84:
74:complex numbers
62:
28:
23:
22:
18:Special divisor
15:
12:
11:
5:
2138:
2128:
2127:
2122:
2105:
2104:
2101:
2100:
2098:
2097:
2092:
2087:
2081:
2079:
2077:Vector bundles
2073:
2072:
2070:
2069:
2064:
2059:
2054:
2049:
2044:
2038:
2032:
2030:
2024:
2023:
2021:
2020:
2015:
2010:
2005:
1999:
1997:
1993:
1992:
1990:
1989:
1984:
1979:
1974:
1969:
1963:
1961:
1957:
1956:
1954:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1912:
1910:
1903:
1899:
1898:
1896:
1895:
1890:
1885:
1879:
1877:
1873:
1872:
1870:
1869:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1813:
1811:
1805:
1804:
1802:
1801:
1796:
1791:
1786:
1781:
1776:
1771:
1766:
1761:
1756:
1751:
1746:
1741:
1736:
1730:
1728:
1722:
1721:
1719:
1718:
1713:
1708:
1703:
1698:
1692:
1690:
1686:
1685:
1682:
1681:
1679:
1678:
1673:
1667:
1665:
1661:
1660:
1658:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1601:
1599:
1595:
1594:
1592:
1591:
1586:
1581:
1576:
1570:
1568:
1561:
1555:
1554:
1552:
1551:
1546:
1544:Riemann sphere
1541:
1536:
1531:
1525:
1523:
1517:
1516:
1508:
1507:
1500:
1493:
1485:
1477:
1476:
1451:
1430:
1405:
1383:
1382:
1380:
1377:
1376:
1375:
1369:
1349:
1343:
1327:
1309:(1): 233–272.
1296:
1264:(2): 269–316.
1247:
1241:
1216:
1200:
1197:
1196:
1195:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1072:
1067:
1036:
1035:
1023:
1020:
1017:
1014:
1009:
1003:
997:
992:
988:
984:
981:
978:
975:
972:
965:
960:
952:
946:
941:
937:
887:
882:
878:
864:
863:
840:
835:
831:
813:David Gieseker
810:
787:
782:
778:
747:
742:
738:
717:
701:
696:
692:
667:William Fulton
664:
644:
639:
635:
605:and dimension
582:
577:
573:
541:
540:
529:
526:
523:
518:
514:
510:
507:
504:
499:
495:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
420:
417:
414:
411:
408:
405:
402:
397:
393:
377:
376:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
186:Picard variety
145:
142:
140:on the curve.
108:associated to
26:
9:
6:
4:
3:
2:
2137:
2126:
2123:
2121:
2118:
2117:
2115:
2096:
2093:
2091:
2088:
2086:
2083:
2082:
2080:
2078:
2074:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2041:
2034:
2033:
2031:
2029:
2028:Singularities
2025:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
2000:
1998:
1994:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1964:
1962:
1958:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1913:
1911:
1907:
1904:
1900:
1894:
1891:
1889:
1886:
1884:
1881:
1880:
1878:
1876:Constructions
1874:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1847:Klein quartic
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1827:Bolza surface
1825:
1823:
1822:Bring's curve
1820:
1818:
1815:
1814:
1812:
1810:
1806:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1754:Conic section
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1734:AF+BG theorem
1732:
1731:
1729:
1727:
1723:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1693:
1691:
1687:
1677:
1674:
1672:
1669:
1668:
1666:
1662:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
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1987:Stable curve
1977:Hodge bundle
1967:ELSV formula
1920:
1769:Fermat curve
1726:Plane curves
1689:Higher genus
1664:Applications
1589:Modular form
1468:. Retrieved
1466:. 2022-08-25
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1883:Dual curve
1511:Topics in
1470:2022-08-28
1445:2201.09445
1424:2022-08-28
1399:1711.04906
1357:Joe Harris
1291:2009-08-22
1278:06.0251.01
1199:References
601:of degree
98:cohomology
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1996:Morphisms
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