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489:(2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping (eds.),
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493:, Nankai Tracts in Mathematics, vol. 10, World Scientific, pp. 63–105 (at §5),
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36:
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442:(2008), "Siegel modular forms and their applications", in Ranestad, Kristian (ed.),
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56:
17:
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28:
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486:
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283:{\displaystyle \pi \colon {\mathcal {C}}_{g}\rightarrow {\mathcal {M}}_{g}}
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40:
572:
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133:
84:
55:. Furthermore, it has applications to the theory of
1163:
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812:
776:
751:
685:
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607:
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317:
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220:
181:
146:
101:
379:{\displaystyle \Lambda _{g}=\pi _{*}\omega _{g}}
584:
8:
993:
651:
591:
577:
569:
417:quasi-coherent sheaf on an algebraic stack
370:
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93:
87:
86:
83:
431:
408:
1014:Clifford's theorem on special divisors
415:Here, "vector bundle" in the sense of
39:, appears in the study of families of
7:
1183:Vector bundles on algebraic curves
1106:Weber's theorem (Algebraic curves)
703:Hasse's theorem on elliptic curves
693:Counting points on elliptic curves
344:
236:. To define the Hodge bundle, let
221:{\displaystyle {\mathcal {M}}_{g}}
182:{\displaystyle {\mathcal {M}}_{g}}
135:
102:{\displaystyle {\mathcal {M}}_{g}}
25:
491:Differential geometry and physics
111:moduli space of algebraic curves
794:Hurwitz's automorphisms theorem
1019:Gonality of an algebraic curve
930:Differential of the first kind
263:
1:
1173:Birkhoff–Grothendieck theorem
872:Nagata's conjecture on curves
743:Schoof–Elkies–Atkin algorithm
617:Five points determine a conic
530:Graduate Texts in Mathematics
450:, pp. 181–245 (at §13),
733:Supersingular elliptic curve
147:{\displaystyle \Lambda _{g}}
940:Riemann's existence theorem
867:Hilbert's sixteenth problem
759:Elliptic curve cryptography
672:Fundamental pair of periods
318:{\displaystyle \omega _{g}}
1234:
1070:Moduli of algebraic curves
444:The 1-2-3 of modular forms
329:. The Hodge bundle is the
456:10.1007/978-3-540-74119-0
294:algebraic curve of genus
230:holomorphic differentials
837:Cayley–Bacharach theorem
764:Elliptic curve primality
524:; Morrison, Ian (1998),
446:, Universitext, Berlin:
327:relative dualizing sheaf
1096:Riemann–Hurwitz formula
1060:Gromov–Witten invariant
920:Compact Riemann surface
708:Mazur's torsion theorem
43:, where it provides an
713:Modular elliptic curve
380:
319:
284:
222:
183:
148:
103:
627:Rational normal curve
381:
333:of this sheaf, i.e.,
320:
285:
223:
184:
149:
104:
1178:Stable vector bundle
1039:Weil reciprocity law
1029:Riemann–Roch theorem
1009:Brill–Noether theory
945:Riemann–Roch theorem
862:Genus–degree formula
723:Mordell–Weil theorem
698:Division polynomials
440:van der Geer, Gerard
340:
302:
240:
201:
162:
131:
82:
990:Structure of curves
882:Quartic plane curve
804:Hyperelliptic curve
784:De Franchis theorem
728:Nagell–Lutz theorem
18:Hodge vector bundle
997:Divisors on curves
789:Faltings's theorem
738:Schoof's algorithm
718:Modularity theorem
376:
315:
280:
218:
179:
144:
99:
1195:
1194:
1191:
1190:
1091:Hasse–Witt matrix
1034:Weierstrass point
981:Smooth completion
950:TeichmĂĽller space
852:Cubic plane curve
772:
771:
686:Arithmetic theory
667:Elliptic integral
662:Elliptic function
551:978-0-387-98429-2
532:, vol. 187,
500:978-981-270-377-4
465:978-3-540-74117-6
120:curves over some
16:(Redirected from
1225:
1218:Algebraic curves
1213:Invariant theory
1024:Jacobian variety
994:
897:Riemann surfaces
887:Real plane curve
847:Cramer's paradox
827:BĂ©zout's theorem
652:
601:algebraic curves
593:
586:
579:
570:
563:
562:
526:Moduli of curves
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228:is the space of
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64:algebraic groups
53:algebraic curves
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1187:
1159:
1150:Delta invariant
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1110:
1079:
1043:
1004:Abel–Jacobi map
985:
959:
955:Torelli theorem
925:Dessin d'enfant
905:Belyi's theorem
891:
877:PlĂĽcker formula
808:
799:Hurwitz surface
768:
747:
681:
655:Analytic theory
647:Elliptic curves
641:
622:Projective line
609:Rational curves
603:
597:
567:
566:
552:
536:, p. 155,
534:Springer-Verlag
520:
519:
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485:
484:
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448:Springer-Verlag
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1167:
1165:Vector bundles
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632:Riemann sphere
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542:10.1007/b98867
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37:W. V. D. Hodge
35:, named after
24:
14:
13:
10:
9:
6:
4:
3:
2:
1230:
1219:
1216:
1214:
1211:
1209:
1208:Moduli theory
1206:
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1184:
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1119:
1117:
1116:Singularities
1113:
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964:Constructions
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943:
941:
938:
936:
935:Klein quartic
933:
931:
928:
926:
923:
921:
918:
916:
915:Bolza surface
913:
911:
910:Bring's curve
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848:
845:
843:
842:Conic section
840:
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823:
822:AF+BG theorem
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637:Twisted cubic
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293:
275:
258:
246:
243:
235:
232:on the curve
231:
213:
196:
192:
174:
157:
156:vector bundle
139:
127:
123:
119:
116:
112:
94:
73:
71:
69:
68:string theory
65:
62:
58:
57:modular forms
54:
50:
49:moduli theory
46:
42:
38:
34:
30:
19:
1124:
1101:Prym variety
1075:Stable curve
1065:Hodge bundle
1064:
1055:ELSV formula
857:Fermat curve
814:Plane curves
777:Higher genus
752:Applications
677:Modular form
525:
516:
490:
481:
443:
434:
411:
397:ELSV formula
295:
233:
194:
126:Hodge bundle
125:
117:
77:
33:Hodge bundle
32:
26:
1130:singularity
976:Polar curve
522:Harris, Joe
487:Liu, Kefeng
331:pushforward
193:at a point
29:mathematics
1202:Categories
971:Dual curve
599:Topics in
426:References
74:Definition
1084:Morphisms
832:Bitangent
368:ω
362:∗
358:π
345:Λ
307:ω
292:universal
264:→
247::
244:π
136:Λ
61:reductive
45:invariant
391:See also
298:and let
1155:Tacnode
1140:Crunode
560:1631825
509:2322389
474:2409679
325:be its
290:be the
109:be the
47:in the
1135:Acnode
1048:Moduli
558:
548:
507:
497:
472:
462:
189:whose
124:. The
122:scheme
41:curves
31:, the
403:Notes
191:fiber
154:is a
115:genus
1145:Cusp
546:ISBN
495:ISBN
460:ISBN
78:Let
66:and
538:doi
452:doi
197:in
158:on
113:of
59:on
51:of
27:In
1204::
556:MR
554:,
544:,
528:,
505:MR
503:,
470:MR
468:,
458:,
70:.
1127:k
1125:A
592:e
585:t
578:v
540::
454::
386:.
372:g
354:=
349:g
311:g
296:g
276:g
270:M
259:g
253:C
234:C
214:g
208:M
195:C
175:g
169:M
140:g
118:g
95:g
89:M
20:)
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