1050:
606:
712:
295:
1559:
of
Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others.
1045:{\displaystyle \sum _{d_{1}+\cdots +d_{n}=3g-3+n}\langle \tau _{d_{1}},\ldots ,\tau _{d_{n}}\rangle \prod _{1\leq i\leq n}{\frac {(2d_{i}-1)!!}{\lambda _{i}^{2d_{i}+1}}}=\sum _{\Gamma \in G_{g,n}}{\frac {2^{-|X_{0}|}}{|{\text{Aut}}\Gamma |}}\prod _{e\in X_{1}}{\frac {2}{\lambda (e)}}}
601:{\displaystyle F(t_{0},t_{1},\ldots )=\sum \langle \tau _{0}^{k_{0}}\tau _{1}^{k_{1}}\cdots \rangle \prod _{i\geq 0}{\frac {t_{i}^{k_{i}}}{k_{i}!}}={\frac {t_{0}^{3}}{6}}+{\frac {t_{1}}{24}}+{\frac {t_{0}t_{2}}{24}}+{\frac {t_{1}^{2}}{24}}+{\frac {t_{0}^{2}t_{3}}{48}}+\cdots }
1320:
1535:
1102:. The function λ is thought of as a function from the marked points to the reals, and extended to edges of the ribbon graph by setting λ of an edge equal to the sum of λ at the two marked points corresponding to each side of the edge.
1409:
1790:, Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14â21, 1991., Houston, TX: Publish or Perish, pp. 235â269,
1187:
90:. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy.
692:
1215:
1427:
1833:
1617:
2449:
1720:
2256:
83:
74:
should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the
1985:
2348:
1945:
1826:
1795:
1766:
1737:
1598:
1331:
1786:(1993), "Algebraic geometry associated with matrix models of two-dimensional gravity", in Goldberg, Lisa R.; Phillips, Anthony V. (eds.),
2414:
2036:
1935:
2404:
2114:
1819:
1122:
2261:
2172:
2182:
2109:
1859:
2079:
1975:
2338:
2302:
2444:
2001:
1914:
182:, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point
629:
2312:
1950:
2358:
2271:
2251:
2187:
2104:
1965:
2006:
1970:
2162:
626:, in other words it satisfies a certain series of partial differential equations corresponding to the basis
1315:{\displaystyle t_{i}={\frac {-{\text{tr }}\Lambda ^{-1-2i}}{1\times 3\times 5\times \cdots \times (2i-1)}}}
2439:
1955:
40:
2333:
2069:
1547:
From this he deduced that exp F is a Ï-function for the KdV hierarchy, thus proving Witten's conjecture.
706:
Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that
1869:
2031:
1980:
2409:
2281:
2192:
1940:
1684:
1636:
1530:{\displaystyle \int X_{ij}X_{kl}d\mu =\delta _{il}\delta _{jk}{\frac {2}{\Lambda _{i}+\Lambda _{j}}}}
2246:
2124:
2089:
2046:
2026:
1564:
36:
2376:
1960:
1626:
1556:
28:
2167:
2147:
2119:
1615:
Kazarian, M. E.; Lando, Sergei K. (2007), "An algebro-geometric proof of Witten's conjecture",
2276:
2223:
2094:
1909:
1904:
1791:
1762:
1733:
1700:
1672:
1652:
1594:
2266:
2152:
2129:
1692:
1644:
1586:
695:
60:
1805:
1776:
1747:
1712:
1664:
1608:
2381:
2197:
2139:
2041:
1864:
1843:
1801:
1772:
1743:
1729:
1708:
1660:
1604:
1582:
112:
79:
71:
2064:
1688:
1640:
17:
1889:
1874:
1851:
1325:
and the probability measure Ό on the positive definite hermitian matrices is given by
70:
Witten's motivation for the conjecture was that two different models of 2-dimensional
2433:
2396:
2177:
2157:
2084:
1879:
1783:
1754:
623:
87:
48:
44:
1540:
which implies that its expansion in terms of
Feynman diagrams is the expression for
2343:
2317:
2307:
2297:
2099:
1919:
1581:, Grundlehren der Mathematischen Wissenschaften , vol. 268, Berlin, New York:
75:
1648:
2218:
2056:
1673:"Intersection theory on the moduli space of curves and the matrix Airy function"
286:
2213:
1590:
1811:
1704:
1656:
2074:
1757:(1991), "Two-dimensional gravity and intersection theory on moduli space",
1728:, Encyclopaedia of Mathematical Sciences, vol. 141, Berlin, New York:
1555:
The Witten conjecture is a special case of a more general relation between
2386:
2371:
1696:
2366:
1631:
1577:
Cornalba, Maurizio; Arbarello, Enrico; Griffiths, Phillip A. (2011),
1404:{\displaystyle d\mu =c_{\Lambda }\exp(-{\text{tr}}X^{2}\Lambda /2)dX}
1788:
Topological methods in modern mathematics (Stony Brook, NY, 1991)
1815:
1761:, vol. 1, Bethlehem, PA: Lehigh Univ., pp. 243â310,
1421:
is a normalizing constant. This measure has the property that
1192:
as Πlends to infinity, where Πand Χ are positive definite
611:
encodes all the intersection indices as its coefficients.
1182:{\displaystyle \log \int \exp(i{\text{tr}}X^{3}/6)d\mu }
614:
Witten's conjecture states that the partition function
86:
for the other is the logarithm of the Ï-function of the
1759:
Surveys in differential geometry (Cambridge, MA, 1990)
1430:
1334:
1218:
1125:
715:
632:
298:
2395:
2357:
2326:
2290:
2239:
2232:
2206:
2138:
2055:
2019:
1994:
1928:
1897:
1888:
1850:
154:is its DeligneâMumford compactification. There are
1529:
1403:
1314:
1181:
1105:By Feynman diagram techniques, this implies that
1044:
686:
600:
1719:Lando, Sergei K.; Zvonkin, Alexander K. (2004),
289:of a line bundle. Witten's generating function
1567:is a generalization of the Witten conjecture.
687:{\displaystyle \{L_{-1},L_{0},L_{1},\ldots \}}
59:. Witten's original conjecture was proved by
1827:
8:
1618:Journal of the American Mathematical Society
819:
773:
681:
633:
393:
343:
2236:
1894:
1834:
1820:
1812:
1055:Here the sum on the right is over the set
64:
1722:Graphs on surfaces and their applications
1630:
1518:
1505:
1495:
1486:
1473:
1451:
1438:
1429:
1384:
1375:
1366:
1348:
1333:
1247:
1238:
1232:
1223:
1217:
1162:
1156:
1147:
1124:
1021:
1013:
1002:
990:
982:
977:
969:
963:
954:
950:
944:
930:
919:
896:
888:
883:
857:
844:
826:
811:
806:
785:
780:
744:
725:
720:
714:
669:
656:
640:
631:
580:
570:
565:
558:
544:
539:
533:
518:
508:
501:
487:
481:
467:
462:
456:
441:
428:
423:
418:
412:
400:
382:
377:
372:
360:
355:
350:
322:
309:
297:
1579:Geometry of algebraic curves. Volume II
2257:Clifford's theorem on special divisors
1677:Communications in Mathematical Physics
56:
52:
1116:,...) is an asymptotic expansion of
1072:of compact Riemann surfaces of genus
7:
2415:Vector bundles on algebraic curves
2349:Weber's theorem (Algebraic curves)
1946:Hasse's theorem on elliptic curves
1936:Counting points on elliptic curves
1515:
1502:
1381:
1349:
1244:
987:
920:
25:
2450:Conjectures that have been proved
1080:marked points. The set of edges
211:ă is the intersection index of Î
2037:Hurwitz's automorphisms theorem
111:is the moduli stack of compact
2262:Gonality of an algebraic curve
2173:Differential of the first kind
1392:
1360:
1306:
1291:
1170:
1141:
1036:
1030:
991:
978:
970:
955:
869:
847:
334:
302:
1:
2405:BirkhoffâGrothendieck theorem
2115:Nagata's conjecture on curves
1986:SchoofâElkiesâAtkin algorithm
1860:Five points determine a conic
1649:10.1090/S0894-0347-07-00566-8
1976:Supersingular elliptic curve
1544:in terms of ribbon graphs.
2183:Riemann's existence theorem
2110:Hilbert's sixteenth problem
2002:Elliptic curve cryptography
1915:Fundamental pair of periods
191:. The intersection index ăÏ
2466:
2313:Moduli of algebraic curves
1671:Kontsevich, Maxim (1992),
1591:10.1007/978-3-540-69392-5
39:of stable classes on the
2080:CayleyâBacharach theorem
2007:Elliptic curve primality
1200:hermitian matrices, and
622:is a Ï-function for the
18:Witten's conjecture
2339:RiemannâHurwitz formula
2303:GromovâWitten invariant
2163:Compact Riemann surface
1951:Mazur's torsion theorem
123:distinct marked points
1956:Modular elliptic curve
1531:
1405:
1316:
1183:
1046:
688:
602:
55:), and generalized in
41:moduli space of curves
35:is a conjecture about
1870:Rational normal curve
1532:
1406:
1317:
1184:
1047:
689:
603:
2410:Stable vector bundle
2282:Weil reciprocity law
2272:RiemannâRoch theorem
2252:BrillâNoether theory
2188:RiemannâRoch theorem
2105:Genusâdegree formula
1966:MordellâWeil theorem
1941:Division polynomials
1428:
1332:
1216:
1123:
713:
630:
296:
37:intersection numbers
2233:Structure of curves
2125:Quartic plane curve
2047:Hyperelliptic curve
2027:De Franchis theorem
1971:NagellâLutz theorem
1689:1992CMaPh.147....1K
1641:2007JAMS...20.1079K
1565:Virasoro conjecture
909:
575:
549:
472:
435:
389:
367:
274:, and 0 if no such
2445:Algebraic geometry
2240:Divisors on curves
2032:Faltings's theorem
1981:Schoof's algorithm
1961:Modularity theorem
1697:10.1007/BF02099526
1557:integrable systems
1527:
1401:
1312:
1179:
1042:
1020:
943:
879:
843:
772:
684:
598:
561:
535:
458:
414:
411:
368:
346:
84:partition function
29:algebraic geometry
2427:
2426:
2423:
2422:
2334:HasseâWitt matrix
2277:Weierstrass point
2224:Smooth completion
2193:TeichmĂŒller space
2095:Cubic plane curve
2015:
2014:
1929:Arithmetic theory
1910:Elliptic integral
1905:Elliptic function
1797:978-0-914098-26-3
1768:978-0-8218-0168-0
1739:978-3-540-00203-1
1600:978-3-540-42688-2
1525:
1369:
1310:
1241:
1150:
1068:of ribbon graphs
1040:
998:
996:
985:
915:
910:
822:
716:
590:
553:
528:
496:
476:
451:
396:
65:Kontsevich (1992)
33:Witten conjecture
16:(Redirected from
2457:
2267:Jacobian variety
2237:
2140:Riemann surfaces
2130:Real plane curve
2090:Cramer's paradox
2070:BĂ©zout's theorem
1895:
1844:algebraic curves
1836:
1829:
1822:
1813:
1808:
1779:
1750:
1727:
1715:
1667:
1634:
1625:(4): 1079â1089,
1611:
1536:
1534:
1533:
1528:
1526:
1524:
1523:
1522:
1510:
1509:
1496:
1494:
1493:
1481:
1480:
1459:
1458:
1446:
1445:
1410:
1408:
1407:
1402:
1388:
1380:
1379:
1370:
1367:
1353:
1352:
1321:
1319:
1318:
1313:
1311:
1309:
1265:
1264:
1263:
1242:
1239:
1233:
1228:
1227:
1188:
1186:
1185:
1180:
1166:
1161:
1160:
1151:
1148:
1051:
1049:
1048:
1043:
1041:
1039:
1022:
1019:
1018:
1017:
997:
995:
994:
986:
983:
981:
975:
974:
973:
968:
967:
958:
945:
942:
941:
940:
911:
908:
901:
900:
887:
878:
862:
861:
845:
842:
818:
817:
816:
815:
792:
791:
790:
789:
771:
749:
748:
730:
729:
696:Virasoro algebra
693:
691:
690:
685:
674:
673:
661:
660:
648:
647:
607:
605:
604:
599:
591:
586:
585:
584:
574:
569:
559:
554:
548:
543:
534:
529:
524:
523:
522:
513:
512:
502:
497:
492:
491:
482:
477:
471:
466:
457:
452:
450:
446:
445:
434:
433:
432:
422:
413:
410:
388:
387:
386:
376:
366:
365:
364:
354:
327:
326:
314:
313:
256:
232:
172:
144:
113:Riemann surfaces
80:algebraic curves
61:Maxim Kontsevich
43:, introduced by
21:
2465:
2464:
2460:
2459:
2458:
2456:
2455:
2454:
2430:
2429:
2428:
2419:
2391:
2382:Delta invariant
2353:
2322:
2286:
2247:AbelâJacobi map
2228:
2202:
2198:Torelli theorem
2168:Dessin d'enfant
2148:Belyi's theorem
2134:
2120:PlĂŒcker formula
2051:
2042:Hurwitz surface
2011:
1990:
1924:
1898:Analytic theory
1890:Elliptic curves
1884:
1865:Projective line
1852:Rational curves
1846:
1840:
1798:
1782:
1769:
1753:
1740:
1730:Springer-Verlag
1725:
1718:
1670:
1614:
1601:
1583:Springer-Verlag
1576:
1573:
1553:
1551:Generalizations
1514:
1501:
1500:
1482:
1469:
1447:
1434:
1426:
1425:
1420:
1371:
1344:
1330:
1329:
1266:
1243:
1234:
1219:
1214:
1213:
1208:
1152:
1121:
1120:
1115:
1101:
1094:
1088:are denoted by
1067:
1026:
1009:
976:
959:
946:
926:
892:
853:
846:
807:
802:
781:
776:
740:
721:
711:
710:
704:
665:
652:
636:
628:
627:
576:
560:
514:
504:
503:
483:
437:
436:
424:
378:
356:
318:
305:
294:
293:
284:
265:
252:
250:
241:
228:
226:
217:
210:
209:
199:
198:
190:
181:
168:
166:
153:
140:
138:
129:
110:
96:
72:quantum gravity
23:
22:
15:
12:
11:
5:
2463:
2461:
2453:
2452:
2447:
2442:
2432:
2431:
2425:
2424:
2421:
2420:
2418:
2417:
2412:
2407:
2401:
2399:
2397:Vector bundles
2393:
2392:
2390:
2389:
2384:
2379:
2374:
2369:
2363:
2361:
2355:
2354:
2352:
2351:
2346:
2341:
2336:
2330:
2328:
2324:
2323:
2321:
2320:
2315:
2310:
2305:
2300:
2294:
2292:
2288:
2287:
2285:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2243:
2241:
2234:
2230:
2229:
2227:
2226:
2221:
2216:
2210:
2208:
2204:
2203:
2201:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2144:
2142:
2136:
2135:
2133:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2061:
2059:
2053:
2052:
2050:
2049:
2044:
2039:
2034:
2029:
2023:
2021:
2017:
2016:
2013:
2012:
2010:
2009:
2004:
1998:
1996:
1992:
1991:
1989:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1932:
1930:
1926:
1925:
1923:
1922:
1917:
1912:
1907:
1901:
1899:
1892:
1886:
1885:
1883:
1882:
1877:
1875:Riemann sphere
1872:
1867:
1862:
1856:
1854:
1848:
1847:
1841:
1839:
1838:
1831:
1824:
1816:
1810:
1809:
1796:
1784:Witten, Edward
1780:
1767:
1755:Witten, Edward
1751:
1738:
1716:
1668:
1612:
1599:
1572:
1569:
1552:
1549:
1538:
1537:
1521:
1517:
1513:
1508:
1504:
1499:
1492:
1489:
1485:
1479:
1476:
1472:
1468:
1465:
1462:
1457:
1454:
1450:
1444:
1441:
1437:
1433:
1418:
1412:
1411:
1400:
1397:
1394:
1391:
1387:
1383:
1378:
1374:
1365:
1362:
1359:
1356:
1351:
1347:
1343:
1340:
1337:
1323:
1322:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1262:
1259:
1256:
1253:
1250:
1246:
1237:
1231:
1226:
1222:
1204:
1190:
1189:
1178:
1175:
1172:
1169:
1165:
1159:
1155:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1113:
1099:
1092:
1084:and points of
1059:
1053:
1052:
1038:
1035:
1032:
1029:
1025:
1016:
1012:
1008:
1005:
1001:
993:
989:
980:
972:
966:
962:
957:
953:
949:
939:
936:
933:
929:
925:
922:
918:
914:
907:
904:
899:
895:
891:
886:
882:
877:
874:
871:
868:
865:
860:
856:
852:
849:
841:
838:
835:
832:
829:
825:
821:
814:
810:
805:
801:
798:
795:
788:
784:
779:
775:
770:
767:
764:
761:
758:
755:
752:
747:
743:
739:
736:
733:
728:
724:
719:
703:
700:
683:
680:
677:
672:
668:
664:
659:
655:
651:
646:
643:
639:
635:
609:
608:
597:
594:
589:
583:
579:
573:
568:
564:
557:
552:
547:
542:
538:
532:
527:
521:
517:
511:
507:
500:
495:
490:
486:
480:
475:
470:
465:
461:
455:
449:
444:
440:
431:
427:
421:
417:
409:
406:
403:
399:
395:
392:
385:
381:
375:
371:
363:
359:
353:
349:
345:
342:
339:
336:
333:
330:
325:
321:
317:
312:
308:
304:
301:
282:
278:exists, where
257:
246:
233:
222:
215:
205:
201:
196:
192:
186:
173:
162:
145:
134:
127:
102:
95:
92:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2462:
2451:
2448:
2446:
2443:
2441:
2440:Moduli theory
2438:
2437:
2435:
2416:
2413:
2411:
2408:
2406:
2403:
2402:
2400:
2398:
2394:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2364:
2362:
2360:
2359:Singularities
2356:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2331:
2329:
2325:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2293:
2289:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2253:
2250:
2248:
2245:
2244:
2242:
2238:
2235:
2231:
2225:
2222:
2220:
2217:
2215:
2212:
2211:
2209:
2207:Constructions
2205:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2178:Klein quartic
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2158:Bolza surface
2156:
2154:
2153:Bring's curve
2151:
2149:
2146:
2145:
2143:
2141:
2137:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2085:Conic section
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2065:AF+BG theorem
2063:
2062:
2060:
2058:
2054:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2024:
2022:
2018:
2008:
2005:
2003:
2000:
1999:
1997:
1993:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1933:
1931:
1927:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1902:
1900:
1896:
1893:
1891:
1887:
1881:
1880:Twisted cubic
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1857:
1855:
1853:
1849:
1845:
1837:
1832:
1830:
1825:
1823:
1818:
1817:
1814:
1807:
1803:
1799:
1793:
1789:
1785:
1781:
1778:
1774:
1770:
1764:
1760:
1756:
1752:
1749:
1745:
1741:
1735:
1731:
1724:
1723:
1717:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1669:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1633:
1628:
1624:
1620:
1619:
1613:
1610:
1606:
1602:
1596:
1592:
1588:
1584:
1580:
1575:
1574:
1570:
1568:
1566:
1561:
1558:
1550:
1548:
1545:
1543:
1519:
1511:
1506:
1497:
1490:
1487:
1483:
1477:
1474:
1470:
1466:
1463:
1460:
1455:
1452:
1448:
1442:
1439:
1435:
1431:
1424:
1423:
1422:
1417:
1398:
1395:
1389:
1385:
1376:
1372:
1363:
1357:
1354:
1345:
1341:
1338:
1335:
1328:
1327:
1326:
1303:
1300:
1297:
1294:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1260:
1257:
1254:
1251:
1248:
1235:
1229:
1224:
1220:
1212:
1211:
1210:
1207:
1203:
1199:
1195:
1176:
1173:
1167:
1163:
1157:
1153:
1144:
1138:
1135:
1132:
1129:
1126:
1119:
1118:
1117:
1112:
1108:
1103:
1098:
1091:
1087:
1083:
1079:
1075:
1071:
1066:
1062:
1058:
1033:
1027:
1023:
1014:
1010:
1006:
1003:
999:
964:
960:
951:
947:
937:
934:
931:
927:
923:
916:
912:
905:
902:
897:
893:
889:
884:
880:
875:
872:
866:
863:
858:
854:
850:
839:
836:
833:
830:
827:
823:
812:
808:
803:
799:
796:
793:
786:
782:
777:
768:
765:
762:
759:
756:
753:
750:
745:
741:
737:
734:
731:
726:
722:
717:
709:
708:
707:
701:
699:
697:
678:
675:
670:
666:
662:
657:
653:
649:
644:
641:
637:
625:
624:KdV hierarchy
621:
617:
612:
595:
592:
587:
581:
577:
571:
566:
562:
555:
550:
545:
540:
536:
530:
525:
519:
515:
509:
505:
498:
493:
488:
484:
478:
473:
468:
463:
459:
453:
447:
442:
438:
429:
425:
419:
415:
407:
404:
401:
397:
390:
383:
379:
373:
369:
361:
357:
351:
347:
340:
337:
331:
328:
323:
319:
315:
310:
306:
299:
292:
291:
290:
288:
285:is the first
281:
277:
273:
269:
264:
260:
255:
249:
245:
240:
236:
231:
225:
221:
214:
208:
204:
195:
189:
185:
180:
176:
171:
165:
161:
158:line bundles
157:
152:
148:
143:
137:
133:
126:
122:
118:
114:
109:
105:
101:
98:Suppose that
93:
91:
89:
88:KdV hierarchy
85:
81:
77:
73:
68:
66:
63:in the paper
62:
58:
57:Witten (1993)
54:
50:
47:in the paper
46:
45:Edward Witten
42:
38:
34:
30:
19:
2344:Prym variety
2318:Stable curve
2308:Hodge bundle
2298:ELSV formula
2100:Fermat curve
2057:Plane curves
2020:Higher genus
1995:Applications
1920:Modular form
1787:
1758:
1721:
1680:
1676:
1632:math/0601760
1622:
1616:
1578:
1562:
1554:
1546:
1541:
1539:
1415:
1413:
1324:
1209:is given by
1205:
1201:
1197:
1193:
1191:
1110:
1106:
1104:
1096:
1089:
1085:
1081:
1077:
1073:
1069:
1064:
1060:
1056:
1054:
705:
619:
615:
613:
610:
279:
275:
271:
267:
262:
258:
253:
247:
243:
238:
234:
229:
223:
219:
212:
206:
202:
193:
187:
183:
178:
174:
169:
163:
159:
155:
150:
146:
141:
135:
131:
124:
120:
116:
107:
103:
99:
97:
76:moduli stack
69:
32:
26:
2219:Polar curve
1683:(1): 1â23,
287:Chern class
2434:Categories
2214:Dual curve
1842:Topics in
1571:References
82:, and the
2327:Morphisms
2075:Bitangent
1705:0010-3616
1657:0894-0347
1516:Λ
1503:Λ
1484:δ
1471:δ
1464:μ
1432:∫
1382:Λ
1364:−
1358:
1350:Λ
1339:μ
1301:−
1289:×
1286:⋯
1283:×
1277:×
1271:×
1255:−
1249:−
1245:Λ
1236:−
1177:μ
1139:
1133:∫
1130:
1028:λ
1007:∈
1000:∏
988:Γ
952:−
924:∈
921:Γ
917:∑
881:λ
864:−
837:≤
831:≤
824:∏
820:⟩
804:τ
797:…
778:τ
774:⟨
760:−
735:⋯
718:∑
679:…
642:−
596:⋯
405:≥
398:∏
394:⟩
391:⋯
370:τ
348:τ
344:⟨
341:∑
332:…
115:of genus
94:Statement
1240:tr
200:, ..., Ï
2387:Tacnode
2372:Crunode
1806:1215968
1777:1144529
1748:2036721
1713:1171758
1685:Bibcode
1665:2328716
1637:Bibcode
1609:2807457
694:of the
242:where ÎŁ
139:, and
51: (
2367:Acnode
2291:Moduli
1804:
1794:
1775:
1765:
1746:
1736:
1711:
1703:
1663:
1655:
1607:
1597:
1414:where
618:= exp
270:â 3 +
49:Witten
31:, the
1726:(PDF)
1627:arXiv
1076:with
702:Proof
251:= dim
227:) on
130:,...,
119:with
2377:Cusp
1792:ISBN
1763:ISBN
1734:ISBN
1701:ISSN
1653:ISSN
1595:ISBN
1563:The
1095:and
167:on
53:1991
1693:doi
1681:147
1645:doi
1587:doi
1355:exp
1196:by
1136:exp
1127:log
984:Aut
266:= 3
78:of
67:.
27:In
2436::
1802:MR
1800:,
1773:MR
1771:,
1744:MR
1742:,
1732:,
1709:MR
1707:,
1699:,
1691:,
1679:,
1675:,
1661:MR
1659:,
1651:,
1643:,
1635:,
1623:20
1621:,
1605:MR
1603:,
1593:,
1585:,
1368:tr
1149:tr
698:.
588:48
551:24
526:24
494:24
1835:e
1828:t
1821:v
1695::
1687::
1647::
1639::
1629::
1589::
1542:F
1520:j
1512:+
1507:i
1498:2
1491:k
1488:j
1478:l
1475:i
1467:=
1461:d
1456:l
1453:k
1449:X
1443:j
1440:i
1436:X
1419:Î
1416:c
1399:X
1396:d
1393:)
1390:2
1386:/
1377:2
1373:X
1361:(
1346:c
1342:=
1336:d
1307:)
1304:1
1298:i
1295:2
1292:(
1280:5
1274:3
1268:1
1261:i
1258:2
1252:1
1230:=
1225:i
1221:t
1206:i
1202:t
1198:N
1194:N
1174:d
1171:)
1168:6
1164:/
1158:3
1154:X
1145:i
1142:(
1114:0
1111:t
1109:(
1107:F
1100:1
1097:X
1093:0
1090:X
1086:X
1082:e
1078:n
1074:g
1070:X
1065:n
1063:,
1061:g
1057:G
1037:)
1034:e
1031:(
1024:2
1015:1
1011:X
1004:e
992:|
979:|
971:|
965:0
961:X
956:|
948:2
938:n
935:,
932:g
928:G
913:=
906:1
903:+
898:i
894:d
890:2
885:i
876:!
873:!
870:)
867:1
859:i
855:d
851:2
848:(
840:n
834:i
828:1
813:n
809:d
800:,
794:,
787:1
783:d
769:n
766:+
763:3
757:g
754:3
751:=
746:n
742:d
738:+
732:+
727:1
723:d
682:}
676:,
671:1
667:L
663:,
658:0
654:L
650:,
645:1
638:L
634:{
620:F
616:Z
593:+
582:3
578:t
572:2
567:0
563:t
556:+
546:2
541:1
537:t
531:+
520:2
516:t
510:0
506:t
499:+
489:1
485:t
479:+
474:6
469:3
464:0
460:t
454:=
448:!
443:i
439:k
430:i
426:k
420:i
416:t
408:0
402:i
384:1
380:k
374:1
362:0
358:k
352:0
338:=
335:)
329:,
324:1
320:t
316:,
311:0
307:t
303:(
300:F
283:1
280:c
276:g
272:n
268:g
263:n
261:,
259:g
254:M
248:i
244:d
239:n
237:,
235:g
230:M
224:i
220:L
218:(
216:1
213:c
207:n
203:d
197:1
194:d
188:i
184:x
179:n
177:,
175:g
170:M
164:i
160:L
156:n
151:n
149:,
147:g
142:M
136:n
132:x
128:1
125:x
121:n
117:g
108:n
106:,
104:g
100:M
20:)
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