668:
3543:
361:
3898:
4944:
unramified outside 0 and ∞. Standard methods of algebraic geometry allow one to find the degree of a map by looking at an infinite fiber and its normal bundle. The result is expressed as an integral of certain characteristic classes over the infinite fiber. In our case this integral happens to be
3197:
4086:
2734:
2964:
2417:
3172:
663:{\displaystyle h_{g;k_{1},\dots ,k_{n}}={\dfrac {m!}{\#{\text{Aut}}(k_{1},\ldots ,k_{n})}}\prod _{i=1}^{n}{\frac {k_{i}^{k_{i}}}{k_{i}!}}\int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}.}
1987:
48:
3601:
1122:
in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the integrals on the right-hand side of the ELSV formula
2102:
3538:{\displaystyle \int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}=\int _{{\overline {\mathcal {M}}}_{1,1}}{\frac {1-\lambda _{1}}{1-k_{1}\psi _{1}}}=\leftk_{1}-\left.}
2562:
3971:
2162:
4879:
1710:
The equivalence between the two definitions of
Hurwitz numbers (counting ramified coverings of the sphere, or counting transitive factorizations) is established by describing a ramified covering by its
839:
4488:
4942:
4745:
4335:
4172:
1334:
2833:
1652:
4791:
4264:
2204:
1845:
1774:
1399:
1002:
242:
2596:
2841:
1789:. (Heuristically this behaves much like complex manifold, except that integrals of characteristic classes that are integers for manifolds are rational numbers for Deligne–Mumford stacks.)
1451:
4664:
1522:
1195:
1117:
192:
898:
3030:
2480:
959:
2307:
4405:
2299:
2235:
776:
300:
3044:
110:
5243:
3893:{\displaystyle h_{1;k}=(k+1)!{\frac {k^{k}}{k!}}\int _{{\overline {\mathcal {M}}}_{1,1}}{\frac {1-\lambda _{1}}{1-k\psi _{1}}}=(k+1)k^{k}\left\{\leftk-\left\right\}.}
1689:
1581:
728:
700:
1885:
350:
1998:
5666:
1723:!, which explains the division by it), and consider the monodromies of the covering about the branch point. This leads to a transitive factorization.
5875:
5395:
5758:
5355:
5236:
4081:{\displaystyle \int _{{\overline {\mathcal {M}}}_{1,1}}\psi _{1}=\int _{{\overline {\mathcal {M}}}_{1,1}}\lambda _{1}={\frac {1}{24}}.}
5835:
5446:
5345:
5825:
2485:
4210:
proved the formula in full generality using the localization techniques. The proof announced by the four initial authors followed (
2113:
4812:
5524:
5229:
4576:
The branching morphism is of finite degree, but has infinite fibers. Our aim is now to compute its degree in two different ways.
783:
4424:
4224:, building on preceding work of several people, gave a unified way to deduce most known results in the intersection theory of
5671:
5582:
5592:
5519:
5005:
Ekedahl, T.; Lando, S.; Shapiro, M.; Vainshtein, A. (2001). "Hurwitz numbers and intersections on moduli spaces of curves".
4579:
The first way is to count the preimages of a generic point in the image. In other words, we count the ramified coverings of
4884:
4687:
4277:
4097:
5269:
1281:
5489:
5385:
5748:
5712:
2774:
2729:{\displaystyle \int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}}
1590:
5411:
5324:
4754:
4227:
2959:{\displaystyle \int _{{\overline {\mathcal {M}}}_{g,n}}(-1)^{j}\lambda _{j}\psi _{1}^{d_{1}}\cdots \psi _{n}^{d_{n}},}
2167:
1808:
1737:
1361:
965:
204:
54:
5722:
5360:
1731:
118:
66:
5860:
5768:
5681:
5661:
5597:
5514:
5375:
1410:
5416:
4948:
Thus the ELSV formula expresses the equality between two ways to compute the degree of the branching morphism.
3177:
was first conjectured by I. P. Goulden and D. M. Jackson. No proof independent from the ELSV formula is known.
5380:
4610:
1468:
1141:
1063:
138:
5572:
5152:
2412:{\displaystyle \psi _{i}=c_{1}({\mathcal {L}}_{i})\in H^{2}({\overline {\mathcal {M}}}_{g,n},\mathbf {Q} ).}
848:
5865:
5365:
2975:
2428:
904:
5743:
5479:
5870:
5279:
4502:
is a smooth map. But it has a natural extension to the space of stable maps. For instance, the value of
4364:
4214:). Now that the space of stable maps to the projective line relative to a point has been constructed by
2273:
2209:
735:
259:
5441:
5390:
4958:
Ekedahl, T.; Lando, S.; Shapiro, M.; Vainshtein, A. (1999). "On
Hurwitz numbers and Hodge integrals".
5830:
5691:
5602:
5350:
5118:
5071:
5024:
4977:
3167:{\displaystyle {\frac {h_{g;k_{1},\dots ,k_{n}}}{m!}}\prod _{i=1}^{n}{\frac {k_{i}!}{k_{i}^{k_{i}}}}}
1872:
58:
32:
5656:
5534:
5499:
5456:
5436:
1027:
85:
81:
73:
88:
43:
5797:
5370:
5198:
5161:
5134:
5108:
5087:
5061:
5040:
5014:
4993:
4967:
4506:
on a node is considered a double branch point, as can be seen by looking at the family of curves
5577:
5557:
5529:
1982:{\displaystyle \lambda _{j}=c_{j}(E)\in H^{2j}({\overline {\mathcal {M}}}_{g,n},\mathbf {Q} ).}
5686:
5633:
5504:
5319:
5314:
5099:
Graber, T.; Vakil, R. (2003). "Hodge integrals and
Hurwitz numbers via virtual localization".
253:
77:
62:
1661:
1553:
707:
679:
5676:
5562:
5539:
5208:
5171:
5126:
5079:
5032:
4985:
1457:
317:
5802:
5607:
5549:
5451:
5274:
5253:
4218:, a proof can be obtained immediately by applying the virtual localization to this space.
1778:
5474:
5122:
5075:
5028:
4981:
5776:
5299:
5284:
5261:
198:
37:
4989:
5854:
5817:
5587:
5567:
5494:
5289:
5138:
5044:
4997:
1715:. More precisely: choose a base point on the sphere, number its preimages from 1 to
5753:
5727:
5717:
5509:
5329:
5091:
1794:
1009:
1005:
307:
2097:{\displaystyle c(E^{*})=1-\lambda _{1}+\lambda _{2}-\cdots +(-1)^{g}\lambda _{g}.}
5628:
5466:
4498:
with multiplicities taken into account. Actually, this definition only works if
1035:
76:
of moduli spaces of curves can be deduced from the ELSV formula, including the
5623:
5176:
5147:
5130:
5083:
17:
5221:
5213:
5186:
5484:
4673:
is to look at the preimage of the most degenerate point, namely, to put all
1712:
5052:
Fantechi, B.; Pandharipande, R. (2002). "Stable maps and branch divisors".
3955:= 1/2 (since there is a unique factorization of the transposition (1 2) in
5036:
2576:+ 1 factors. We expand this product, extract from it the part of degree 3
2572:(the dimension of the moduli space). Thus the integrand is a product of
5807:
5792:
4607:
more fixed simple branch points. This is precisely the
Hurwitz number
5787:
5203:
5113:
5066:
5019:
4972:
5187:"Stable Morphisms to Singular Schemes and Relative Stable Morphisms"
197:
as the number of ramified coverings of the complex projective line (
5166:
1524:
is the number of transitive factorizations of identity of type (
5225:
2557:{\displaystyle 1+k_{i}\psi _{i}+k_{i}^{2}\psi _{i}^{2}+\cdots }
3903:
On the other hand, according to
Example A, the Hurwitz number
4211:
4183:
2157:{\displaystyle {\mathcal {L}}_{1},\ldots ,{\mathcal {L}}_{n}}
1049:
is the first Chern class of the cotangent line bundle to the
4891:
4874:{\displaystyle z\mapsto z^{k_{1}},\dots ,z\mapsto z^{k_{n}}}
4762:
4694:
4437:
4284:
4235:
4029:
3984:
3844:
3786:
3673:
3595:
According to
Example B, the ELSV formula in this case reads
3494:
3429:
3339:
3210:
2854:
2609:
2374:
2340:
2280:
2216:
2175:
2143:
2120:
1944:
1816:
1745:
973:
540:
1404:
equals the identity permutation and the group generated by
834:{\displaystyle \#\operatorname {Aut} (k_{1},\ldots ,k_{n})}
4483:{\displaystyle f\in {\mathcal {M}}_{g;k_{1},\dots ,k_{n}}}
4809:
rational components on which the stable map has the form
310:. Here if a covering has a nontrivial automorphism group
3965:
Plugging these two values into the ELSV formula we find
1200:
also have a definition in purely algebraic terms. With
4937:{\displaystyle {\mathcal {M}}_{g;k_{1},\dots ,k_{n}}}
4887:
4815:
4757:
4740:{\displaystyle {\mathcal {M}}_{g;k_{1},\dots ,k_{n}}}
4690:
4613:
4427:
4367:
4330:{\displaystyle {\mathcal {M}}_{g;k_{1},\dots ,k_{n}}}
4280:
4230:
4187:
4167:{\displaystyle h_{1;k}={\frac {(k^{2}-1)k^{k}}{24}}.}
4100:
3974:
3604:
3200:
3047:
2978:
2844:
2777:
2599:
2488:
2431:
2310:
2276:
2212:
2170:
2116:
2001:
1888:
1811:
1740:
1664:
1593:
1556:
1471:
1413:
1364:
1284:
1144:
1066:
968:
907:
851:
786:
738:
710:
682:
411:
364:
320:
262:
207:
141:
91:
1329:{\displaystyle (\tau _{1},\dots ,\tau _{m},\sigma )}
61:, is an equality between a Hurwitz number (counting
5816:
5767:
5736:
5700:
5649:
5642:
5616:
5548:
5465:
5429:
5404:
5338:
5307:
5298:
5260:
1339:
is a transitive factorization of identity of type (
4945:equal to the right-hand side of the ELSV formula.
4936:
4873:
4785:
4739:
4658:
4482:
4399:
4329:
4258:
4166:
4080:
3892:
3537:
3166:
3024:
2958:
2827:
2728:
2556:
2474:
2411:
2293:
2229:
2198:
2156:
2096:
1981:
1839:
1768:
1683:
1646:
1575:
1516:
1445:
1393:
1328:
1189:
1111:
996:
953:
892:
833:
770:
722:
694:
662:
344:
294:
236:
186:
104:
2828:{\displaystyle k_{1}^{d_{1}}\cdots k_{n}^{d_{n}}}
1647:{\displaystyle (\tau _{1},\dots ,\tau _{k+2g-1})}
4786:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
4259:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
2199:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
1840:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
1769:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
1394:{\displaystyle \tau _{1}\cdots \tau _{m}\sigma }
997:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
237:{\displaystyle \mathbb {P} ^{1}(\mathbb {C} ))}
1695:times the number of factorizations of a given
1587:! times the number of lists of transpositions
5237:
8:
4207:
5646:
5304:
5244:
5230:
5222:
3941:= 0 (since there are no transpositions in
1446:{\displaystyle \tau _{1},\dots ,\tau _{m}}
5212:
5202:
5175:
5165:
5112:
5065:
5018:
4971:
4960:Comptes Rendus de l'Académie des Sciences
4926:
4907:
4896:
4890:
4889:
4886:
4863:
4858:
4831:
4826:
4814:
4797:marked points we send this curve to 0 in
4771:
4761:
4759:
4756:
4729:
4710:
4699:
4693:
4692:
4689:
4648:
4629:
4618:
4612:
4472:
4453:
4442:
4436:
4435:
4426:
4391:
4372:
4366:
4319:
4300:
4289:
4283:
4282:
4279:
4244:
4234:
4232:
4229:
4149:
4130:
4120:
4105:
4099:
4065:
4056:
4038:
4028:
4026:
4024:
4011:
3993:
3983:
3981:
3979:
3973:
3962:into a product of three transpositions).
3871:
3853:
3843:
3841:
3839:
3813:
3795:
3785:
3783:
3781:
3761:
3730:
3709:
3696:
3682:
3672:
3670:
3668:
3648:
3642:
3609:
3603:
3521:
3503:
3493:
3491:
3489:
3471:
3456:
3438:
3428:
3426:
3424:
3403:
3393:
3375:
3362:
3348:
3338:
3336:
3334:
3315:
3305:
3280:
3270:
3246:
3233:
3219:
3209:
3207:
3205:
3199:
3154:
3149:
3144:
3130:
3123:
3117:
3106:
3084:
3065:
3054:
3048:
3046:
3013:
2977:
2945:
2940:
2935:
2920:
2915:
2910:
2900:
2890:
2863:
2853:
2851:
2849:
2843:
2817:
2812:
2807:
2792:
2787:
2782:
2776:
2714:
2704:
2679:
2669:
2645:
2632:
2618:
2608:
2606:
2604:
2598:
2542:
2537:
2527:
2522:
2509:
2499:
2487:
2463:
2453:
2435:
2430:
2398:
2383:
2373:
2371:
2361:
2345:
2339:
2338:
2328:
2315:
2309:
2285:
2279:
2278:
2275:
2221:
2215:
2214:
2211:
2184:
2174:
2172:
2169:
2148:
2142:
2141:
2125:
2119:
2118:
2115:
2085:
2075:
2047:
2034:
2012:
2000:
1968:
1953:
1943:
1941:
1928:
1906:
1893:
1887:
1825:
1815:
1813:
1810:
1754:
1744:
1742:
1739:
1669:
1663:
1620:
1601:
1592:
1561:
1555:
1506:
1487:
1476:
1470:
1437:
1418:
1412:
1382:
1369:
1363:
1311:
1292:
1283:
1246:be transpositions in the symmetric group
1179:
1160:
1149:
1143:
1101:
1082:
1071:
1065:
982:
972:
970:
967:
921:
906:
878:
859:
850:
822:
803:
785:
762:
743:
737:
709:
681:
645:
635:
610:
600:
576:
563:
549:
539:
537:
535:
519:
506:
501:
496:
490:
484:
473:
456:
437:
425:
410:
399:
380:
369:
363:
337:
329:
324:
319:
286:
267:
261:
224:
223:
214:
210:
209:
206:
176:
157:
146:
140:
96:
90:
4659:{\displaystyle h_{g;k_{1},\dots ,k_{n}}}
4221:
3917:times the number of ways to decompose a
2755:, whose monomials have degrees between 3
2584:and integrate it over the moduli space.
1517:{\displaystyle h_{g;k_{1},\dots ,k_{n}}}
1190:{\displaystyle h_{g;k_{1},\dots ,k_{n}}}
1124:
1112:{\displaystyle h_{g;k_{1},\dots ,k_{n}}}
187:{\displaystyle h_{g;k_{1},\dots ,k_{n}}}
65:of the sphere) and an integral over the
3561:= 1. To simplify the notation, denote
2739:is a symmetric polynomial in variables
5667:Clifford's theorem on special divisors
2564:, where the sum can be cut at degree 3
893:{\displaystyle (k_{1},\ldots ,k_{n});}
841:is the number of automorphisms of the
4669:The second way to find the degree of
7:
4881:. Thus we obtain all stable maps in
4793:. Indeed, given a stable curve with
3025:{\displaystyle j=3g-3+n-\sum d_{i}.}
2475:{\displaystyle 1/(1-k_{i}\psi _{i})}
1805:vector bundle over the moduli space
954:{\displaystyle m=\sum k_{i}+n+2g-2;}
4188:Fantechi & Pandharipande (2002)
3934:+ 1 transpositions. In particular,
1879:. Its Chern classes are denoted by
244:that are connected curves of genus
72:Several fundamental results in the
5836:Vector bundles on algebraic curves
5759:Weber's theorem (Algebraic curves)
5356:Hasse's theorem on elliptic curves
5346:Counting points on elliptic curves
5148:"KP hierarchy for Hodge integrals"
4805:) and attach to its marked points
4400:{\displaystyle k_{1},\dots ,k_{n}}
4215:
4182:The ELSV formula was announced by
2771:. The coefficient of the monomial
2294:{\displaystyle {\mathcal {L}}_{i}}
2230:{\displaystyle {\mathcal {L}}_{i}}
1727:The integral over the moduli space
787:
771:{\displaystyle k_{1},\dots ,k_{n}}
422:
295:{\displaystyle k_{1},\dots ,k_{n}}
25:
3038:The polynomiality of the numbers
673:Here the notation is as follows:
314:it should be counted with weight
5191:Journal of Differential Geometry
2399:
1969:
5447:Hurwitz's automorphisms theorem
4751:isomorphic to the moduli space
4677:branch points together at 0 in
4587:) with a branch point of type (
4206:= 1 (with the corrected sign).
31:, named after its four authors
5876:Theorems in algebraic geometry
5672:Gonality of an algebraic curve
5583:Differential of the first kind
4851:
4819:
4684:The preimage of this point in
4186:, but with an erroneous sign.
4142:
4123:
3921:-cycle in the symmetric group
3754:
3742:
3636:
3624:
3321:
3292:
3286:
3257:
3252:
3239:
2887:
2877:
2720:
2691:
2685:
2656:
2651:
2638:
2469:
2440:
2403:
2367:
2351:
2334:
2072:
2062:
2018:
2005:
1973:
1937:
1918:
1912:
1871:marked points is the space of
1641:
1594:
1323:
1285:
884:
852:
828:
796:
651:
622:
616:
587:
582:
569:
462:
430:
338:
330:
231:
228:
220:
1:
5826:Birkhoff–Grothendieck theorem
5525:Nagata's conjecture on curves
5396:Schoof–Elkies–Atkin algorithm
5270:Five points determine a conic
4990:10.1016/S0764-4442(99)80435-2
2590:It follows that the integral
2588:The integral as a polynomial.
1719:(this introduces a factor of
119:Gopakumar–Mariño–Vafa formula
67:moduli space of stable curves
5386:Supersingular elliptic curve
4766:
4337:be the space of stable maps
4239:
4033:
3988:
3848:
3790:
3677:
3498:
3433:
3343:
3214:
2858:
2613:
2378:
2179:
1948:
1820:
1749:
977:
544:
355:The ELSV formula then reads
105:{\displaystyle \lambda _{g}}
5593:Riemann's existence theorem
5520:Hilbert's sixteenth problem
5412:Elliptic curve cryptography
5325:Fundamental pair of periods
2270:. The first Chern class of
2257:) is the cotangent line to
1259:numbered cycles of lengths
5892:
5723:Moduli of algebraic curves
4559:tend towards the value of
4550:→ 0, two branch points of
1847:whose fiber over a curve (
1038:of its dual vector bundle;
252:numbered preimages of the
5177:10.1016/j.aim.2008.10.017
4490:the unordered set of its
4208:Graber & Vakil (2003)
1699:-cycle into a product of
1255:and σ a permutation with
702:is a nonnegative integer;
117:It is generalized by the
5490:Cayley–Bacharach theorem
5417:Elliptic curve primality
5146:Kazarian, Maxim (2009).
5007:Inventiones Mathematicae
4747:is an infinite fiber of
1781:of (complex) dimension 3
1658:-cycle. In other words,
5749:Riemann–Hurwitz formula
5713:Gromov–Witten invariant
5573:Compact Riemann surface
5361:Mazur's torsion theorem
5153:Advances in Mathematics
5131:10.1023/A:1021791611677
5084:10.1023/A:1014347115536
4521:and the family of maps
4266:from the ELSV formula.
2110:Introduce line bundles
1684:{\displaystyle h_{g;k}}
1576:{\displaystyle h_{g;k}}
723:{\displaystyle n\geq 1}
695:{\displaystyle g\geq 0}
5366:Modular elliptic curve
5214:10.4310/jdg/1090348132
5101:Compositio Mathematica
5054:Compositio Mathematica
4938:
4875:
4787:
4741:
4660:
4513:given by the equation
4484:
4401:
4331:
4260:
4168:
4082:
3894:
3539:
3168:
3122:
3026:
2960:
2829:
2730:
2558:
2476:
2413:
2295:
2231:
2200:
2158:
2098:
1983:
1841:
1770:
1685:
1648:
1577:
1518:
1447:
1395:
1330:
1191:
1113:
998:
955:
894:
835:
778:are positive integers;
772:
730:is a positive integer;
724:
696:
664:
489:
346:
296:
256:having multiplicities
238:
188:
106:
5280:Rational normal curve
5037:10.1007/s002220100164
4939:
4876:
4788:
4742:
4661:
4485:
4419:Lyashko–Looijenga map
4402:
4332:
4261:
4184:Ekedahl et al. (1999)
4169:
4091:From which we deduce
4083:
3895:
3540:
3169:
3102:
3027:
2961:
2830:
2731:
2559:
2477:
2414:
2296:
2232:
2201:
2159:
2099:
1984:
1873:abelian differentials
1842:
1779:Deligne–Mumford stack
1771:
1686:
1649:
1578:
1519:
1448:
1396:
1331:
1192:
1114:
999:
956:
895:
836:
773:
725:
697:
665:
469:
347:
345:{\displaystyle 1/|G|}
297:
239:
189:
107:
5831:Stable vector bundle
5692:Weil reciprocity law
5682:Riemann–Roch theorem
5662:Brill–Noether theory
5598:Riemann–Roch theorem
5515:Genus–degree formula
5376:Mordell–Weil theorem
5351:Division polynomials
4885:
4813:
4755:
4688:
4611:
4425:
4365:
4278:
4228:
4098:
3972:
3602:
3198:
3045:
2976:
2842:
2775:
2597:
2486:
2429:
2308:
2274:
2210:
2168:
2114:
1999:
1886:
1809:
1738:
1707:− 1 transpositions.
1662:
1591:
1554:
1469:
1411:
1362:
1282:
1142:
1135:The Hurwitz numbers
1064:
966:
905:
849:
784:
736:
708:
680:
362:
318:
260:
205:
139:
89:
82:Virasoro constraints
27:In mathematics, the
5643:Structure of curves
5535:Quartic plane curve
5457:Hyperelliptic curve
5437:De Franchis theorem
5381:Nagell–Lutz theorem
5123:2000math......3028G
5076:1999math......5104F
5029:2001InMat.146..297E
4982:1999CRASM.328.1175E
4212:Ekedahl et al. 2001
3161:
2952:
2927:
2824:
2799:
2547:
2532:
1654:whose product is a
1131:The Hurwitz numbers
1028:Hodge vector bundle
513:
74:intersection theory
5650:Divisors on curves
5442:Faltings's theorem
5391:Schoof's algorithm
5371:Modularity theorem
4934:
4871:
4783:
4737:
4656:
4480:
4412:branching morphism
4397:
4327:
4256:
4164:
4078:
3930:into a product of
3890:
3535:
3164:
3140:
3022:
2956:
2931:
2906:
2825:
2803:
2778:
2726:
2554:
2533:
2518:
2482:is interpreted as
2472:
2409:
2291:
2227:
2196:
2154:
2094:
1979:
1837:
1766:
1681:
1644:
1573:
1514:
1443:
1391:
1326:
1187:
1109:
994:
951:
890:
831:
768:
720:
692:
660:
492:
467:
342:
292:
234:
184:
102:
63:ramified coverings
5848:
5847:
5844:
5843:
5744:Hasse–Witt matrix
5687:Weierstrass point
5634:Smooth completion
5603:Teichmüller space
5505:Cubic plane curve
5425:
5424:
5339:Arithmetic theory
5320:Elliptic integral
5315:Elliptic function
4966:(12): 1175–1180.
4769:
4494:branch points in
4242:
4159:
4073:
4036:
3991:
3851:
3793:
3737:
3680:
3662:
3501:
3436:
3410:
3346:
3325:
3217:
3162:
3100:
2861:
2724:
2616:
2381:
2182:
1951:
1823:
1752:
1355:) if the product
1053:-th marked point.
980:
655:
547:
529:
466:
428:
254:point at infinity
78:Witten conjecture
16:(Redirected from
5883:
5861:Algebraic curves
5677:Jacobian variety
5647:
5550:Riemann surfaces
5540:Real plane curve
5500:Cramer's paradox
5480:Bézout's theorem
5305:
5254:algebraic curves
5246:
5239:
5232:
5223:
5218:
5216:
5206:
5185:Li, Jun (2001).
5181:
5179:
5169:
5142:
5116:
5095:
5069:
5048:
5022:
5001:
4975:
4943:
4941:
4940:
4935:
4933:
4932:
4931:
4930:
4912:
4911:
4895:
4894:
4880:
4878:
4877:
4872:
4870:
4869:
4868:
4867:
4838:
4837:
4836:
4835:
4792:
4790:
4789:
4784:
4782:
4781:
4770:
4765:
4760:
4746:
4744:
4743:
4738:
4736:
4735:
4734:
4733:
4715:
4714:
4698:
4697:
4665:
4663:
4662:
4657:
4655:
4654:
4653:
4652:
4634:
4633:
4489:
4487:
4486:
4481:
4479:
4478:
4477:
4476:
4458:
4457:
4441:
4440:
4406:
4404:
4403:
4398:
4396:
4395:
4377:
4376:
4361:poles of orders
4336:
4334:
4333:
4328:
4326:
4325:
4324:
4323:
4305:
4304:
4288:
4287:
4265:
4263:
4262:
4257:
4255:
4254:
4243:
4238:
4233:
4173:
4171:
4170:
4165:
4160:
4155:
4154:
4153:
4135:
4134:
4121:
4116:
4115:
4087:
4085:
4084:
4079:
4074:
4066:
4061:
4060:
4051:
4050:
4049:
4048:
4037:
4032:
4027:
4016:
4015:
4006:
4005:
4004:
4003:
3992:
3987:
3982:
3899:
3897:
3896:
3891:
3886:
3882:
3881:
3877:
3876:
3875:
3866:
3865:
3864:
3863:
3852:
3847:
3842:
3823:
3819:
3818:
3817:
3808:
3807:
3806:
3805:
3794:
3789:
3784:
3766:
3765:
3738:
3736:
3735:
3734:
3715:
3714:
3713:
3697:
3695:
3694:
3693:
3692:
3681:
3676:
3671:
3663:
3661:
3653:
3652:
3643:
3620:
3619:
3544:
3542:
3541:
3536:
3531:
3527:
3526:
3525:
3516:
3515:
3514:
3513:
3502:
3497:
3492:
3476:
3475:
3466:
3462:
3461:
3460:
3451:
3450:
3449:
3448:
3437:
3432:
3427:
3411:
3409:
3408:
3407:
3398:
3397:
3381:
3380:
3379:
3363:
3361:
3360:
3359:
3358:
3347:
3342:
3337:
3326:
3324:
3320:
3319:
3310:
3309:
3285:
3284:
3275:
3274:
3255:
3251:
3250:
3234:
3232:
3231:
3230:
3229:
3218:
3213:
3208:
3173:
3171:
3170:
3165:
3163:
3160:
3159:
3158:
3148:
3139:
3135:
3134:
3124:
3121:
3116:
3101:
3099:
3091:
3090:
3089:
3088:
3070:
3069:
3049:
3031:
3029:
3028:
3023:
3018:
3017:
2965:
2963:
2962:
2957:
2951:
2950:
2949:
2939:
2926:
2925:
2924:
2914:
2905:
2904:
2895:
2894:
2876:
2875:
2874:
2873:
2862:
2857:
2852:
2834:
2832:
2831:
2826:
2823:
2822:
2821:
2811:
2798:
2797:
2796:
2786:
2735:
2733:
2732:
2727:
2725:
2723:
2719:
2718:
2709:
2708:
2684:
2683:
2674:
2673:
2654:
2650:
2649:
2633:
2631:
2630:
2629:
2628:
2617:
2612:
2607:
2563:
2561:
2560:
2555:
2546:
2541:
2531:
2526:
2514:
2513:
2504:
2503:
2481:
2479:
2478:
2473:
2468:
2467:
2458:
2457:
2439:
2418:
2416:
2415:
2410:
2402:
2394:
2393:
2382:
2377:
2372:
2366:
2365:
2350:
2349:
2344:
2343:
2333:
2332:
2320:
2319:
2300:
2298:
2297:
2292:
2290:
2289:
2284:
2283:
2236:
2234:
2233:
2228:
2226:
2225:
2220:
2219:
2205:
2203:
2202:
2197:
2195:
2194:
2183:
2178:
2173:
2163:
2161:
2160:
2155:
2153:
2152:
2147:
2146:
2130:
2129:
2124:
2123:
2103:
2101:
2100:
2095:
2090:
2089:
2080:
2079:
2052:
2051:
2039:
2038:
2017:
2016:
1988:
1986:
1985:
1980:
1972:
1964:
1963:
1952:
1947:
1942:
1936:
1935:
1911:
1910:
1898:
1897:
1846:
1844:
1843:
1838:
1836:
1835:
1824:
1819:
1814:
1775:
1773:
1772:
1767:
1765:
1764:
1753:
1748:
1743:
1733:The moduli space
1690:
1688:
1687:
1682:
1680:
1679:
1653:
1651:
1650:
1645:
1640:
1639:
1606:
1605:
1582:
1580:
1579:
1574:
1572:
1571:
1523:
1521:
1520:
1515:
1513:
1512:
1511:
1510:
1492:
1491:
1452:
1450:
1449:
1444:
1442:
1441:
1423:
1422:
1400:
1398:
1397:
1392:
1387:
1386:
1374:
1373:
1335:
1333:
1332:
1327:
1316:
1315:
1297:
1296:
1196:
1194:
1193:
1188:
1186:
1185:
1184:
1183:
1165:
1164:
1118:
1116:
1115:
1110:
1108:
1107:
1106:
1105:
1087:
1086:
1003:
1001:
1000:
995:
993:
992:
981:
976:
971:
960:
958:
957:
952:
926:
925:
899:
897:
896:
891:
883:
882:
864:
863:
840:
838:
837:
832:
827:
826:
808:
807:
777:
775:
774:
769:
767:
766:
748:
747:
729:
727:
726:
721:
701:
699:
698:
693:
669:
667:
666:
661:
656:
654:
650:
649:
640:
639:
615:
614:
605:
604:
585:
581:
580:
564:
562:
561:
560:
559:
548:
543:
538:
530:
528:
524:
523:
512:
511:
510:
500:
491:
488:
483:
468:
465:
461:
460:
442:
441:
429:
426:
420:
412:
406:
405:
404:
403:
385:
384:
351:
349:
348:
343:
341:
333:
328:
301:
299:
298:
293:
291:
290:
272:
271:
243:
241:
240:
235:
227:
219:
218:
213:
193:
191:
190:
185:
183:
182:
181:
180:
162:
161:
111:
109:
108:
103:
101:
100:
52:
41:
21:
5891:
5890:
5886:
5885:
5884:
5882:
5881:
5880:
5851:
5850:
5849:
5840:
5812:
5803:Delta invariant
5781:
5763:
5732:
5696:
5657:Abel–Jacobi map
5638:
5612:
5608:Torelli theorem
5578:Dessin d'enfant
5558:Belyi's theorem
5544:
5530:Plücker formula
5461:
5452:Hurwitz surface
5421:
5400:
5334:
5308:Analytic theory
5300:Elliptic curves
5294:
5275:Projective line
5262:Rational curves
5256:
5250:
5184:
5145:
5098:
5051:
5004:
4957:
4954:
4922:
4903:
4888:
4883:
4882:
4859:
4854:
4827:
4822:
4811:
4810:
4758:
4753:
4752:
4725:
4706:
4691:
4686:
4685:
4644:
4625:
4614:
4609:
4608:
4602:
4593:
4572:
4566:at the node of
4565:
4558:
4529:
4511:
4468:
4449:
4434:
4423:
4422:
4387:
4368:
4363:
4362:
4315:
4296:
4281:
4276:
4275:
4272:
4231:
4226:
4225:
4222:Kazarian (2009)
4205:
4196:
4180:
4145:
4126:
4122:
4101:
4096:
4095:
4052:
4025:
4020:
4007:
3980:
3975:
3970:
3969:
3961:
3954:
3947:
3940:
3929:
3912:
3867:
3840:
3835:
3834:
3830:
3809:
3782:
3777:
3776:
3772:
3771:
3767:
3757:
3726:
3716:
3705:
3698:
3669:
3664:
3654:
3644:
3605:
3600:
3599:
3567:
3551:
3517:
3490:
3485:
3484:
3480:
3467:
3452:
3425:
3420:
3419:
3415:
3399:
3389:
3382:
3371:
3364:
3335:
3330:
3311:
3301:
3276:
3266:
3256:
3242:
3235:
3206:
3201:
3196:
3195:
3150:
3126:
3125:
3092:
3080:
3061:
3050:
3043:
3042:
3009:
2974:
2973:
2941:
2916:
2896:
2886:
2850:
2845:
2840:
2839:
2813:
2788:
2773:
2772:
2754:
2745:
2710:
2700:
2675:
2665:
2655:
2641:
2634:
2605:
2600:
2595:
2594:
2505:
2495:
2484:
2483:
2459:
2449:
2427:
2426:
2370:
2357:
2337:
2324:
2311:
2306:
2305:
2277:
2272:
2271:
2269:
2256:
2247:
2213:
2208:
2207:
2206:. The fiber of
2171:
2166:
2165:
2140:
2117:
2112:
2111:
2081:
2071:
2043:
2030:
2008:
1997:
1996:
1940:
1924:
1902:
1889:
1884:
1883:
1866:
1857:
1812:
1807:
1806:
1741:
1736:
1735:
1729:
1665:
1660:
1659:
1616:
1597:
1589:
1588:
1557:
1552:
1551:
1539:
1530:
1502:
1483:
1472:
1467:
1466:
1433:
1414:
1409:
1408:
1378:
1365:
1360:
1359:
1354:
1345:
1307:
1288:
1280:
1279:
1274:
1265:
1254:
1245:
1239:
1219:
1210:
1175:
1156:
1145:
1140:
1139:
1133:
1097:
1078:
1067:
1062:
1061:
1048:
969:
964:
963:
917:
903:
902:
874:
855:
847:
846:
818:
799:
782:
781:
758:
739:
734:
733:
706:
705:
678:
677:
641:
631:
606:
596:
586:
572:
565:
536:
531:
515:
514:
502:
452:
433:
421:
413:
395:
376:
365:
360:
359:
316:
315:
282:
263:
258:
257:
208:
203:
202:
172:
153:
142:
137:
136:
127:
92:
87:
86:
59:Alek Vainshtein
55:Michael Shapiro
46:
35:
33:Torsten Ekedahl
23:
22:
15:
12:
11:
5:
5889:
5887:
5879:
5878:
5873:
5868:
5863:
5853:
5852:
5846:
5845:
5842:
5841:
5839:
5838:
5833:
5828:
5822:
5820:
5818:Vector bundles
5814:
5813:
5811:
5810:
5805:
5800:
5795:
5790:
5785:
5779:
5773:
5771:
5765:
5764:
5762:
5761:
5756:
5751:
5746:
5740:
5738:
5734:
5733:
5731:
5730:
5725:
5720:
5715:
5710:
5704:
5702:
5698:
5697:
5695:
5694:
5689:
5684:
5679:
5674:
5669:
5664:
5659:
5653:
5651:
5644:
5640:
5639:
5637:
5636:
5631:
5626:
5620:
5618:
5614:
5613:
5611:
5610:
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5565:
5560:
5554:
5552:
5546:
5545:
5543:
5542:
5537:
5532:
5527:
5522:
5517:
5512:
5507:
5502:
5497:
5492:
5487:
5482:
5477:
5471:
5469:
5463:
5462:
5460:
5459:
5454:
5449:
5444:
5439:
5433:
5431:
5427:
5426:
5423:
5422:
5420:
5419:
5414:
5408:
5406:
5402:
5401:
5399:
5398:
5393:
5388:
5383:
5378:
5373:
5368:
5363:
5358:
5353:
5348:
5342:
5340:
5336:
5335:
5333:
5332:
5327:
5322:
5317:
5311:
5309:
5302:
5296:
5295:
5293:
5292:
5287:
5285:Riemann sphere
5282:
5277:
5272:
5266:
5264:
5258:
5257:
5251:
5249:
5248:
5241:
5234:
5226:
5220:
5219:
5197:(3): 509–578.
5182:
5143:
5096:
5060:(3): 345–364.
5049:
5013:(2): 297–327.
5002:
4953:
4950:
4929:
4925:
4921:
4918:
4915:
4910:
4906:
4902:
4899:
4893:
4866:
4862:
4857:
4853:
4850:
4847:
4844:
4841:
4834:
4830:
4825:
4821:
4818:
4780:
4777:
4774:
4768:
4764:
4732:
4728:
4724:
4721:
4718:
4713:
4709:
4705:
4702:
4696:
4651:
4647:
4643:
4640:
4637:
4632:
4628:
4624:
4621:
4617:
4598:
4591:
4570:
4563:
4554:
4525:
4509:
4475:
4471:
4467:
4464:
4461:
4456:
4452:
4448:
4445:
4439:
4433:
4430:
4394:
4390:
4386:
4383:
4380:
4375:
4371:
4322:
4318:
4314:
4311:
4308:
4303:
4299:
4295:
4292:
4286:
4271:
4268:
4253:
4250:
4247:
4241:
4237:
4201:
4194:
4190:proved it for
4179:
4176:
4175:
4174:
4163:
4158:
4152:
4148:
4144:
4141:
4138:
4133:
4129:
4125:
4119:
4114:
4111:
4108:
4104:
4089:
4088:
4077:
4072:
4069:
4064:
4059:
4055:
4047:
4044:
4041:
4035:
4031:
4023:
4019:
4014:
4010:
4002:
3999:
3996:
3990:
3986:
3978:
3959:
3952:
3945:
3938:
3925:
3907:
3901:
3900:
3889:
3885:
3880:
3874:
3870:
3862:
3859:
3856:
3850:
3846:
3838:
3833:
3829:
3826:
3822:
3816:
3812:
3804:
3801:
3798:
3792:
3788:
3780:
3775:
3770:
3764:
3760:
3756:
3753:
3750:
3747:
3744:
3741:
3733:
3729:
3725:
3722:
3719:
3712:
3708:
3704:
3701:
3691:
3688:
3685:
3679:
3675:
3667:
3660:
3657:
3651:
3647:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3618:
3615:
3612:
3608:
3565:
3550:
3547:
3546:
3545:
3534:
3530:
3524:
3520:
3512:
3509:
3506:
3500:
3496:
3488:
3483:
3479:
3474:
3470:
3465:
3459:
3455:
3447:
3444:
3441:
3435:
3431:
3423:
3418:
3414:
3406:
3402:
3396:
3392:
3388:
3385:
3378:
3374:
3370:
3367:
3357:
3354:
3351:
3345:
3341:
3333:
3329:
3323:
3318:
3314:
3308:
3304:
3300:
3297:
3294:
3291:
3288:
3283:
3279:
3273:
3269:
3265:
3262:
3259:
3254:
3249:
3245:
3241:
3238:
3228:
3225:
3222:
3216:
3212:
3204:
3175:
3174:
3157:
3153:
3147:
3143:
3138:
3133:
3129:
3120:
3115:
3112:
3109:
3105:
3098:
3095:
3087:
3083:
3079:
3076:
3073:
3068:
3064:
3060:
3057:
3053:
3033:
3032:
3021:
3016:
3012:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2967:
2966:
2955:
2948:
2944:
2938:
2934:
2930:
2923:
2919:
2913:
2909:
2903:
2899:
2893:
2889:
2885:
2882:
2879:
2872:
2869:
2866:
2860:
2856:
2848:
2820:
2816:
2810:
2806:
2802:
2795:
2791:
2785:
2781:
2750:
2743:
2737:
2736:
2722:
2717:
2713:
2707:
2703:
2699:
2696:
2693:
2690:
2687:
2682:
2678:
2672:
2668:
2664:
2661:
2658:
2653:
2648:
2644:
2640:
2637:
2627:
2624:
2621:
2615:
2611:
2603:
2553:
2550:
2545:
2540:
2536:
2530:
2525:
2521:
2517:
2512:
2508:
2502:
2498:
2494:
2491:
2471:
2466:
2462:
2456:
2452:
2448:
2445:
2442:
2438:
2434:
2423:The integrand.
2420:
2419:
2408:
2405:
2401:
2397:
2392:
2389:
2386:
2380:
2376:
2369:
2364:
2360:
2356:
2353:
2348:
2342:
2336:
2331:
2327:
2323:
2318:
2314:
2301:is denoted by
2288:
2282:
2265:
2252:
2245:
2237:over a curve (
2224:
2218:
2193:
2190:
2187:
2181:
2177:
2151:
2145:
2139:
2136:
2133:
2128:
2122:
2108:The ψ-classes.
2105:
2104:
2093:
2088:
2084:
2078:
2074:
2070:
2067:
2064:
2061:
2058:
2055:
2050:
2046:
2042:
2037:
2033:
2029:
2026:
2023:
2020:
2015:
2011:
2007:
2004:
1990:
1989:
1978:
1975:
1971:
1967:
1962:
1959:
1956:
1950:
1946:
1939:
1934:
1931:
1927:
1923:
1920:
1917:
1914:
1909:
1905:
1901:
1896:
1892:
1862:
1855:
1834:
1831:
1828:
1822:
1818:
1763:
1760:
1757:
1751:
1747:
1728:
1725:
1678:
1675:
1672:
1668:
1643:
1638:
1635:
1632:
1629:
1626:
1623:
1619:
1615:
1612:
1609:
1604:
1600:
1596:
1570:
1567:
1564:
1560:
1535:
1528:
1509:
1505:
1501:
1498:
1495:
1490:
1486:
1482:
1479:
1475:
1454:
1453:
1440:
1436:
1432:
1429:
1426:
1421:
1417:
1402:
1401:
1390:
1385:
1381:
1377:
1372:
1368:
1350:
1343:
1337:
1336:
1325:
1322:
1319:
1314:
1310:
1306:
1303:
1300:
1295:
1291:
1287:
1270:
1263:
1250:
1241:
1237:
1215:
1208:
1198:
1197:
1182:
1178:
1174:
1171:
1168:
1163:
1159:
1155:
1152:
1148:
1132:
1129:
1120:
1119:
1104:
1100:
1096:
1093:
1090:
1085:
1081:
1077:
1074:
1070:
1055:
1054:
1044:
1039:
1021:
1020:marked points;
991:
988:
985:
979:
975:
961:
950:
947:
944:
941:
938:
935:
932:
929:
924:
920:
916:
913:
910:
900:
889:
886:
881:
877:
873:
870:
867:
862:
858:
854:
830:
825:
821:
817:
814:
811:
806:
802:
798:
795:
792:
789:
779:
765:
761:
757:
754:
751:
746:
742:
731:
719:
716:
713:
703:
691:
688:
685:
671:
670:
659:
653:
648:
644:
638:
634:
630:
627:
624:
621:
618:
613:
609:
603:
599:
595:
592:
589:
584:
579:
575:
571:
568:
558:
555:
552:
546:
542:
534:
527:
522:
518:
509:
505:
499:
495:
487:
482:
479:
476:
472:
464:
459:
455:
451:
448:
445:
440:
436:
432:
424:
419:
416:
409:
402:
398:
394:
391:
388:
383:
379:
375:
372:
368:
340:
336:
332:
327:
323:
289:
285:
281:
278:
275:
270:
266:
233:
230:
226:
222:
217:
212:
199:Riemann sphere
195:
194:
179:
175:
171:
168:
165:
160:
156:
152:
149:
145:
131:Hurwitz number
126:
123:
99:
95:
24:
18:Hurwitz number
14:
13:
10:
9:
6:
4:
3:
2:
5888:
5877:
5874:
5872:
5869:
5867:
5866:Moduli theory
5864:
5862:
5859:
5858:
5856:
5837:
5834:
5832:
5829:
5827:
5824:
5823:
5821:
5819:
5815:
5809:
5806:
5804:
5801:
5799:
5796:
5794:
5791:
5789:
5786:
5784:
5782:
5775:
5774:
5772:
5770:
5769:Singularities
5766:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5741:
5739:
5735:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5705:
5703:
5699:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5675:
5673:
5670:
5668:
5665:
5663:
5660:
5658:
5655:
5654:
5652:
5648:
5645:
5641:
5635:
5632:
5630:
5627:
5625:
5622:
5621:
5619:
5617:Constructions
5615:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5588:Klein quartic
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5568:Bolza surface
5566:
5564:
5563:Bring's curve
5561:
5559:
5556:
5555:
5553:
5551:
5547:
5541:
5538:
5536:
5533:
5531:
5528:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5495:Conic section
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5475:AF+BG theorem
5473:
5472:
5470:
5468:
5464:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5434:
5432:
5428:
5418:
5415:
5413:
5410:
5409:
5407:
5403:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5344:
5343:
5341:
5337:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5312:
5310:
5306:
5303:
5301:
5297:
5291:
5290:Twisted cubic
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5267:
5265:
5263:
5259:
5255:
5247:
5242:
5240:
5235:
5233:
5228:
5227:
5224:
5215:
5210:
5205:
5200:
5196:
5192:
5188:
5183:
5178:
5173:
5168:
5163:
5159:
5155:
5154:
5149:
5144:
5140:
5136:
5132:
5128:
5124:
5120:
5115:
5110:
5106:
5102:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5068:
5063:
5059:
5055:
5050:
5046:
5042:
5038:
5034:
5030:
5026:
5021:
5016:
5012:
5008:
5003:
4999:
4995:
4991:
4987:
4983:
4979:
4974:
4969:
4965:
4961:
4956:
4955:
4951:
4949:
4946:
4927:
4923:
4919:
4916:
4913:
4908:
4904:
4900:
4897:
4864:
4860:
4855:
4848:
4845:
4842:
4839:
4832:
4828:
4823:
4816:
4808:
4804:
4800:
4796:
4778:
4775:
4772:
4750:
4730:
4726:
4722:
4719:
4716:
4711:
4707:
4703:
4700:
4682:
4680:
4676:
4672:
4667:
4649:
4645:
4641:
4638:
4635:
4630:
4626:
4622:
4619:
4615:
4606:
4601:
4597:
4590:
4586:
4582:
4577:
4574:
4569:
4562:
4557:
4553:
4549:
4545:
4541:
4537:
4533:
4528:
4524:
4520:
4516:
4512:
4505:
4501:
4497:
4493:
4473:
4469:
4465:
4462:
4459:
4454:
4450:
4446:
4443:
4431:
4428:
4420:
4416:
4413:
4408:
4392:
4388:
4384:
4381:
4378:
4373:
4369:
4360:
4356:
4352:
4348:
4344:
4341:from a genus
4340:
4320:
4316:
4312:
4309:
4306:
4301:
4297:
4293:
4290:
4270:Idea of proof
4269:
4267:
4251:
4248:
4245:
4223:
4219:
4217:
4213:
4209:
4204:
4200:
4193:
4189:
4185:
4177:
4161:
4156:
4150:
4146:
4139:
4136:
4131:
4127:
4117:
4112:
4109:
4106:
4102:
4094:
4093:
4092:
4075:
4070:
4067:
4062:
4057:
4053:
4045:
4042:
4039:
4021:
4017:
4012:
4008:
4000:
3997:
3994:
3976:
3968:
3967:
3966:
3963:
3958:
3951:
3944:
3937:
3933:
3928:
3924:
3920:
3916:
3911:
3906:
3887:
3883:
3878:
3872:
3868:
3860:
3857:
3854:
3836:
3831:
3827:
3824:
3820:
3814:
3810:
3802:
3799:
3796:
3778:
3773:
3768:
3762:
3758:
3751:
3748:
3745:
3739:
3731:
3727:
3723:
3720:
3717:
3710:
3706:
3702:
3699:
3689:
3686:
3683:
3665:
3658:
3655:
3649:
3645:
3639:
3633:
3630:
3627:
3621:
3616:
3613:
3610:
3606:
3598:
3597:
3596:
3593:
3591:
3587:
3583:
3579:
3575:
3571:
3564:
3560:
3556:
3548:
3532:
3528:
3522:
3518:
3510:
3507:
3504:
3486:
3481:
3477:
3472:
3468:
3463:
3457:
3453:
3445:
3442:
3439:
3421:
3416:
3412:
3404:
3400:
3394:
3390:
3386:
3383:
3376:
3372:
3368:
3365:
3355:
3352:
3349:
3331:
3327:
3316:
3312:
3306:
3302:
3298:
3295:
3289:
3281:
3277:
3271:
3267:
3263:
3260:
3247:
3243:
3236:
3226:
3223:
3220:
3202:
3194:
3193:
3192:
3190:
3186:
3182:
3178:
3155:
3151:
3145:
3141:
3136:
3131:
3127:
3118:
3113:
3110:
3107:
3103:
3096:
3093:
3085:
3081:
3077:
3074:
3071:
3066:
3062:
3058:
3055:
3051:
3041:
3040:
3039:
3037:
3019:
3014:
3010:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2972:
2971:
2970:
2953:
2946:
2942:
2936:
2932:
2928:
2921:
2917:
2911:
2907:
2901:
2897:
2891:
2883:
2880:
2870:
2867:
2864:
2846:
2838:
2837:
2836:
2818:
2814:
2808:
2804:
2800:
2793:
2789:
2783:
2779:
2770:
2766:
2762:
2758:
2753:
2749:
2742:
2715:
2711:
2705:
2701:
2697:
2694:
2688:
2680:
2676:
2670:
2666:
2662:
2659:
2646:
2642:
2635:
2625:
2622:
2619:
2601:
2593:
2592:
2591:
2589:
2585:
2583:
2579:
2575:
2571:
2567:
2551:
2548:
2543:
2538:
2534:
2528:
2523:
2519:
2515:
2510:
2506:
2500:
2496:
2492:
2489:
2464:
2460:
2454:
2450:
2446:
2443:
2436:
2432:
2425:The fraction
2424:
2406:
2395:
2390:
2387:
2384:
2362:
2358:
2354:
2346:
2329:
2325:
2321:
2316:
2312:
2304:
2303:
2302:
2286:
2268:
2264:
2260:
2255:
2251:
2244:
2240:
2222:
2191:
2188:
2185:
2149:
2137:
2134:
2131:
2126:
2109:
2091:
2086:
2082:
2076:
2068:
2065:
2059:
2056:
2053:
2048:
2044:
2040:
2035:
2031:
2027:
2024:
2021:
2013:
2009:
2002:
1995:
1994:
1993:
1976:
1965:
1960:
1957:
1954:
1932:
1929:
1925:
1921:
1915:
1907:
1903:
1899:
1894:
1890:
1882:
1881:
1880:
1878:
1874:
1870:
1865:
1861:
1854:
1850:
1832:
1829:
1826:
1804:
1800:
1797:
1796:
1790:
1788:
1784:
1780:
1776:
1761:
1758:
1755:
1734:
1726:
1724:
1722:
1718:
1714:
1708:
1706:
1702:
1698:
1694:
1676:
1673:
1670:
1666:
1657:
1636:
1633:
1630:
1627:
1624:
1621:
1617:
1613:
1610:
1607:
1602:
1598:
1586:
1568:
1565:
1562:
1558:
1549:
1545:
1543:
1540:) divided by
1538:
1534:
1527:
1507:
1503:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1473:
1465:
1461:
1459:
1438:
1434:
1430:
1427:
1424:
1419:
1415:
1407:
1406:
1405:
1388:
1383:
1379:
1375:
1370:
1366:
1358:
1357:
1356:
1353:
1349:
1342:
1320:
1317:
1312:
1308:
1304:
1301:
1298:
1293:
1289:
1278:
1277:
1276:
1273:
1269:
1262:
1258:
1253:
1249:
1244:
1235:
1231:
1227:
1223:
1218:
1214:
1207:
1203:
1180:
1176:
1172:
1169:
1166:
1161:
1157:
1153:
1150:
1146:
1138:
1137:
1136:
1130:
1128:
1126:
1125:Kazarian 2009
1102:
1098:
1094:
1091:
1088:
1083:
1079:
1075:
1072:
1068:
1060:
1059:
1058:
1052:
1047:
1043:
1040:
1037:
1033:
1029:
1025:
1022:
1019:
1015:
1011:
1010:stable curves
1007:
989:
986:
983:
962:
948:
945:
942:
939:
936:
933:
930:
927:
922:
918:
914:
911:
908:
901:
887:
879:
875:
871:
868:
865:
860:
856:
844:
823:
819:
815:
812:
809:
804:
800:
793:
790:
780:
763:
759:
755:
752:
749:
744:
740:
732:
717:
714:
711:
704:
689:
686:
683:
676:
675:
674:
657:
646:
642:
636:
632:
628:
625:
619:
611:
607:
601:
597:
593:
590:
577:
573:
566:
556:
553:
550:
532:
525:
520:
516:
507:
503:
497:
493:
485:
480:
477:
474:
470:
457:
453:
449:
446:
443:
438:
434:
417:
414:
407:
400:
396:
392:
389:
386:
381:
377:
373:
370:
366:
358:
357:
356:
353:
334:
325:
321:
313:
309:
308:branch points
305:
287:
283:
279:
276:
273:
268:
264:
255:
251:
247:
215:
200:
177:
173:
169:
166:
163:
158:
154:
150:
147:
143:
135:
134:
133:
132:
124:
122:
120:
115:
113:
97:
93:
83:
79:
75:
70:
68:
64:
60:
56:
50:
45:
39:
34:
30:
19:
5871:Permutations
5777:
5754:Prym variety
5728:Stable curve
5718:Hodge bundle
5708:ELSV formula
5707:
5510:Fermat curve
5467:Plane curves
5430:Higher genus
5405:Applications
5330:Modular form
5204:math/0009097
5194:
5190:
5157:
5151:
5114:math/0003028
5107:(1): 25–36.
5104:
5100:
5067:math/9905104
5057:
5053:
5020:math/0004096
5010:
5006:
4973:math/9902104
4963:
4959:
4947:
4806:
4802:
4798:
4794:
4748:
4683:
4678:
4674:
4670:
4668:
4604:
4599:
4595:
4588:
4584:
4580:
4578:
4575:
4567:
4560:
4555:
4551:
4547:
4543:
4539:
4535:
4531:
4526:
4522:
4518:
4514:
4507:
4503:
4499:
4495:
4491:
4418:
4414:
4411:
4409:
4358:
4357:has exactly
4354:
4353:) such that
4350:
4346:
4342:
4338:
4273:
4220:
4202:
4198:
4191:
4181:
4090:
3964:
3956:
3949:
3942:
3935:
3931:
3926:
3922:
3918:
3914:
3909:
3904:
3902:
3594:
3589:
3585:
3581:
3577:
3573:
3569:
3562:
3558:
3554:
3552:
3188:
3184:
3180:
3179:
3176:
3035:
3034:
2968:
2768:
2764:
2760:
2756:
2751:
2747:
2740:
2738:
2587:
2586:
2581:
2577:
2573:
2569:
2565:
2422:
2421:
2266:
2262:
2258:
2253:
2249:
2242:
2238:
2107:
2106:
1991:
1876:
1868:
1863:
1859:
1852:
1848:
1802:
1801:is the rank
1798:
1795:Hodge bundle
1792:
1791:
1786:
1782:
1777:is a smooth
1732:
1730:
1720:
1716:
1709:
1704:
1700:
1696:
1692:
1655:
1584:
1547:
1546:
1541:
1536:
1532:
1525:
1463:
1462:
1455:
1403:
1351:
1347:
1340:
1338:
1271:
1267:
1260:
1256:
1251:
1247:
1242:
1233:
1229:
1225:
1221:
1216:
1212:
1205:
1201:
1199:
1134:
1121:
1057:The numbers
1056:
1050:
1045:
1041:
1031:
1023:
1017:
1013:
1006:moduli space
842:
672:
354:
311:
306:more simple
303:
249:
245:
196:
130:
128:
116:
71:
44:Sergei Lando
29:ELSV formula
28:
26:
5783:singularity
5629:Polar curve
5160:(1): 1–21.
4603:) at ∞ and
4421:assigns to
1550:The number
1464:Definition.
1036:Chern class
129:Define the
125:The formula
112:-conjecture
47: [
36: [
5855:Categories
5624:Dual curve
5252:Topics in
4952:References
3572:. We have
3191:= 1. Then
3181:Example B.
1548:Example A.
1458:transitive
1236:− 2, let τ
1034:the total
84:, and the
5737:Morphisms
5485:Bitangent
5167:0809.3263
4917:…
4852:↦
4843:…
4820:↦
4767:¯
4720:…
4639:…
4463:…
4432:∈
4382:…
4345:curve to
4310:…
4240:¯
4216:Li (2001)
4137:−
4054:λ
4034:¯
4022:∫
4009:ψ
3989:¯
3977:∫
3948:), while
3913:equals 1/
3869:λ
3849:¯
3837:∫
3828:−
3811:ψ
3791:¯
3779:∫
3728:ψ
3721:−
3707:λ
3703:−
3678:¯
3666:∫
3519:λ
3499:¯
3487:∫
3478:−
3454:ψ
3434:¯
3422:∫
3401:ψ
3387:−
3373:λ
3369:−
3344:¯
3332:∫
3313:ψ
3299:−
3290:⋯
3278:ψ
3264:−
3248:∗
3215:¯
3203:∫
3104:∏
3075:…
3007:∑
3004:−
2992:−
2933:ψ
2929:⋯
2908:ψ
2898:λ
2881:−
2859:¯
2847:∫
2801:⋯
2712:ψ
2698:−
2689:⋯
2677:ψ
2663:−
2647:∗
2614:¯
2602:∫
2552:⋯
2535:ψ
2507:ψ
2461:ψ
2447:−
2379:¯
2355:∈
2313:ψ
2180:¯
2135:…
2083:λ
2066:−
2057:⋯
2054:−
2045:λ
2032:λ
2028:−
2014:∗
1949:¯
1922:∈
1891:λ
1821:¯
1750:¯
1713:monodromy
1634:−
1618:τ
1611:…
1599:τ
1497:…
1435:τ
1428:…
1416:τ
1389:σ
1380:τ
1376:⋯
1367:τ
1321:σ
1309:τ
1302:…
1290:τ
1170:…
1092:…
1012:of genus
978:¯
943:−
915:∑
869:…
813:…
794:
788:#
753:…
715:≥
687:≥
643:ψ
629:−
620:⋯
608:ψ
594:−
578:∗
545:¯
533:∫
471:∏
447:…
423:#
390:…
277:…
167:…
94:λ
5139:15706096
5045:10881259
4998:15218497
4197:= ... =
1992:We have
1240:, ..., τ
1211:+ ... +
5808:Tacnode
5793:Crunode
5119:Bibcode
5092:1124032
5072:Bibcode
5025:Bibcode
4978:Bibcode
4594:, ...,
4417:or the
4178:History
3549:Example
3036:Remark.
2969:where
2835:equals
2746:, ...,
2248:, ...,
1867:) with
1858:, ...,
1531:, ...,
1346:, ...,
1275:. Then
1266:, ...,
1026:is the
1004:is the
845:-tuple
248:, with
5788:Acnode
5701:Moduli
5137:
5090:
5043:
4996:
3588:− 2 =
2767:− 3 +
2759:− 3 +
2580:− 3 +
2568:− 3 +
1785:− 3 +
80:, the
5199:arXiv
5162:arXiv
5135:S2CID
5109:arXiv
5088:S2CID
5062:arXiv
5041:S2CID
5015:arXiv
4994:S2CID
4968:arXiv
4546:. As
3592:+ 1.
2763:and 2
2164:over
1691:is 1/
1583:is 1/
1032:c(E*)
1016:with
51:]
40:]
5798:Cusp
4538:) =
4410:The
4274:Let
3953:1, 2
3939:1, 1
3553:Let
3183:Let
1793:The
1220:and
1030:and
302:and
5209:doi
5172:doi
5158:221
5127:doi
5105:135
5080:doi
5058:130
5033:doi
5011:146
4986:doi
4964:328
3908:1,
3584:+ 2
3568:by
2261:at
1875:on
1703:+ 2
1544:!.
1456:is
1232:+ 2
1127:).
1008:of
791:Aut
427:Aut
5857::
5207:.
5195:57
5193:.
5189:.
5170:.
5156:.
5150:.
5133:.
5125:.
5117:.
5103:.
5086:.
5078:.
5070:.
5056:.
5039:.
5031:.
5023:.
5009:.
4992:.
4984:.
4976:.
4962:.
4749:br
4681:.
4671:br
4666:.
4573:.
4542:+
4534:,
4517:=
4515:xy
4415:br
4407:.
4157:24
4071:24
3580:+
3576:=
3557:=
3187:=
2241:,
1851:,
1460:.
1228:+
1224:=
1204:=
352:.
201:,
121:.
114:.
69:.
57:,
53:,
49:ru
42:,
38:sv
5780:k
5778:A
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5217:.
5211::
5201::
5180:.
5174::
5164::
5141:.
5129::
5121::
5111::
5094:.
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5064::
5047:.
5035::
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5000:.
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4162:.
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4128:k
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4118:=
4113:k
4110:;
4107:1
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4076:.
4068:1
4063:=
4058:1
4046:1
4043:,
4040:1
4030:M
4018:=
4013:1
4001:1
3998:,
3995:1
3985:M
3960:2
3957:S
3950:h
3946:1
3943:S
3936:h
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3927:k
3923:S
3919:k
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3888:.
3884:}
3879:]
3873:1
3861:1
3858:,
3855:1
3845:M
3832:[
3825:k
3821:]
3815:1
3803:1
3800:,
3797:1
3787:M
3774:[
3769:{
3763:k
3759:k
3755:)
3752:1
3749:+
3746:k
3743:(
3740:=
3732:1
3724:k
3718:1
3711:1
3700:1
3690:1
3687:,
3684:1
3674:M
3659:!
3656:k
3650:k
3646:k
3640:!
3637:)
3634:1
3631:+
3628:k
3625:(
3622:=
3617:k
3614:;
3611:1
3607:h
3590:k
3586:g
3582:n
3578:K
3574:m
3570:k
3566:1
3563:k
3559:g
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3533:.
3529:]
3523:1
3511:1
3508:,
3505:1
3495:M
3482:[
3473:1
3469:k
3464:]
3458:1
3446:1
3443:,
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3417:[
3413:=
3405:1
3395:1
3391:k
3384:1
3377:1
3366:1
3356:1
3353:,
3350:1
3340:M
3328:=
3322:)
3317:n
3307:n
3303:k
3296:1
3293:(
3287:)
3282:1
3272:1
3268:k
3261:1
3258:(
3253:)
3244:E
3240:(
3237:c
3227:n
3224:,
3221:g
3211:M
3189:n
3185:g
3156:i
3152:k
3146:i
3142:k
3137:!
3132:i
3128:k
3119:n
3114:1
3111:=
3108:i
3097:!
3094:m
3086:n
3082:k
3078:,
3072:,
3067:1
3063:k
3059:;
3056:g
3052:h
3020:.
3015:i
3011:d
3001:n
2998:+
2995:3
2989:g
2986:3
2983:=
2980:j
2954:,
2947:n
2943:d
2937:n
2922:1
2918:d
2912:1
2902:j
2892:j
2888:)
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2878:(
2871:n
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2865:g
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2761:n
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2748:k
2744:1
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2721:)
2716:n
2706:n
2702:k
2695:1
2692:(
2686:)
2681:1
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2660:1
2657:(
2652:)
2643:E
2639:(
2636:c
2626:n
2623:,
2620:g
2610:M
2582:n
2578:g
2574:n
2570:n
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2549:+
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2441:(
2437:/
2433:1
2407:.
2404:)
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2375:M
2368:(
2363:2
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2352:)
2347:i
2341:L
2335:(
2330:1
2326:c
2322:=
2317:i
2287:i
2281:L
2267:i
2263:x
2259:C
2254:n
2250:x
2246:1
2243:x
2239:C
2223:i
2217:L
2192:n
2189:,
2186:g
2176:M
2150:n
2144:L
2138:,
2132:,
2127:1
2121:L
2092:.
2087:g
2077:g
2073:)
2069:1
2063:(
2060:+
2049:2
2041:+
2036:1
2025:1
2022:=
2019:)
2010:E
2006:(
2003:c
1977:.
1974:)
1970:Q
1966:,
1961:n
1958:,
1955:g
1945:M
1938:(
1933:j
1930:2
1926:H
1919:)
1916:E
1913:(
1908:j
1904:c
1900:=
1895:j
1877:C
1869:n
1864:n
1860:x
1856:1
1853:x
1849:C
1833:n
1830:,
1827:g
1817:M
1803:g
1799:E
1787:n
1783:g
1762:n
1759:,
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1721:K
1717:K
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1642:)
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1595:(
1585:k
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1500:,
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1272:n
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1024:E
1018:n
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990:n
987:,
984:g
974:M
949:;
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940:g
937:2
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919:k
912:=
909:m
888:;
885:)
880:n
876:k
872:,
866:,
861:1
857:k
853:(
843:n
829:)
824:n
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658:.
652:)
647:n
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574:E
570:(
567:c
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526:!
521:i
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478:=
475:i
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418:!
415:m
408:=
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371:g
367:h
339:|
335:G
331:|
326:/
322:1
312:G
304:m
288:n
284:k
280:,
274:,
269:1
265:k
250:n
246:g
232:)
229:)
225:C
221:(
216:1
211:P
178:n
174:k
170:,
164:,
159:1
155:k
151:;
148:g
144:h
98:g
20:)
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