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are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary
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Creative
Commons Attribution-ShareAlike 3.0 Unported License
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596:. Geometry of algebraic curves. vol 1 Springer,
274:Adjunction formula § Applications to curves
268:The genus–degree formula can be proven from the
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445:
424:
8:
565:This article incorporates material from the
614:, Principles of algebraic geometry, Wiley,
137:{\displaystyle g={\frac {1}{2}}(d-1)(d-2).}
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27:Theorem in classical algebraic geometry
1107:Clifford's theorem on special divisors
403:{\displaystyle g={\binom {d-1}{n}},\,}
7:
253:{\displaystyle {\frac {1}{2}}r(r-1)}
199:. If the curve is non-singular the
1265:Vector bundles on algebraic curves
1199:Weber's theorem (Algebraic curves)
796:Hasse's theorem on elliptic curves
786:Counting points on elliptic curves
491:Introduction to Algebraic Geometry
455:{\displaystyle {\tbinom {d-1}{n}}}
428:
373:
25:
336:{\displaystyle \mathbb {P} ^{n}}
192:{\displaystyle \mathbb {P} ^{2}}
887:Hurwitz's automorphisms theorem
573:", which is licensed under the
1112:Gonality of an algebraic curve
1023:Differential of the first kind
247:
235:
147:Here "plane curve" means that
128:
116:
113:
101:
1:
1255:Birkhoff–Grothendieck theorem
965:Nagata's conjecture on curves
836:Schoof–Elkies–Atkin algorithm
710:Five points determine a conic
826:Supersingular elliptic curve
644:Kulikov, Viktor S. (2001) ,
1033:Riemann's existence theorem
960:Hilbert's sixteenth problem
852:Elliptic curve cryptography
765:Fundamental pair of periods
651:Encyclopedia of Mathematics
1306:
1163:Moduli of algebraic curves
540:, chapter V, example 1.5.1
167:is a closed curve in the
930:Cayley–Bacharach theorem
857:Elliptic curve primality
1189:Riemann–Hurwitz formula
1153:Gromov–Witten invariant
1013:Compact Riemann surface
801:Mazur's torsion theorem
622:, chapter 2, section 1.
495:Oxford University Press
215:decreases the genus by
806:Modular elliptic curve
483:Semple, John Greenlees
456:
404:
337:
301:
254:
193:
161:
138:
62:
720:Rational normal curve
588:, Maurizio Cornalba,
457:
405:
350:the formula becomes:
338:
302:
255:
194:
162:
139:
63:
1260:Stable vector bundle
1132:Weil reciprocity law
1122:Riemann–Roch theorem
1102:Brill–Noether theory
1038:Riemann–Roch theorem
955:Genus–degree formula
816:Mordell–Weil theorem
791:Division polynomials
571:Genus degree formula
464:binomial coefficient
417:
357:
318:
291:
219:
174:
151:
82:
52:
36:genus–degree formula
1083:Structure of curves
975:Quartic plane curve
897:Hyperelliptic curve
877:De Franchis theorem
821:Nagell–Lutz theorem
532:, Springer GTM 52,
284:For a non-singular
272:; for details, see
38:relates the degree
1090:Divisors on curves
882:Faltings's theorem
831:Schoof's algorithm
811:Modularity theorem
646:"Genus of a curve"
630:Algebraic geometry
577:but not under the
526:Algebraic geometry
497:. pp. 53–54.
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450:
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270:adjunction formula
250:
189:
157:
134:
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32:algebraic geometry
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1276:
1273:
1272:
1184:Hasse–Witt matrix
1127:Weierstrass point
1074:Smooth completion
1043:Teichmüller space
945:Cubic plane curve
865:
864:
779:Arithmetic theory
760:Elliptic integral
755:Elliptic function
608:Phillip Griffiths
590:Phillip Griffiths
493:(1985 ed.).
443:
388:
300:{\displaystyle H}
230:
160:{\displaystyle C}
99:
75:via the formula:
61:{\displaystyle C}
16:(Redirected from
1297:
1290:Algebraic curves
1117:Jacobian variety
1087:
990:Riemann surfaces
980:Real plane curve
940:Cramer's paradox
920:Bézout's theorem
745:
694:algebraic curves
686:
679:
672:
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626:Robin Hartshorne
586:Enrico Arbarello
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530:Robin Hartshorne
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1097:Abel–Jacobi map
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1048:Torelli theorem
1018:Dessin d'enfant
998:Belyi's theorem
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970:Plücker formula
901:
892:Hurwitz surface
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748:Analytic theory
740:Elliptic curves
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715:Projective line
702:Rational curves
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280:Generalization
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1057:Constructions
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935:Conic section
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915:AF+BG theorem
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730:Twisted cubic
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638:0-387-90244-9
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621:
620:0-471-05059-8
617:
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609:
606:
604:, appendix A.
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602:0-387-90997-4
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538:0-387-90244-9
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504:0-19-853363-2
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487:Roth, Leonard
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55:
48:
45:
41:
37:
33:
30:In classical
19:
18:Genus formula
1194:Prym variety
1168:Stable curve
1158:Hodge bundle
1148:ELSV formula
954:
950:Fermat curve
907:Plane curves
870:Higher genus
845:Applications
770:Modular form
649:
632:, Springer,
629:
564:
525:
521:
490:
477:
412:
347:
308:
286:hypersurface
283:
267:
212:
146:
72:
39:
35:
29:
1069:Polar curve
567:Citizendium
209:singularity
47:plane curve
44:irreducible
1064:Dual curve
692:Topics in
612:Joe Harris
594:Joe Harris
559:References
307:of degree
1177:Morphisms
925:Bitangent
656:EMS Press
569:article "
434:−
379:−
242:−
123:−
108:−
68:with its
1284:Category
628:(1977):
547:See also
203:and the
1237:Tacnode
1222:Crunode
513:0814690
462:is the
311:in the
1217:Acnode
1141:Moduli
636:
618:
600:
536:
511:
501:
413:where
42:of an
34:, the
470:Notes
264:Proof
1227:Cusp
634:ISBN
616:ISBN
610:and
598:ISBN
579:GFDL
534:ISBN
499:ISBN
343:of
1286::
654:,
648:,
592:,
528:,
509:MR
507:.
489:.
485:;
466:.
276:.
260:.
685:e
678:t
671:v
640:.
581:.
515:.
446:)
441:n
437:1
431:d
425:(
397:,
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386:n
382:1
376:d
370:(
364:=
361:g
348:g
329:n
324:P
309:d
295:H
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245:1
239:r
236:(
233:r
228:2
225:1
213:r
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180:P
155:C
132:.
129:)
126:2
120:d
117:(
114:)
111:1
105:d
102:(
97:2
94:1
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86:g
73:g
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40:d
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.