33:
5099:
4884:
550:
1613:
One can also consider the coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was introduced. In fact, the idea of a moduli stack was introduced by
Deligne and Mumford in an attempt to
1953:
5168:
One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points, pairwise distinct and distinct from the nodes. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli
5094:{\displaystyle {\begin{aligned}\partial {\overline {\mathcal {M}}}_{2}&=\Delta _{0}^{*}\coprod \Delta _{1}^{*}\\&={\overline {\mathcal {M}}}_{1,2}/(\mathbb {Z} /2)\coprod ({\overline {\mathcal {M}}}_{1}\times {\overline {\mathcal {M}}}_{1})/(\mathbb {Z} /2)\end{aligned}}}
2620:
4758:
395:. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of
4499:
3662:
4410:
3135:
432:
2504:
4631:
3394:
1794:
3777:
3969:
1159:
1374:
5611:
748:
2512:
5452:
3494:
4134:
4294:
5780:
3844:
5158:
5827:
4638:
2671:
4889:
1799:
437:
2134:
5541:
5400:
5326:
5246:
3567:
This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus. The non-hyperelliptic curves are all given by plane curves of degree 4 (using the
2707:
4255:
1286:
1018:
306:
2308:
2987:
2740:
3037:
2045:
783:
3557:
2825:
2421:
The genus 1 case is one of the first well-understood cases of moduli spaces, at least over the complex numbers, because isomorphism classes of elliptic curves are classified by the
5501:
of the corresponding component, edges correspond to nodes of the curve and the half-edges correspond to the markings. The closure of the locus of curves with a given dual graph in
5204:
2862:
2405:
858:
6115:
2753:
Note that most authors consider the case of genus one curves with one marked point as the origin of the group since otherwise the stabilizer group in a hypothetical moduli space
4211:
4168:
2782:
2368:
2188:
1576:
1049:
956:
631:
584:
424:
240:
4422:
4005:
2337:
2266:
2237:
1727:
4539:
1779:
3171:
6138:
Girard, Martine; Kohel, David R. (2006), Hess, Florian; Pauli, Sebastian; Pohst, Michael (eds.), "Classification of Genus 3 Curves in
Special Strata of the Moduli Space",
5483:
3577:
3429:
1983:
4322:
2827:
given by the curve, since elliptic curves have an
Abelian group structure. This adds unneeded technical complexity to this hypothetical moduli space. On the other hand,
4817:
1410:
1195:
4031:
3893:
17:
5727:
5694:
4057:
3197:
1228:
5281:
1641:
393:
266:
5646:
5493:
with vertices labelled by nonnegative integers and allowed to have loops, multiple edges and also numbered half-edges. Here the vertices of the graph correspond to
3045:
1545:
1518:
1491:
1464:
1437:
1080:
658:
545:{\displaystyle {\begin{aligned}\dim({\text{space of genus 0 curves}})-\dim({\text{group of automorphisms}})&=0-\dim(\mathrm {PGL} (2))\\&=-3.\end{aligned}}}
367:
338:
3867:
2068:
2010:
4876:
4844:
4778:
4314:
3514:
2208:
2157:
1698:
1674:
2430:
1614:
prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.
925:
4546:
3221:
6791:. IRMA Lectures in Mathematics and Theoretical Physics. Vol. 17. Zürich, Switzerland: European Mathematical Society Publishing House. pp. 667–716.
1948:{\displaystyle {\begin{aligned}H^{1}(C,T_{C})&\cong H^{0}(C,\omega _{C}\otimes T_{C}^{\vee })\\&\cong H^{0}(C,\omega _{C}^{\otimes 2})\end{aligned}}}
272:
if it is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms. The resulting stack is denoted
268:, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is
555:
Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack
6905:
3674:
6294:
6298:
3905:
6021:
203:
proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").
1088:
6644:
5886:
1291:
5782:. Stable curves whose dual graph is a tree are called "compact type" (because the Jacobian is compact) and their moduli space is denoted
2615:{\displaystyle {\mathcal {M}}_{1,1}|_{\mathbb {C} }={\mathcal {M}}_{1,1}\times _{{\text{Spec}}(\mathbb {Z} )}{\text{Spec}}(\mathbb {C} )}
97:
50:
5550:
670:
6782:
Zvonkine, Dimitri (2012). "An introduction to moduli spaces of curves and their intersection theory". In
Papadopoulos, Athanase (ed.).
69:
7328:
5848:
6783:
3434:
76:
4065:
6558:"Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes"
2709:
by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over
7420:
7057:
7017:
6898:
6804:
6765:
6738:
6706:
6511:
6439:
6403:
6173:
5997:
5966:
4753:{\displaystyle \Delta _{g/2}\cong ({\overline {\mathcal {M}}}_{g/2}\times {\overline {\mathcal {M}}}_{g/2})/(\mathbb {Z} /2)}
4260:
5732:
3803:
633:
is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus
83:
5341:
5111:
6878:
5785:
2882:
2625:
6506:. Shafarevich, Igor Rostislavovich, 1923-2017, Artin, Michael, Tate, John Torrence, 1925-2019. Boston: Birkhäuser. 1983.
6498:
5405:
2076:
7486:
7108:
7007:
3849:
and it was long expected this would be true in all genera. In fact, Severi had proved this to be true for genera up to
65:
5504:
5363:
5289:
5209:
7476:
6842:
6602:
6471:
3895:
all such moduli spaces are of general type, meaning they are not unirational. They accomplished this by studying the
2676:
116:
4224:
1233:
275:
7186:
6891:
2274:
5329:
2899:
2712:
2416:
7333:
7254:
7244:
7181:
6837:. Advanced Lectures in Mathematics (ALM). Vol. 24. Somerville, MA: International Press. pp. 293–330.
5613:
of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex
2995:
965:
54:
6931:
2015:
1643:; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.
753:
3519:
2787:
7516:
7151:
7047:
6210:
5490:
6380:
5172:
4494:{\displaystyle \Delta _{h}^{*}\cong {\overline {\mathcal {M}}}_{h}\times {\overline {\mathcal {M}}}_{g-h}}
2830:
2373:
791:
7410:
7374:
7073:
6986:
6086:
3212:
2890:
161:. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding
4185:
4142:
2756:
2342:
2162:
1550:
1023:
962:, Deligne and Mumford show this stack is smooth and use the stack of isomorphisms between stable curves
930:
605:
558:
398:
214:
90:
7022:
6730:
6690:
6535:
6372:
6316:
6500:
Arithmetic and geometry: papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday
3977:
2313:
2242:
2213:
1703:
7430:
3657:{\displaystyle \operatorname {Hilb} _{\mathbb {P} ^{2}}^{8t-4}\cong \mathbb {P} ^{{\binom {6}{4}}-1}}
3208:
2865:
1052:
959:
4504:
1735:
7343:
7323:
7259:
7176:
7078:
7037:
6658:
6321:
4405:{\displaystyle \partial {\overline {\mathcal {M}}}_{g}=\coprod _{0\leq h\leq (g/2)}\Delta _{h}^{*}}
3143:
1986:
7234:
7042:
6557:
3402:
2673:
is just the affine line, but it can be compactified to a stack with underlying topological space
1961:
196:
43:
2889:, so the moduli space can be determined completely from the branch locus of the curve using the
170:
7511:
7027:
6653:
6553:
1598:
1383:
1168:
7141:
6636:
4010:
3872:
3130:{\displaystyle \mathbb {A} ^{3}\setminus (\Delta _{a,b}\cup \Delta _{a,c}\cup \Delta _{b,c}),}
166:
7405:
7103:
7052:
6941:
6040:
5953:, Lecture Notes in Mathematics, vol. 349, Springer Berlin Heidelberg, pp. 143–316,
5699:
5666:
5494:
5457:
4036:
3176:
1597:, follows from a theorem on stable reduction on curves. This can be found using a theorem of
1547:
is irreducible. From the general theory of algebraic stacks, this implies the stack quotient
1200:
5251:
1620:
372:
245:
7481:
7353:
7012:
6852:
6814:
6612:
6330:
6249:
6183:
6153:
5624:
3568:
1523:
1496:
1469:
1442:
1415:
1058:
636:
146:
7264:
6576:
4791:
343:
314:
242:
classifies families of smooth projective curves, together with their isomorphisms. When
8:
7318:
7196:
7161:
7118:
7098:
6823:
5925:
2886:
2499:{\displaystyle j:{\mathcal {M}}_{1,1}|_{\mathbb {C} }\to \mathbb {A} _{\mathbb {C} }^{1}}
188:
184:
6334:
6157:
4626:{\displaystyle \Delta _{0}^{*}\cong {\overline {\mathcal {M}}}_{g-1,2}/(\mathbb {Z} /2)}
3852:
3389:{\displaystyle z^{2}=ax^{6}+bx^{5}y+cx^{4}y^{2}+dx^{3}y^{3}+ex^{2}y^{4}+fxy^{5}+gy^{6},}
2050:
1992:
7448:
7239:
7219:
7032:
6671:
6529:
6477:
6354:
6288:
6253:
6219:
6187:
6143:
6118:
6065:
6015:
5929:
5903:
5881:
4861:
4829:
4763:
4299:
3499:
2193:
2142:
1683:
1677:
1659:
786:
130:
7191:
5928:(2011). "Tautological and non-tautological cohomology of the moduli space of curves".
5729:
and the graph is a tree) are called "rational tail" and their moduli space is denoted
3571:), which are parameterized by the smooth locus in the Hilbert scheme of hypersurfaces
863:
7348:
7295:
7166:
6981:
6976:
6838:
6800:
6761:
6734:
6702:
6616:
6598:
6517:
6507:
6467:
6435:
6419:
6399:
6358:
6346:
6276:
6237:
6169:
6057:
6003:
5993:
5962:
5843:
5838:
3896:
6675:
6463:
6257:
6191:
6142:, vol. 4076, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 346–360,
5907:
7338:
7224:
7201:
6792:
6753:
6663:
6572:
6481:
6459:
6427:
6391:
6338:
6229:
6161:
6049:
5985:
5954:
5895:
5498:
2747:
340:; hence a stable nodal curve can be completely specified by choosing the values of
200:
180:
7453:
7269:
7211:
7113:
6936:
6915:
6848:
6810:
6698:
6608:
6273:
Sulla classificazione delle curve algebriche e sul teorema d'esistenza di
Riemann
6245:
6179:
4849:
A pair of curves of the same genus connected at a double point up to permutation.
2269:
1602:
1520:, all components must be contained in a single component, hence the coarse space
192:
177:
154:
150:
6431:
6395:
5340:
modular forms are sections of line bundles on the stack of elliptic curves with
3496:. Then, the locus of sections which contain no triple root contains every curve
7136:
6961:
6946:
6923:
6628:
6590:
6458:. Proceedings of Symposia in Pure Mathematics. Vol. 80. pp. 125–147.
6312:
6035:
5873:
5544:
2743:
665:
595:
6757:
3772:{\displaystyle {\mathcal {M}}_{3}=\coprod {\mathcal {M}}_{3}^{\mathrm {hyp} }}
7505:
7468:
7249:
7229:
7156:
6951:
6883:
6827:
6632:
6586:
6521:
6376:
6350:
6280:
6241:
6061:
6007:
5877:
5169:
stacks of smooth (or stable) genus g curves with n marked points are denoted
4823:
4179:
1785:
1590:
1586:
599:
6828:"Tautological and non-tautological cohomology of the moduli space of curves"
6620:
6454:
Farkas, Gavril (2009). "The global geometry of the moduli space of curves".
6319:(1987). "The Kodaira dimension of the moduli space of curves of genus ?23".
4178:
This is significant geometrically because it implies any linear system on a
7415:
7389:
7379:
7369:
7171:
6991:
5333:
5105:
Further analysis of these strata can be used to give the generators of the
3964:{\displaystyle \kappa _{g}=\mathrm {Kod} ({\overline {\mathcal {M}}}_{g}),}
1377:
661:
269:
158:
7290:
7128:
6879:"Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology"
6867:
6722:
6233:
3794:
2422:
750:
of tri-canonically embedded curves (from the embedding of the very ample
6165:
2893:. Since an arbitrary genus 2 curve is given by a polynomial of the form
7285:
6796:
6667:
6342:
6205:
6069:
5973:
5958:
5899:
1154:{\displaystyle H_{g}^{o}\coprod H_{g,1}\coprod \cdots \coprod H_{g,n}}
7146:
6748:
Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A. (2011).
6718:
6148:
5106:
3215:, a hyperelliptic curve can be described as a polynomial of the form
1369:{\displaystyle {\mathcal {M}}_{g}^{0}=H_{g}^{0}/\mathrm {PGL} (5g-6)}
6208:(2015). "The Chow ring of the moduli space of curves of genus six".
6053:
1617:
The coarse moduli spaces have the same dimension as the stacks when
32:
6123:
3793:
In all of the previous cases, the moduli spaces can be found to be
1594:
1439:
must contain non-singular curves. Consequently, the singular locus
165:
and the moduli space is different. One also distinguishes between
6224:
5934:
5606:{\displaystyle \prod _{v}{\overline {\mathcal {M}}}_{g_{v},n_{v}}}
5402:
is that their boundary can be described in terms of moduli spaces
743:{\displaystyle \mathrm {Hilb} _{\mathbb {P} ^{5g-5-1}}^{P_{g}(n)}}
7458:
7443:
1605:, and showing its equivalence to the stable reduction of curves.
6752:. Grundlehren der mathematischen Wissenschaften. Vol. 268.
7438:
6270:
5663:
Stable curves whose dual graph contains a vertex labelled by
5485:. Given a marked, stable, nodal curve one can associate its
2239:. The only technical difficulty is the automorphism group of
1466:
is connected, hence it is contained in a single component of
3489:{\displaystyle \Gamma (\mathbb {P} (3,1),{\mathcal {O}}(g))}
6747:
6637:"The irreducibility of the space of curves of given genus"
5882:"The irreducibility of the space of curves of given genus"
4129:{\displaystyle \kappa _{g}=3g-3=\dim({\mathcal {M}}_{g}),}
308:. Both moduli stacks carry universal families of curves.
1412:, none of them would be complete. Also, any component of
6420:"On the Kodaira Dimension of the Moduli Space of Curves"
6381:"On the Kodaira Dimension of the Moduli Space of Curves"
5648:
equal to the number of outgoing edges and half-edges at
5360:
An important property of the compactified moduli spaces
5336:
are sections of line bundles on this stack, and level
4289:{\displaystyle \partial {\overline {\mathcal {M}}}_{g}}
6083:
Larson, Eric (2019-04-17). "The integral Chow ring of
6038:(1960). "Arithmetic Variety of Moduli for Genus Two".
5775:{\displaystyle {\mathcal {M}}_{g,n}^{\mathrm {r.t.} }}
5328:
of genus 1 curves with one marked point. This is the
3839:{\displaystyle \mathbb {P} ^{n}\to {\mathcal {M}}_{g}}
3668:
Then, the moduli space is stratified by the substacks
1656:
Determining the geometry of the moduli space of genus
1646:
6868:"Topology and geometry of the moduli space of curves"
6775:
6426:, New York, NY: Springer New York, pp. 171–234,
6390:, New York, NY: Springer New York, pp. 171–234,
6089:
5788:
5735:
5702:
5669:
5627:
5553:
5507:
5497:
of the nodal curve, the labelling of a vertex is the
5460:
5408:
5366:
5292:
5254:
5212:
5175:
5153:{\displaystyle A^{*}({\overline {\mathcal {M}}}_{2})}
5114:
4887:
4864:
4832:
4794:
4766:
4641:
4549:
4507:
4425:
4325:
4302:
4263:
4227:
4188:
4145:
4068:
4039:
4013:
3980:
3908:
3875:
3855:
3806:
3677:
3580:
3522:
3502:
3437:
3405:
3224:
3179:
3146:
3048:
2998:
2902:
2833:
2790:
2759:
2715:
2679:
2628:
2515:
2433:
2376:
2345:
2316:
2277:
2245:
2216:
2196:
2165:
2145:
2079:
2053:
2018:
1995:
1964:
1797:
1738:
1706:
1686:
1662:
1623:
1553:
1526:
1499:
1472:
1445:
1418:
1386:
1294:
1236:
1203:
1171:
1091:
1061:
1026:
968:
933:
866:
794:
756:
673:
639:
608:
561:
435:
401:
375:
346:
317:
278:
248:
217:
6585:
5822:{\displaystyle {\mathcal {M}}_{g,n}^{\mathrm {c.} }}
2666:{\displaystyle {\mathcal {M}}_{1,1}|_{\mathbb {C} }}
6689:
3797:, meaning there exists a dominant rational morphism
57:. Unsourced material may be challenged and removed.
6821:
6109:
5923:
5821:
5774:
5721:
5688:
5640:
5605:
5535:
5477:
5447:{\displaystyle {\overline {\mathcal {M}}}_{g',n'}}
5446:
5394:
5320:
5286:A case of particular interest is the moduli stack
5275:
5240:
5198:
5152:
5093:
4870:
4838:
4811:
4772:
4752:
4625:
4533:
4493:
4404:
4308:
4288:
4249:
4205:
4162:
4128:
4051:
4025:
3999:
3963:
3887:
3861:
3838:
3771:
3656:
3551:
3508:
3488:
3423:
3388:
3191:
3165:
3129:
3039:, the parameter space for such curves is given by
3031:
2981:
2856:
2819:
2776:
2734:
2701:
2665:
2614:
2498:
2399:
2362:
2331:
2302:
2260:
2231:
2202:
2182:
2151:
2129:{\displaystyle H^{0}(C,\omega _{C}^{\otimes 2})=0}
2128:
2062:
2039:
2004:
1977:
1947:
1773:
1721:
1692:
1668:
1635:
1570:
1539:
1512:
1493:. Furthermore, because every component intersects
1485:
1458:
1431:
1404:
1368:
1280:
1222:
1189:
1153:
1074:
1043:
1012:
950:
919:
852:
777:
742:
652:
625:
589:
578:
544:
418:
387:
361:
332:
300:
260:
234:
6552:
206:
7503:
5621:taken from the labelling and number of markings
5536:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
5395:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
5321:{\displaystyle {\overline {\mathcal {M}}}_{1,1}}
5241:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
4784:The curves lying above these loci correspond to
4633:where the action permutes the two marked points.
153:) whose points represent isomorphism classes of
5948:
5660:plus the number of closed cycles in the graph.
2702:{\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}
6913:
6627:
6597:(3rd enl. ed.). Berlin: Springer-Verlag.
6311:
5872:
4250:{\displaystyle {\overline {\mathcal {M}}}_{g}}
1281:{\displaystyle S^{*}=H_{g}\setminus H_{g}^{o}}
301:{\displaystyle {\overline {\mathcal {M}}}_{g}}
6899:
6417:
6371:
5951:Les schémas de modules de courbes elliptiques
4853:
4257:has a natural stratification on the boundary
4216:
3640:
3627:
2070:, hence there are no global sections, meaning
1197:is the subscheme of smooth stable curves and
6683:
6293:: CS1 maint: multiple names: authors list (
6275:. Tipografia della R. Accademia dei Lincei.
3788:
3202:
2303:{\displaystyle {\text{PGL}}(2,\mathbb {C} )}
1989:shows the degree of the canonical bundle is
176:The most basic problem is that of moduli of
6833:. In Farkas, Gavril; Morrison, Ian (eds.).
6137:
4846:curve at a single double point singularity.
2982:{\displaystyle y^{2}-x(x-1)(x-a)(x-b)(x-c)}
2735:{\displaystyle {\text{Spec}}(\mathbb {Z} )}
2190:is just a single point, and the only genus
2139:showing there are no deformations of genus
1380:). If there existed multiple components of
6906:
6892:
6785:Handbook of TeichmĂĽller Theory, Volume III
6297:) CS1 maint: numeric names: authors list (
6203:
6020:: CS1 maint: location missing publisher (
5984:
5348:(roughly a marking of the points of order
5163:
1055:. Moreover, they find a stratification of
6717:
6657:
6223:
6147:
6122:
5933:
5072:
4998:
4878:case, there is a stratification given by
4735:
4608:
3809:
3619:
3588:
3445:
3051:
3032:{\displaystyle a,b,c\in \mathbb {A} ^{1}}
3019:
2876:
2784:would have stabilizer group at the point
2725:
2688:
2682:
2657:
2605:
2587:
2544:
2485:
2479:
2468:
2319:
2293:
2248:
2219:
1709:
1013:{\displaystyle \mathrm {Isom} _{S}(C,C')}
692:
117:Learn how and when to remove this message
6781:
4173:
3869:. Although, it turns out that for genus
2310:, which rigidifies once three points on
2040:{\displaystyle \omega _{C}^{\otimes 2}}
1608:
778:{\displaystyle \omega _{C}^{\otimes 3}}
594:It is a non-trivial theorem, proved by
14:
7504:
7329:Clifford's theorem on special divisors
6453:
6271:Severi, Francesco, 1879-1961. (1915).
6082:
4296:whose points represent singular genus
3783:
3552:{\displaystyle \in {\mathcal {M}}_{2}}
2820:{\displaystyle \in {\mathcal {M}}_{1}}
1788:this cohomology group is isomorphic to
1051:has finite stabilizers, hence it is a
927:is a construction of the moduli space
18:Deligne–Mumford moduli space of curves
6887:
6872:The American Institute of Mathematics
6546:
6493:
6491:
6034:
6727:Arithmetic Moduli of Elliptic Curves
6645:Publications Mathématiques de l'IHÉS
6418:Harris, Joe; Mumford, David (1982),
5974:http://publications.ias.edu/node/367
5919:
5917:
5887:Publications Mathématiques de l'IHÉS
5868:
5866:
5864:
5355:
5199:{\displaystyle {\mathcal {M}}_{g,n}}
2857:{\displaystyle {\mathcal {M}}_{1,1}}
2400:{\displaystyle {\mathcal {M}}_{0,3}}
853:{\displaystyle P_{g}(n)=(6n-1)(g-1)}
55:adding citations to reliable sources
26:
6110:{\displaystyle {\overline {M}}_{2}}
4182:cannot contain the universal curve
1680:. The number of moduli for a genus
1647:Examples of low genus moduli spaces
24:
7487:Vector bundles on algebraic curves
7421:Weber's theorem (Algebraic curves)
7018:Hasse's theorem on elliptic curves
7008:Counting points on elliptic curves
6776:Cohomology and intersection theory
6488:
5949:Deligne, P.; Rapoport, M. (1973),
5810:
5792:
5763:
5757:
5739:
5568:
5512:
5413:
5371:
5297:
5217:
5179:
5132:
5045:
5023:
4968:
4940:
4922:
4899:
4892:
4700:
4670:
4643:
4572:
4551:
4470:
4448:
4427:
4388:
4333:
4326:
4316:curves. It decomposes into strata
4271:
4264:
4232:
4206:{\displaystyle {\mathcal {C}}_{g}}
4192:
4163:{\displaystyle {\mathcal {M}}_{g}}
4149:
4109:
3940:
3929:
3926:
3923:
3825:
3763:
3760:
3757:
3745:
3720:
3717:
3714:
3681:
3631:
3538:
3469:
3438:
3148:
3106:
3087:
3068:
2837:
2806:
2777:{\displaystyle {\mathcal {M}}_{1}}
2763:
2632:
2556:
2519:
2443:
2380:
2363:{\displaystyle {\mathcal {M}}_{0}}
2349:
2183:{\displaystyle {\mathcal {M}}_{0}}
2169:
1729:, is given by the cohomology group
1601:regarding the stable reduction of
1571:{\displaystyle {\mathcal {M}}_{g}}
1557:
1344:
1341:
1338:
1298:
1044:{\displaystyle {\mathcal {M}}_{g}}
1030:
980:
977:
974:
971:
951:{\displaystyle {\mathcal {M}}_{g}}
937:
892:
889:
886:
685:
682:
679:
676:
626:{\displaystyle {\mathcal {M}}_{g}}
612:
579:{\displaystyle {\mathcal {M}}_{1}}
565:
506:
503:
500:
419:{\displaystyle {\mathcal {M}}_{0}}
405:
283:
235:{\displaystyle {\mathcal {M}}_{g}}
221:
157:. It is thus a special case of a
145:is a geometric space (typically a
25:
7528:
6860:
5914:
5861:
5849:Grothendieck–Riemann–Roch theorem
3061:
1288:. They analyze the components of
1260:
311:Both stacks above have dimension
4000:{\displaystyle \kappa _{g}>0}
2339:are fixed, so most authors take
2332:{\displaystyle \mathbb {P} ^{1}}
2261:{\displaystyle \mathbb {P} ^{1}}
2232:{\displaystyle \mathbb {P} ^{1}}
1722:{\displaystyle \mathbb {P} ^{1}}
1676:curves can be established using
31:
7109:Hurwitz's automorphisms theorem
6750:Geometry of Algebraic Curves II
6447:
6411:
6365:
6305:
5696:(hence all other vertices have
3431:are parameters for sections of
2417:Moduli stack of elliptic curves
1230:is an irreducible component of
590:Construction and irreducibility
42:needs additional citations for
7334:Gonality of an algebraic curve
7245:Differential of the first kind
6264:
6197:
6131:
6076:
6028:
5978:
5942:
5147:
5125:
5084:
5068:
5060:
5016:
5010:
4994:
4747:
4731:
4723:
4663:
4620:
4604:
4534:{\displaystyle 1\leq h<g/2}
4382:
4368:
4120:
4103:
3955:
3933:
3819:
3736:
3733:
3730:
3724:
3695:
3529:
3523:
3483:
3480:
3474:
3461:
3449:
3441:
3121:
3064:
2976:
2964:
2961:
2949:
2946:
2934:
2931:
2919:
2797:
2791:
2729:
2721:
2651:
2609:
2601:
2591:
2583:
2538:
2474:
2462:
2297:
2283:
2117:
2090:
1938:
1911:
1888:
1851:
1831:
1812:
1774:{\displaystyle H^{1}(C,T_{C})}
1768:
1749:
1363:
1348:
1007:
990:
914:
911:
896:
867:
847:
835:
832:
817:
811:
805:
735:
729:
519:
516:
510:
496:
474:
466:
454:
446:
207:Moduli stacks of stable curves
199:of the given genus, for which
195:these correspond precisely to
13:
1:
7477:Birkhoff–Grothendieck theorem
7187:Nagata's conjecture on curves
7058:Schoof–Elkies–Atkin algorithm
6932:Five points determine a conic
5854:
3166:{\displaystyle \Delta _{i,j}}
1581:
173:for the same moduli problem.
7048:Supersingular elliptic curve
6096:
5572:
5516:
5417:
5375:
5301:
5221:
5136:
5049:
5027:
4972:
4903:
4819:connected at a double point.
4704:
4674:
4576:
4474:
4452:
4337:
4275:
4236:
3944:
3899:of the coarse moduli spaces
2742:was originally completed by
785:for every curve) which have
287:
66:"Moduli of algebraic curves"
7:
7255:Riemann's existence theorem
7182:Hilbert's sixteenth problem
7074:Elliptic curve cryptography
6987:Fundamental pair of periods
6464:10.1090/pspum/080.1/2483934
6432:10.1007/978-1-4757-4265-7_8
6396:10.1007/978-1-4757-4265-7_8
5832:
3424:{\displaystyle a,\ldots ,f}
1978:{\displaystyle \omega _{C}}
10:
7533:
7385:Moduli of algebraic curves
6835:Handbook of Moduli, Vol. I
6731:Princeton University Press
6595:Geometric invariant theory
4854:Stratification for genus 2
4217:Stratification of boundary
3562:
2992:for some uniquely defined
2871:
2414:
2410:
1651:
7467:
7429:
7398:
7362:
7311:
7304:
7278:
7210:
7127:
7091:
7066:
7000:
6969:
6960:
6922:
6758:10.1007/978-3-540-69392-5
6684:Books on moduli of curves
6140:Algorithmic Number Theory
3789:Unirationality conjecture
3209:weighted projective space
3203:Weighted projective space
3173:corresponds to the locus
2885:that all such curves are
1405:{\displaystyle H_{g}^{o}}
1190:{\displaystyle H_{g}^{o}}
7152:Cayley–Bacharach theorem
7079:Elliptic curve primality
6693:; Morrison, Ian (1998).
6322:Inventiones Mathematicae
5330:stack of elliptic curves
4026:{\displaystyle g\geq 23}
3888:{\displaystyle g\geq 23}
1958:for the dualizing sheaf
602:, that the moduli stack
197:compact Riemann surfaces
7411:Riemann–Hurwitz formula
7375:Gromov–Witten invariant
7235:Compact Riemann surface
7023:Mazur's torsion theorem
6554:Grothendieck, Alexander
5722:{\displaystyle g_{v}=0}
5689:{\displaystyle g_{v}=g}
5478:{\displaystyle g'<g}
5164:Moduli of marked curves
4052:{\displaystyle g>23}
3516:represented by a point
3213:Riemann–Hurwitz formula
3192:{\displaystyle i\neq j}
2891:Riemann–Hurwitz formula
1223:{\displaystyle H_{g,i}}
451:space of genus 0 curves
7028:Modular elliptic curve
6579:. Exposés No. 7 and 8.
6565:SĂ©minaire Henri Cartan
6534:: CS1 maint: others (
6111:
5823:
5776:
5723:
5690:
5642:
5607:
5537:
5495:irreducible components
5479:
5448:
5396:
5322:
5277:
5276:{\displaystyle 3g-3+n}
5248:), and have dimension
5242:
5200:
5154:
5095:
4872:
4840:
4813:
4774:
4754:
4627:
4535:
4495:
4406:
4310:
4290:
4251:
4207:
4164:
4130:
4053:
4027:
4001:
3965:
3889:
3863:
3847:
3840:
3773:
3658:
3553:
3510:
3490:
3425:
3390:
3193:
3167:
3131:
3033:
2983:
2877:Affine parameter space
2858:
2821:
2778:
2736:
2703:
2667:
2616:
2507:
2500:
2401:
2364:
2333:
2304:
2262:
2233:
2204:
2184:
2153:
2137:
2130:
2064:
2041:
2006:
1979:
1956:
1949:
1782:
1775:
1723:
1694:
1670:
1637:
1636:{\displaystyle g>1}
1572:
1541:
1514:
1487:
1460:
1433:
1406:
1370:
1282:
1224:
1191:
1155:
1076:
1045:
1014:
952:
921:
854:
779:
744:
654:
627:
580:
546:
471:group of automorphisms
420:
389:
388:{\displaystyle g>1}
363:
334:
302:
262:
261:{\displaystyle g>1}
236:
6942:Rational normal curve
6591:Kirwan, Frances Clare
6112:
6041:Annals of Mathematics
5824:
5777:
5724:
5691:
5643:
5641:{\displaystyle n_{v}}
5608:
5543:is isomorphic to the
5538:
5480:
5449:
5397:
5323:
5278:
5243:
5201:
5155:
5096:
4873:
4841:
4814:
4775:
4755:
4628:
4536:
4496:
4407:
4311:
4291:
4252:
4208:
4174:Geometric implication
4165:
4131:
4054:
4028:
4002:
3966:
3890:
3864:
3841:
3799:
3774:
3659:
3554:
3511:
3491:
3426:
3391:
3194:
3168:
3132:
3034:
2984:
2866:Deligne–Mumford stack
2859:
2822:
2779:
2737:
2704:
2668:
2617:
2501:
2426:
2402:
2365:
2334:
2305:
2263:
2234:
2205:
2185:
2154:
2131:
2072:
2065:
2042:
2007:
1980:
1950:
1790:
1776:
1731:
1724:
1695:
1671:
1638:
1573:
1542:
1540:{\displaystyle H_{g}}
1515:
1513:{\displaystyle S^{*}}
1488:
1486:{\displaystyle H_{g}}
1461:
1459:{\displaystyle S^{*}}
1434:
1432:{\displaystyle H_{g}}
1407:
1371:
1283:
1225:
1192:
1156:
1077:
1075:{\displaystyle H_{g}}
1053:Deligne–Mumford stack
1046:
1015:
953:
922:
855:
780:
745:
655:
653:{\displaystyle H_{g}}
628:
581:
547:
421:
390:
364:
335:
303:
263:
237:
7482:Stable vector bundle
7354:Weil reciprocity law
7344:Riemann–Roch theorem
7324:Brill–Noether theory
7260:Riemann–Roch theorem
7177:Genus–degree formula
7038:Mordell–Weil theorem
7013:Division polynomials
6824:Pandharipande, Rahul
6234:10.14231/ag-2015-006
6087:
5926:Pandharipande, Rahul
5786:
5733:
5700:
5667:
5625:
5551:
5505:
5458:
5406:
5364:
5290:
5252:
5210:
5173:
5112:
4885:
4862:
4830:
4812:{\displaystyle C,C'}
4792:
4764:
4639:
4547:
4505:
4423:
4323:
4300:
4261:
4225:
4186:
4170:is of general type.
4143:
4066:
4037:
4011:
3978:
3906:
3873:
3853:
3804:
3675:
3578:
3569:genus degree formula
3520:
3500:
3435:
3403:
3222:
3177:
3144:
3046:
2996:
2900:
2831:
2788:
2757:
2713:
2677:
2626:
2513:
2431:
2374:
2343:
2314:
2275:
2243:
2214:
2194:
2163:
2159:curves. This proves
2143:
2077:
2051:
2016:
1993:
1962:
1795:
1736:
1704:
1684:
1660:
1621:
1609:Coarse moduli spaces
1551:
1524:
1497:
1470:
1443:
1416:
1384:
1292:
1234:
1201:
1169:
1089:
1059:
1024:
966:
931:
864:
792:
754:
671:
637:
606:
559:
433:
399:
373:
362:{\displaystyle 3g-3}
344:
333:{\displaystyle 3g-3}
315:
276:
246:
215:
171:coarse moduli spaces
51:improve this article
7517:Algebraic varieties
7305:Structure of curves
7197:Quartic plane curve
7119:Hyperelliptic curve
7099:De Franchis theorem
7043:Nagell–Lutz theorem
6335:1987InMat..90..359E
6166:10.1007/11792086_25
6158:2006math......3555G
5818:
5771:
5656:is the sum of the g
5652:. The total genus
4953:
4935:
4564:
4440:
4401:
3784:Birational geometry
3768:
3613:
2881:In genus 2 it is a
2698:
2495:
2210:curves is given by
2116:
2036:
2012:, so the degree of
1937:
1887:
1401:
1331:
1313:
1277:
1186:
1106:
774:
739:
7312:Divisors on curves
7104:Faltings's theorem
7053:Schoof's algorithm
7033:Modularity theorem
6668:10.1007/bf02684599
6547:Classic references
6456:Algebraic Geometry
6343:10.1007/bf01388710
6211:Algebraic Geometry
6107:
5990:Algebraic geometry
5959:10.1007/bfb0066716
5900:10.1007/BF02684599
5819:
5789:
5772:
5736:
5719:
5686:
5638:
5603:
5563:
5533:
5475:
5444:
5392:
5318:
5273:
5238:
5196:
5150:
5091:
5089:
4939:
4921:
4868:
4836:
4809:
4770:
4750:
4623:
4550:
4531:
4491:
4426:
4402:
4387:
4386:
4306:
4286:
4247:
4203:
4160:
4126:
4049:
4023:
3997:
3961:
3885:
3862:{\displaystyle 10}
3859:
3836:
3769:
3742:
3654:
3581:
3549:
3506:
3486:
3421:
3386:
3189:
3163:
3127:
3029:
2979:
2854:
2817:
2774:
2732:
2699:
2680:
2663:
2612:
2496:
2477:
2397:
2360:
2329:
2300:
2258:
2229:
2200:
2180:
2149:
2126:
2099:
2063:{\displaystyle -4}
2060:
2037:
2019:
2005:{\displaystyle -2}
2002:
1975:
1945:
1943:
1920:
1873:
1771:
1719:
1690:
1678:deformation Theory
1666:
1633:
1568:
1537:
1510:
1483:
1456:
1429:
1402:
1387:
1366:
1317:
1295:
1278:
1263:
1220:
1187:
1172:
1151:
1092:
1072:
1041:
1010:
960:deformation theory
948:
917:
860:. Then, the stack
850:
787:Hilbert polynomial
775:
757:
740:
674:
650:
623:
576:
542:
540:
416:
385:
359:
330:
298:
258:
232:
183:curves of a fixed
131:algebraic geometry
7499:
7498:
7495:
7494:
7406:Hasse–Witt matrix
7349:Weierstrass point
7296:Smooth completion
7265:TeichmĂĽller space
7167:Cubic plane curve
7087:
7086:
7001:Arithmetic theory
6982:Elliptic integral
6977:Elliptic function
6806:978-3-03719-103-3
6767:978-3-540-42688-2
6740:978-0-691-08352-0
6708:978-0-387-98429-2
6589:; Fogarty, John;
6513:978-1-4757-9286-7
6441:978-1-4419-1936-6
6405:978-1-4419-1936-6
6175:978-3-540-36075-9
6099:
5999:978-1-4757-3849-0
5986:Hartshorne, Robin
5968:978-3-540-06558-6
5844:Tautological ring
5839:Witten conjecture
5575:
5554:
5519:
5420:
5378:
5356:Boundary geometry
5304:
5224:
5139:
5052:
5030:
4975:
4906:
4871:{\displaystyle 2}
4839:{\displaystyle g}
4788:A pair of curves
4773:{\displaystyle g}
4707:
4677:
4579:
4477:
4455:
4351:
4340:
4309:{\displaystyle g}
4278:
4239:
4221:The moduli space
3947:
3897:Kodaira dimension
3638:
3509:{\displaystyle C}
2719:
2622:. Topologically,
2599:
2581:
2281:
2203:{\displaystyle 0}
2152:{\displaystyle 0}
1693:{\displaystyle 0}
1669:{\displaystyle 0}
1603:Abelian varieties
586:has dimension 0.
472:
452:
369:parameters, when
290:
211:The moduli stack
127:
126:
119:
101:
16:(Redirected from
7524:
7339:Jacobian variety
7309:
7308:
7212:Riemann surfaces
7202:Real plane curve
7162:Cramer's paradox
7142:BĂ©zout's theorem
6967:
6966:
6916:algebraic curves
6908:
6901:
6894:
6885:
6884:
6875:
6856:
6832:
6818:
6797:10.4171/103-1/12
6790:
6771:
6744:
6719:Katz, Nicholas M
6712:
6695:Moduli of Curves
6679:
6661:
6641:
6624:
6580:
6562:
6540:
6539:
6533:
6525:
6505:
6495:
6486:
6485:
6451:
6445:
6444:
6415:
6409:
6408:
6385:
6369:
6363:
6362:
6309:
6303:
6302:
6292:
6284:
6268:
6262:
6261:
6227:
6201:
6195:
6194:
6151:
6135:
6129:
6128:
6126:
6116:
6114:
6113:
6108:
6106:
6105:
6100:
6092:
6080:
6074:
6073:
6032:
6026:
6025:
6019:
6011:
5988:(29 June 2013).
5982:
5976:
5971:
5946:
5940:
5939:
5937:
5921:
5912:
5911:
5870:
5828:
5826:
5825:
5820:
5817:
5816:
5807:
5796:
5795:
5781:
5779:
5778:
5773:
5770:
5769:
5754:
5743:
5742:
5728:
5726:
5725:
5720:
5712:
5711:
5695:
5693:
5692:
5687:
5679:
5678:
5647:
5645:
5644:
5639:
5637:
5636:
5612:
5610:
5609:
5604:
5602:
5601:
5600:
5599:
5587:
5586:
5576:
5571:
5566:
5562:
5542:
5540:
5539:
5534:
5532:
5531:
5520:
5515:
5510:
5499:arithmetic genus
5484:
5482:
5481:
5476:
5468:
5453:
5451:
5450:
5445:
5443:
5442:
5441:
5430:
5421:
5416:
5411:
5401:
5399:
5398:
5393:
5391:
5390:
5379:
5374:
5369:
5327:
5325:
5324:
5319:
5317:
5316:
5305:
5300:
5295:
5282:
5280:
5279:
5274:
5247:
5245:
5244:
5239:
5237:
5236:
5225:
5220:
5215:
5205:
5203:
5202:
5197:
5195:
5194:
5183:
5182:
5159:
5157:
5156:
5151:
5146:
5145:
5140:
5135:
5130:
5124:
5123:
5100:
5098:
5097:
5092:
5090:
5080:
5075:
5067:
5059:
5058:
5053:
5048:
5043:
5037:
5036:
5031:
5026:
5021:
5006:
5001:
4993:
4988:
4987:
4976:
4971:
4966:
4957:
4952:
4947:
4934:
4929:
4913:
4912:
4907:
4902:
4897:
4877:
4875:
4874:
4869:
4845:
4843:
4842:
4837:
4818:
4816:
4815:
4810:
4808:
4779:
4777:
4776:
4771:
4759:
4757:
4756:
4751:
4743:
4738:
4730:
4722:
4721:
4717:
4708:
4703:
4698:
4692:
4691:
4687:
4678:
4673:
4668:
4659:
4658:
4654:
4632:
4630:
4629:
4624:
4616:
4611:
4603:
4598:
4597:
4580:
4575:
4570:
4563:
4558:
4540:
4538:
4537:
4532:
4527:
4500:
4498:
4497:
4492:
4490:
4489:
4478:
4473:
4468:
4462:
4461:
4456:
4451:
4446:
4439:
4434:
4411:
4409:
4408:
4403:
4400:
4395:
4385:
4378:
4347:
4346:
4341:
4336:
4331:
4315:
4313:
4312:
4307:
4295:
4293:
4292:
4287:
4285:
4284:
4279:
4274:
4269:
4256:
4254:
4253:
4248:
4246:
4245:
4240:
4235:
4230:
4212:
4210:
4209:
4204:
4202:
4201:
4196:
4195:
4169:
4167:
4166:
4161:
4159:
4158:
4153:
4152:
4135:
4133:
4132:
4127:
4119:
4118:
4113:
4112:
4078:
4077:
4058:
4056:
4055:
4050:
4033:. In fact, for
4032:
4030:
4029:
4024:
4006:
4004:
4003:
3998:
3990:
3989:
3970:
3968:
3967:
3962:
3954:
3953:
3948:
3943:
3938:
3932:
3918:
3917:
3894:
3892:
3891:
3886:
3868:
3866:
3865:
3860:
3845:
3843:
3842:
3837:
3835:
3834:
3829:
3828:
3818:
3817:
3812:
3778:
3776:
3775:
3770:
3767:
3766:
3754:
3749:
3748:
3723:
3712:
3707:
3706:
3691:
3690:
3685:
3684:
3663:
3661:
3660:
3655:
3653:
3652:
3645:
3644:
3643:
3630:
3622:
3612:
3598:
3597:
3596:
3591:
3558:
3556:
3555:
3550:
3548:
3547:
3542:
3541:
3515:
3513:
3512:
3507:
3495:
3493:
3492:
3487:
3473:
3472:
3448:
3430:
3428:
3427:
3422:
3395:
3393:
3392:
3387:
3382:
3381:
3366:
3365:
3347:
3346:
3337:
3336:
3321:
3320:
3311:
3310:
3295:
3294:
3285:
3284:
3266:
3265:
3250:
3249:
3234:
3233:
3198:
3196:
3195:
3190:
3172:
3170:
3169:
3164:
3162:
3161:
3136:
3134:
3133:
3128:
3120:
3119:
3101:
3100:
3082:
3081:
3060:
3059:
3054:
3038:
3036:
3035:
3030:
3028:
3027:
3022:
2988:
2986:
2985:
2980:
2912:
2911:
2883:classical result
2863:
2861:
2860:
2855:
2853:
2852:
2841:
2840:
2826:
2824:
2823:
2818:
2816:
2815:
2810:
2809:
2783:
2781:
2780:
2775:
2773:
2772:
2767:
2766:
2741:
2739:
2738:
2733:
2728:
2720:
2717:
2708:
2706:
2705:
2700:
2697:
2692:
2691:
2685:
2672:
2670:
2669:
2664:
2662:
2661:
2660:
2654:
2648:
2647:
2636:
2635:
2621:
2619:
2618:
2613:
2608:
2600:
2597:
2595:
2594:
2590:
2582:
2579:
2572:
2571:
2560:
2559:
2549:
2548:
2547:
2541:
2535:
2534:
2523:
2522:
2505:
2503:
2502:
2497:
2494:
2489:
2488:
2482:
2473:
2472:
2471:
2465:
2459:
2458:
2447:
2446:
2406:
2404:
2403:
2398:
2396:
2395:
2384:
2383:
2369:
2367:
2366:
2361:
2359:
2358:
2353:
2352:
2338:
2336:
2335:
2330:
2328:
2327:
2322:
2309:
2307:
2306:
2301:
2296:
2282:
2279:
2267:
2265:
2264:
2259:
2257:
2256:
2251:
2238:
2236:
2235:
2230:
2228:
2227:
2222:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2179:
2178:
2173:
2172:
2158:
2156:
2155:
2150:
2135:
2133:
2132:
2127:
2115:
2107:
2089:
2088:
2069:
2067:
2066:
2061:
2046:
2044:
2043:
2038:
2035:
2027:
2011:
2009:
2008:
2003:
1984:
1982:
1981:
1976:
1974:
1973:
1954:
1952:
1951:
1946:
1944:
1936:
1928:
1910:
1909:
1894:
1886:
1881:
1869:
1868:
1850:
1849:
1830:
1829:
1811:
1810:
1780:
1778:
1777:
1772:
1767:
1766:
1748:
1747:
1728:
1726:
1725:
1720:
1718:
1717:
1712:
1699:
1697:
1696:
1691:
1675:
1673:
1672:
1667:
1642:
1640:
1639:
1634:
1578:is irreducible.
1577:
1575:
1574:
1569:
1567:
1566:
1561:
1560:
1546:
1544:
1543:
1538:
1536:
1535:
1519:
1517:
1516:
1511:
1509:
1508:
1492:
1490:
1489:
1484:
1482:
1481:
1465:
1463:
1462:
1457:
1455:
1454:
1438:
1436:
1435:
1430:
1428:
1427:
1411:
1409:
1408:
1403:
1400:
1395:
1375:
1373:
1372:
1367:
1347:
1336:
1330:
1325:
1312:
1307:
1302:
1301:
1287:
1285:
1284:
1279:
1276:
1271:
1259:
1258:
1246:
1245:
1229:
1227:
1226:
1221:
1219:
1218:
1196:
1194:
1193:
1188:
1185:
1180:
1160:
1158:
1157:
1152:
1150:
1149:
1125:
1124:
1105:
1100:
1081:
1079:
1078:
1073:
1071:
1070:
1050:
1048:
1047:
1042:
1040:
1039:
1034:
1033:
1019:
1017:
1016:
1011:
1006:
989:
988:
983:
957:
955:
954:
949:
947:
946:
941:
940:
926:
924:
923:
920:{\displaystyle }
918:
895:
884:
879:
878:
859:
857:
856:
851:
804:
803:
784:
782:
781:
776:
773:
765:
749:
747:
746:
741:
738:
728:
727:
717:
716:
715:
695:
688:
659:
657:
656:
651:
649:
648:
632:
630:
629:
624:
622:
621:
616:
615:
585:
583:
582:
577:
575:
574:
569:
568:
551:
549:
548:
543:
541:
525:
509:
473:
470:
453:
450:
425:
423:
422:
417:
415:
414:
409:
408:
394:
392:
391:
386:
368:
366:
365:
360:
339:
337:
336:
331:
307:
305:
304:
299:
297:
296:
291:
286:
281:
267:
265:
264:
259:
241:
239:
238:
233:
231:
230:
225:
224:
201:Bernhard Riemann
155:algebraic curves
122:
115:
111:
108:
102:
100:
59:
35:
27:
21:
7532:
7531:
7527:
7526:
7525:
7523:
7522:
7521:
7502:
7501:
7500:
7491:
7463:
7454:Delta invariant
7425:
7394:
7358:
7319:Abel–Jacobi map
7300:
7274:
7270:Torelli theorem
7240:Dessin d'enfant
7220:Belyi's theorem
7206:
7192:PlĂĽcker formula
7123:
7114:Hurwitz surface
7083:
7062:
6996:
6970:Analytic theory
6962:Elliptic curves
6956:
6937:Projective line
6924:Rational curves
6918:
6912:
6866:
6863:
6845:
6830:
6807:
6788:
6778:
6768:
6741:
6709:
6699:Springer Verlag
6686:
6639:
6629:Deligne, Pierre
6605:
6560:
6549:
6544:
6543:
6527:
6526:
6514:
6503:
6497:
6496:
6489:
6474:
6452:
6448:
6442:
6424:Selected Papers
6416:
6412:
6406:
6388:Selected Papers
6383:
6370:
6366:
6313:Eisenbud, David
6310:
6306:
6286:
6285:
6269:
6265:
6204:Penev, Nikola;
6202:
6198:
6176:
6136:
6132:
6101:
6091:
6090:
6088:
6085:
6084:
6081:
6077:
6054:10.2307/1970233
6036:Igusa, Jun-Ichi
6033:
6029:
6013:
6012:
6000:
5983:
5979:
5969:
5947:
5943:
5922:
5915:
5874:Deligne, Pierre
5871:
5862:
5857:
5835:
5809:
5808:
5797:
5791:
5790:
5787:
5784:
5783:
5756:
5755:
5744:
5738:
5737:
5734:
5731:
5730:
5707:
5703:
5701:
5698:
5697:
5674:
5670:
5668:
5665:
5664:
5659:
5632:
5628:
5626:
5623:
5622:
5620:
5595:
5591:
5582:
5578:
5577:
5567:
5565:
5564:
5558:
5552:
5549:
5548:
5521:
5511:
5509:
5508:
5506:
5503:
5502:
5461:
5459:
5456:
5455:
5434:
5423:
5422:
5412:
5410:
5409:
5407:
5404:
5403:
5380:
5370:
5368:
5367:
5365:
5362:
5361:
5358:
5306:
5296:
5294:
5293:
5291:
5288:
5287:
5253:
5250:
5249:
5226:
5216:
5214:
5213:
5211:
5208:
5207:
5184:
5178:
5177:
5176:
5174:
5171:
5170:
5166:
5141:
5131:
5129:
5128:
5119:
5115:
5113:
5110:
5109:
5088:
5087:
5076:
5071:
5063:
5054:
5044:
5042:
5041:
5032:
5022:
5020:
5019:
5002:
4997:
4989:
4977:
4967:
4965:
4964:
4955:
4954:
4948:
4943:
4930:
4925:
4914:
4908:
4898:
4896:
4895:
4888:
4886:
4883:
4882:
4863:
4860:
4859:
4856:
4831:
4828:
4827:
4801:
4793:
4790:
4789:
4765:
4762:
4761:
4739:
4734:
4726:
4713:
4709:
4699:
4697:
4696:
4683:
4679:
4669:
4667:
4666:
4650:
4646:
4642:
4640:
4637:
4636:
4612:
4607:
4599:
4581:
4571:
4569:
4568:
4559:
4554:
4548:
4545:
4544:
4523:
4506:
4503:
4502:
4479:
4469:
4467:
4466:
4457:
4447:
4445:
4444:
4435:
4430:
4424:
4421:
4420:
4396:
4391:
4374:
4355:
4342:
4332:
4330:
4329:
4324:
4321:
4320:
4301:
4298:
4297:
4280:
4270:
4268:
4267:
4262:
4259:
4258:
4241:
4231:
4229:
4228:
4226:
4223:
4222:
4219:
4197:
4191:
4190:
4189:
4187:
4184:
4183:
4176:
4154:
4148:
4147:
4146:
4144:
4141:
4140:
4114:
4108:
4107:
4106:
4073:
4069:
4067:
4064:
4063:
4038:
4035:
4034:
4012:
4009:
4008:
3985:
3981:
3979:
3976:
3975:
3949:
3939:
3937:
3936:
3922:
3913:
3909:
3907:
3904:
3903:
3874:
3871:
3870:
3854:
3851:
3850:
3830:
3824:
3823:
3822:
3813:
3808:
3807:
3805:
3802:
3801:
3791:
3786:
3756:
3755:
3750:
3744:
3743:
3713:
3708:
3702:
3698:
3686:
3680:
3679:
3678:
3676:
3673:
3672:
3639:
3626:
3625:
3624:
3623:
3618:
3617:
3599:
3592:
3587:
3586:
3585:
3579:
3576:
3575:
3565:
3543:
3537:
3536:
3535:
3521:
3518:
3517:
3501:
3498:
3497:
3468:
3467:
3444:
3436:
3433:
3432:
3404:
3401:
3400:
3377:
3373:
3361:
3357:
3342:
3338:
3332:
3328:
3316:
3312:
3306:
3302:
3290:
3286:
3280:
3276:
3261:
3257:
3245:
3241:
3229:
3225:
3223:
3220:
3219:
3205:
3178:
3175:
3174:
3151:
3147:
3145:
3142:
3141:
3109:
3105:
3090:
3086:
3071:
3067:
3055:
3050:
3049:
3047:
3044:
3043:
3023:
3018:
3017:
2997:
2994:
2993:
2907:
2903:
2901:
2898:
2897:
2879:
2874:
2842:
2836:
2835:
2834:
2832:
2829:
2828:
2811:
2805:
2804:
2803:
2789:
2786:
2785:
2768:
2762:
2761:
2760:
2758:
2755:
2754:
2724:
2716:
2714:
2711:
2710:
2693:
2687:
2686:
2681:
2678:
2675:
2674:
2656:
2655:
2650:
2649:
2637:
2631:
2630:
2629:
2627:
2624:
2623:
2604:
2596:
2586:
2578:
2577:
2573:
2561:
2555:
2554:
2553:
2543:
2542:
2537:
2536:
2524:
2518:
2517:
2516:
2514:
2511:
2510:
2490:
2484:
2483:
2478:
2467:
2466:
2461:
2460:
2448:
2442:
2441:
2440:
2432:
2429:
2428:
2419:
2413:
2385:
2379:
2378:
2377:
2375:
2372:
2371:
2354:
2348:
2347:
2346:
2344:
2341:
2340:
2323:
2318:
2317:
2315:
2312:
2311:
2292:
2278:
2276:
2273:
2272:
2270:algebraic group
2252:
2247:
2246:
2244:
2241:
2240:
2223:
2218:
2217:
2215:
2212:
2211:
2195:
2192:
2191:
2174:
2168:
2167:
2166:
2164:
2161:
2160:
2144:
2141:
2140:
2108:
2103:
2084:
2080:
2078:
2075:
2074:
2052:
2049:
2048:
2028:
2023:
2017:
2014:
2013:
1994:
1991:
1990:
1969:
1965:
1963:
1960:
1959:
1942:
1941:
1929:
1924:
1905:
1901:
1892:
1891:
1882:
1877:
1864:
1860:
1845:
1841:
1834:
1825:
1821:
1806:
1802:
1798:
1796:
1793:
1792:
1762:
1758:
1743:
1739:
1737:
1734:
1733:
1713:
1708:
1707:
1705:
1702:
1701:
1685:
1682:
1681:
1661:
1658:
1657:
1654:
1649:
1622:
1619:
1618:
1611:
1584:
1562:
1556:
1555:
1554:
1552:
1549:
1548:
1531:
1527:
1525:
1522:
1521:
1504:
1500:
1498:
1495:
1494:
1477:
1473:
1471:
1468:
1467:
1450:
1446:
1444:
1441:
1440:
1423:
1419:
1417:
1414:
1413:
1396:
1391:
1385:
1382:
1381:
1337:
1332:
1326:
1321:
1308:
1303:
1297:
1296:
1293:
1290:
1289:
1272:
1267:
1254:
1250:
1241:
1237:
1235:
1232:
1231:
1208:
1204:
1202:
1199:
1198:
1181:
1176:
1170:
1167:
1166:
1139:
1135:
1114:
1110:
1101:
1096:
1090:
1087:
1086:
1066:
1062:
1060:
1057:
1056:
1035:
1029:
1028:
1027:
1025:
1022:
1021:
1020:, to show that
999:
984:
970:
969:
967:
964:
963:
942:
936:
935:
934:
932:
929:
928:
885:
880:
874:
870:
865:
862:
861:
799:
795:
793:
790:
789:
766:
761:
755:
752:
751:
723:
719:
718:
696:
691:
690:
689:
675:
672:
669:
668:
644:
640:
638:
635:
634:
617:
611:
610:
609:
607:
604:
603:
592:
570:
564:
563:
562:
560:
557:
556:
539:
538:
523:
522:
499:
477:
469:
449:
436:
434:
431:
430:
410:
404:
403:
402:
400:
397:
396:
374:
371:
370:
345:
342:
341:
316:
313:
312:
292:
282:
280:
279:
277:
274:
273:
247:
244:
243:
226:
220:
219:
218:
216:
213:
212:
209:
193:complex numbers
151:algebraic stack
135:moduli space of
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
7530:
7520:
7519:
7514:
7497:
7496:
7493:
7492:
7490:
7489:
7484:
7479:
7473:
7471:
7469:Vector bundles
7465:
7464:
7462:
7461:
7456:
7451:
7446:
7441:
7435:
7433:
7427:
7426:
7424:
7423:
7418:
7413:
7408:
7402:
7400:
7396:
7395:
7393:
7392:
7387:
7382:
7377:
7372:
7366:
7364:
7360:
7359:
7357:
7356:
7351:
7346:
7341:
7336:
7331:
7326:
7321:
7315:
7313:
7306:
7302:
7301:
7299:
7298:
7293:
7288:
7282:
7280:
7276:
7275:
7273:
7272:
7267:
7262:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7222:
7216:
7214:
7208:
7207:
7205:
7204:
7199:
7194:
7189:
7184:
7179:
7174:
7169:
7164:
7159:
7154:
7149:
7144:
7139:
7133:
7131:
7125:
7124:
7122:
7121:
7116:
7111:
7106:
7101:
7095:
7093:
7089:
7088:
7085:
7084:
7082:
7081:
7076:
7070:
7068:
7064:
7063:
7061:
7060:
7055:
7050:
7045:
7040:
7035:
7030:
7025:
7020:
7015:
7010:
7004:
7002:
6998:
6997:
6995:
6994:
6989:
6984:
6979:
6973:
6971:
6964:
6958:
6957:
6955:
6954:
6949:
6947:Riemann sphere
6944:
6939:
6934:
6928:
6926:
6920:
6919:
6911:
6910:
6903:
6896:
6888:
6882:
6881:
6876:
6862:
6861:External links
6859:
6858:
6857:
6843:
6822:Faber, Carel;
6819:
6805:
6777:
6774:
6773:
6772:
6766:
6745:
6739:
6714:
6713:
6707:
6685:
6682:
6681:
6680:
6659:10.1.1.589.288
6633:Mumford, David
6625:
6603:
6587:Mumford, David
6582:
6581:
6548:
6545:
6542:
6541:
6512:
6487:
6472:
6446:
6440:
6410:
6404:
6377:Mumford, David
6364:
6329:(2): 359–387.
6304:
6263:
6218:(1): 123–136.
6196:
6174:
6130:
6104:
6098:
6095:
6075:
6048:(3): 612–649.
6027:
5998:
5977:
5967:
5941:
5924:Faber, Carel;
5913:
5878:Mumford, David
5859:
5858:
5856:
5853:
5852:
5851:
5846:
5841:
5834:
5831:
5815:
5812:
5806:
5803:
5800:
5794:
5768:
5765:
5762:
5759:
5753:
5750:
5747:
5741:
5718:
5715:
5710:
5706:
5685:
5682:
5677:
5673:
5657:
5635:
5631:
5618:
5598:
5594:
5590:
5585:
5581:
5574:
5570:
5561:
5557:
5545:stack quotient
5530:
5527:
5524:
5518:
5514:
5474:
5471:
5467:
5464:
5440:
5437:
5433:
5429:
5426:
5419:
5415:
5389:
5386:
5383:
5377:
5373:
5357:
5354:
5315:
5312:
5309:
5303:
5299:
5272:
5269:
5266:
5263:
5260:
5257:
5235:
5232:
5229:
5223:
5219:
5193:
5190:
5187:
5181:
5165:
5162:
5149:
5144:
5138:
5134:
5127:
5122:
5118:
5103:
5102:
5086:
5083:
5079:
5074:
5070:
5066:
5062:
5057:
5051:
5047:
5040:
5035:
5029:
5025:
5018:
5015:
5012:
5009:
5005:
5000:
4996:
4992:
4986:
4983:
4980:
4974:
4970:
4963:
4960:
4958:
4956:
4951:
4946:
4942:
4938:
4933:
4928:
4924:
4920:
4917:
4915:
4911:
4905:
4901:
4894:
4891:
4890:
4867:
4858:For the genus
4855:
4852:
4851:
4850:
4847:
4835:
4820:
4807:
4804:
4800:
4797:
4782:
4781:
4769:
4749:
4746:
4742:
4737:
4733:
4729:
4725:
4720:
4716:
4712:
4706:
4702:
4695:
4690:
4686:
4682:
4676:
4672:
4665:
4662:
4657:
4653:
4649:
4645:
4634:
4622:
4619:
4615:
4610:
4606:
4602:
4596:
4593:
4590:
4587:
4584:
4578:
4574:
4567:
4562:
4557:
4553:
4542:
4530:
4526:
4522:
4519:
4516:
4513:
4510:
4488:
4485:
4482:
4476:
4472:
4465:
4460:
4454:
4450:
4443:
4438:
4433:
4429:
4414:
4413:
4399:
4394:
4390:
4384:
4381:
4377:
4373:
4370:
4367:
4364:
4361:
4358:
4354:
4350:
4345:
4339:
4335:
4328:
4305:
4283:
4277:
4273:
4266:
4244:
4238:
4234:
4218:
4215:
4200:
4194:
4175:
4172:
4157:
4151:
4137:
4136:
4125:
4122:
4117:
4111:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4076:
4072:
4048:
4045:
4042:
4022:
4019:
4016:
3996:
3993:
3988:
3984:
3972:
3971:
3960:
3957:
3952:
3946:
3942:
3935:
3931:
3928:
3925:
3921:
3916:
3912:
3884:
3881:
3878:
3858:
3833:
3827:
3821:
3816:
3811:
3790:
3787:
3785:
3782:
3781:
3780:
3765:
3762:
3759:
3753:
3747:
3741:
3738:
3735:
3732:
3729:
3726:
3722:
3719:
3716:
3711:
3705:
3701:
3697:
3694:
3689:
3683:
3666:
3665:
3651:
3648:
3642:
3637:
3634:
3629:
3621:
3616:
3611:
3608:
3605:
3602:
3595:
3590:
3584:
3564:
3561:
3546:
3540:
3534:
3531:
3528:
3525:
3505:
3485:
3482:
3479:
3476:
3471:
3466:
3463:
3460:
3457:
3454:
3451:
3447:
3443:
3440:
3420:
3417:
3414:
3411:
3408:
3397:
3396:
3385:
3380:
3376:
3372:
3369:
3364:
3360:
3356:
3353:
3350:
3345:
3341:
3335:
3331:
3327:
3324:
3319:
3315:
3309:
3305:
3301:
3298:
3293:
3289:
3283:
3279:
3275:
3272:
3269:
3264:
3260:
3256:
3253:
3248:
3244:
3240:
3237:
3232:
3228:
3204:
3201:
3188:
3185:
3182:
3160:
3157:
3154:
3150:
3138:
3137:
3126:
3123:
3118:
3115:
3112:
3108:
3104:
3099:
3096:
3093:
3089:
3085:
3080:
3077:
3074:
3070:
3066:
3063:
3058:
3053:
3026:
3021:
3016:
3013:
3010:
3007:
3004:
3001:
2990:
2989:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2910:
2906:
2878:
2875:
2873:
2870:
2851:
2848:
2845:
2839:
2814:
2808:
2802:
2799:
2796:
2793:
2771:
2765:
2731:
2727:
2723:
2696:
2690:
2684:
2659:
2653:
2646:
2643:
2640:
2634:
2611:
2607:
2603:
2593:
2589:
2585:
2576:
2570:
2567:
2564:
2558:
2552:
2546:
2540:
2533:
2530:
2527:
2521:
2493:
2487:
2481:
2476:
2470:
2464:
2457:
2454:
2451:
2445:
2439:
2436:
2415:Main article:
2412:
2409:
2394:
2391:
2388:
2382:
2357:
2351:
2326:
2321:
2299:
2295:
2291:
2288:
2285:
2255:
2250:
2226:
2221:
2199:
2177:
2171:
2148:
2125:
2122:
2119:
2114:
2111:
2106:
2102:
2098:
2095:
2092:
2087:
2083:
2059:
2056:
2034:
2031:
2026:
2022:
2001:
1998:
1972:
1968:
1940:
1935:
1932:
1927:
1923:
1919:
1916:
1913:
1908:
1904:
1900:
1897:
1895:
1893:
1890:
1885:
1880:
1876:
1872:
1867:
1863:
1859:
1856:
1853:
1848:
1844:
1840:
1837:
1835:
1833:
1828:
1824:
1820:
1817:
1814:
1809:
1805:
1801:
1800:
1770:
1765:
1761:
1757:
1754:
1751:
1746:
1742:
1716:
1711:
1689:
1665:
1653:
1650:
1648:
1645:
1632:
1629:
1626:
1610:
1607:
1583:
1580:
1565:
1559:
1534:
1530:
1507:
1503:
1480:
1476:
1453:
1449:
1426:
1422:
1399:
1394:
1390:
1365:
1362:
1359:
1356:
1353:
1350:
1346:
1343:
1340:
1335:
1329:
1324:
1320:
1316:
1311:
1306:
1300:
1275:
1270:
1266:
1262:
1257:
1253:
1249:
1244:
1240:
1217:
1214:
1211:
1207:
1184:
1179:
1175:
1163:
1162:
1148:
1145:
1142:
1138:
1134:
1131:
1128:
1123:
1120:
1117:
1113:
1109:
1104:
1099:
1095:
1069:
1065:
1038:
1032:
1009:
1005:
1002:
998:
995:
992:
987:
982:
979:
976:
973:
945:
939:
916:
913:
910:
907:
904:
901:
898:
894:
891:
888:
883:
877:
873:
869:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
802:
798:
772:
769:
764:
760:
737:
734:
731:
726:
722:
714:
711:
708:
705:
702:
699:
694:
687:
684:
681:
678:
666:Hilbert scheme
647:
643:
620:
614:
596:Pierre Deligne
591:
588:
573:
567:
553:
552:
537:
534:
531:
528:
526:
524:
521:
518:
515:
512:
508:
505:
502:
498:
495:
492:
489:
486:
483:
480:
478:
476:
468:
465:
462:
459:
456:
448:
445:
442:
439:
438:
413:
407:
384:
381:
378:
358:
355:
352:
349:
329:
326:
323:
320:
295:
289:
285:
257:
254:
251:
229:
223:
208:
205:
163:moduli problem
125:
124:
39:
37:
30:
9:
6:
4:
3:
2:
7529:
7518:
7515:
7513:
7512:Moduli theory
7510:
7509:
7507:
7488:
7485:
7483:
7480:
7478:
7475:
7474:
7472:
7470:
7466:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7436:
7434:
7432:
7431:Singularities
7428:
7422:
7419:
7417:
7414:
7412:
7409:
7407:
7404:
7403:
7401:
7397:
7391:
7388:
7386:
7383:
7381:
7378:
7376:
7373:
7371:
7368:
7367:
7365:
7361:
7355:
7352:
7350:
7347:
7345:
7342:
7340:
7337:
7335:
7332:
7330:
7327:
7325:
7322:
7320:
7317:
7316:
7314:
7310:
7307:
7303:
7297:
7294:
7292:
7289:
7287:
7284:
7283:
7281:
7279:Constructions
7277:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7250:Klein quartic
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7230:Bolza surface
7228:
7226:
7225:Bring's curve
7223:
7221:
7218:
7217:
7215:
7213:
7209:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7160:
7158:
7157:Conic section
7155:
7153:
7150:
7148:
7145:
7143:
7140:
7138:
7137:AF+BG theorem
7135:
7134:
7132:
7130:
7126:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7096:
7094:
7090:
7080:
7077:
7075:
7072:
7071:
7069:
7065:
7059:
7056:
7054:
7051:
7049:
7046:
7044:
7041:
7039:
7036:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7005:
7003:
6999:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6974:
6972:
6968:
6965:
6963:
6959:
6953:
6952:Twisted cubic
6950:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6929:
6927:
6925:
6921:
6917:
6909:
6904:
6902:
6897:
6895:
6890:
6889:
6886:
6880:
6877:
6873:
6869:
6865:
6864:
6854:
6850:
6846:
6844:9781571462572
6840:
6836:
6829:
6825:
6820:
6816:
6812:
6808:
6802:
6798:
6794:
6787:
6786:
6780:
6779:
6769:
6763:
6759:
6755:
6751:
6746:
6742:
6736:
6732:
6728:
6724:
6720:
6716:
6715:
6710:
6704:
6700:
6696:
6692:
6688:
6687:
6677:
6673:
6669:
6665:
6660:
6655:
6651:
6647:
6646:
6638:
6634:
6630:
6626:
6622:
6618:
6614:
6610:
6606:
6604:3-540-56963-4
6600:
6596:
6592:
6588:
6584:
6583:
6578:
6574:
6570:
6566:
6559:
6556:(1960–1961).
6555:
6551:
6550:
6537:
6531:
6523:
6519:
6515:
6509:
6502:
6501:
6494:
6492:
6483:
6479:
6475:
6473:9780821847022
6469:
6465:
6461:
6457:
6450:
6443:
6437:
6433:
6429:
6425:
6421:
6414:
6407:
6401:
6397:
6393:
6389:
6382:
6378:
6374:
6368:
6360:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6324:
6323:
6318:
6314:
6308:
6300:
6296:
6290:
6282:
6278:
6274:
6267:
6259:
6255:
6251:
6247:
6243:
6239:
6235:
6231:
6226:
6221:
6217:
6213:
6212:
6207:
6200:
6193:
6189:
6185:
6181:
6177:
6171:
6167:
6163:
6159:
6155:
6150:
6145:
6141:
6134:
6125:
6120:
6102:
6093:
6079:
6071:
6067:
6063:
6059:
6055:
6051:
6047:
6043:
6042:
6037:
6031:
6023:
6017:
6009:
6005:
6001:
5995:
5991:
5987:
5981:
5975:
5970:
5964:
5960:
5956:
5952:
5945:
5936:
5931:
5927:
5920:
5918:
5909:
5905:
5901:
5897:
5893:
5889:
5888:
5883:
5879:
5875:
5869:
5867:
5865:
5860:
5850:
5847:
5845:
5842:
5840:
5837:
5836:
5830:
5813:
5804:
5801:
5798:
5766:
5760:
5751:
5748:
5745:
5716:
5713:
5708:
5704:
5683:
5680:
5675:
5671:
5661:
5655:
5651:
5633:
5629:
5616:
5596:
5592:
5588:
5583:
5579:
5559:
5555:
5547:of a product
5546:
5528:
5525:
5522:
5500:
5496:
5492:
5488:
5472:
5469:
5465:
5462:
5438:
5435:
5431:
5427:
5424:
5387:
5384:
5381:
5353:
5351:
5347:
5345:
5339:
5335:
5334:modular forms
5331:
5313:
5310:
5307:
5284:
5270:
5267:
5264:
5261:
5258:
5255:
5233:
5230:
5227:
5191:
5188:
5185:
5161:
5142:
5120:
5116:
5108:
5081:
5077:
5064:
5055:
5038:
5033:
5013:
5007:
5003:
4990:
4984:
4981:
4978:
4961:
4959:
4949:
4944:
4936:
4931:
4926:
4918:
4916:
4909:
4881:
4880:
4879:
4865:
4848:
4833:
4825:
4824:normalization
4821:
4805:
4802:
4798:
4795:
4787:
4786:
4785:
4767:
4744:
4740:
4727:
4718:
4714:
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4693:
4688:
4684:
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4655:
4651:
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4635:
4617:
4613:
4600:
4594:
4591:
4588:
4585:
4582:
4565:
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4555:
4543:
4528:
4524:
4520:
4517:
4514:
4511:
4508:
4486:
4483:
4480:
4463:
4458:
4441:
4436:
4431:
4419:
4418:
4417:
4397:
4392:
4379:
4375:
4371:
4365:
4362:
4359:
4356:
4352:
4348:
4343:
4319:
4318:
4317:
4303:
4281:
4242:
4214:
4198:
4181:
4180:ruled variety
4171:
4155:
4123:
4115:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4074:
4070:
4062:
4061:
4060:
4046:
4043:
4040:
4020:
4017:
4014:
3994:
3991:
3986:
3982:
3958:
3950:
3919:
3914:
3910:
3902:
3901:
3900:
3898:
3882:
3879:
3876:
3856:
3846:
3831:
3814:
3798:
3796:
3751:
3739:
3727:
3709:
3703:
3699:
3692:
3687:
3671:
3670:
3669:
3649:
3646:
3635:
3632:
3614:
3609:
3606:
3603:
3600:
3593:
3582:
3574:
3573:
3572:
3570:
3560:
3544:
3532:
3526:
3503:
3477:
3464:
3458:
3455:
3452:
3418:
3415:
3412:
3409:
3406:
3383:
3378:
3374:
3370:
3367:
3362:
3358:
3354:
3351:
3348:
3343:
3339:
3333:
3329:
3325:
3322:
3317:
3313:
3307:
3303:
3299:
3296:
3291:
3287:
3281:
3277:
3273:
3270:
3267:
3262:
3258:
3254:
3251:
3246:
3242:
3238:
3235:
3230:
3226:
3218:
3217:
3216:
3214:
3210:
3200:
3186:
3183:
3180:
3158:
3155:
3152:
3124:
3116:
3113:
3110:
3102:
3097:
3094:
3091:
3083:
3078:
3075:
3072:
3056:
3042:
3041:
3040:
3024:
3014:
3011:
3008:
3005:
3002:
2999:
2973:
2970:
2967:
2958:
2955:
2952:
2943:
2940:
2937:
2928:
2925:
2922:
2916:
2913:
2908:
2904:
2896:
2895:
2894:
2892:
2888:
2887:hyperelliptic
2884:
2869:
2867:
2849:
2846:
2843:
2812:
2800:
2794:
2769:
2751:
2749:
2745:
2694:
2644:
2641:
2638:
2574:
2568:
2565:
2562:
2550:
2531:
2528:
2525:
2506:
2491:
2455:
2452:
2449:
2437:
2434:
2425:
2424:
2418:
2408:
2392:
2389:
2386:
2355:
2324:
2289:
2286:
2271:
2253:
2224:
2197:
2175:
2146:
2136:
2123:
2120:
2112:
2109:
2104:
2100:
2096:
2093:
2085:
2081:
2071:
2057:
2054:
2032:
2029:
2024:
2020:
1999:
1996:
1988:
1985:. But, using
1970:
1966:
1955:
1933:
1930:
1925:
1921:
1917:
1914:
1906:
1902:
1898:
1896:
1883:
1878:
1874:
1870:
1865:
1861:
1857:
1854:
1846:
1842:
1838:
1836:
1826:
1822:
1818:
1815:
1807:
1803:
1789:
1787:
1786:Serre duality
1781:
1763:
1759:
1755:
1752:
1744:
1740:
1730:
1714:
1687:
1679:
1663:
1644:
1630:
1627:
1624:
1615:
1606:
1604:
1600:
1596:
1592:
1588:
1579:
1563:
1532:
1528:
1505:
1501:
1478:
1474:
1451:
1447:
1424:
1420:
1397:
1392:
1388:
1379:
1360:
1357:
1354:
1351:
1333:
1327:
1322:
1318:
1314:
1309:
1304:
1273:
1268:
1264:
1255:
1251:
1247:
1242:
1238:
1215:
1212:
1209:
1205:
1182:
1177:
1173:
1146:
1143:
1140:
1136:
1132:
1129:
1126:
1121:
1118:
1115:
1111:
1107:
1102:
1097:
1093:
1085:
1084:
1083:
1067:
1063:
1054:
1036:
1003:
1000:
996:
993:
985:
961:
943:
908:
905:
902:
899:
881:
875:
871:
844:
841:
838:
829:
826:
823:
820:
814:
808:
800:
796:
788:
770:
767:
762:
758:
732:
724:
720:
712:
709:
706:
703:
700:
697:
667:
663:
662:stable curves
645:
641:
618:
601:
600:David Mumford
597:
587:
571:
535:
532:
529:
527:
513:
493:
490:
487:
484:
481:
479:
463:
460:
457:
443:
440:
429:
428:
427:
411:
382:
379:
376:
356:
353:
350:
347:
327:
324:
321:
318:
309:
293:
271:
255:
252:
249:
227:
204:
202:
198:
194:
190:
186:
182:
179:
174:
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
121:
118:
110:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
7416:Prym variety
7390:Stable curve
7384:
7380:Hodge bundle
7370:ELSV formula
7172:Fermat curve
7129:Plane curves
7092:Higher genus
7067:Applications
6992:Modular form
6871:
6834:
6784:
6749:
6726:
6723:Mazur, Barry
6694:
6649:
6643:
6594:
6571:(1). Paris.
6568:
6564:
6499:
6455:
6449:
6423:
6413:
6387:
6367:
6326:
6320:
6307:
6272:
6266:
6215:
6209:
6199:
6149:math/0603555
6139:
6133:
6078:
6045:
6039:
6030:
5992:. New York.
5989:
5980:
5950:
5944:
5891:
5885:
5662:
5653:
5649:
5614:
5486:
5359:
5349:
5343:
5337:
5285:
5167:
5104:
4857:
4783:
4415:
4220:
4177:
4138:
3973:
3848:
3800:
3792:
3667:
3566:
3398:
3206:
3139:
2991:
2880:
2864:is a smooth
2752:
2508:
2427:
2420:
2138:
2073:
1987:Riemann–Roch
1957:
1791:
1783:
1732:
1700:curve, e.g.
1655:
1616:
1612:
1599:Grothendieck
1585:
1378:GIT quotient
1164:
593:
554:
426:is equal to
310:
210:
175:
162:
159:moduli space
142:
138:
134:
128:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
7291:Polar curve
6691:Harris, Joe
6373:Harris, Joe
6317:Harris, Joe
6206:Vakil, Ravi
5617:has genus g
5454:for genera
4826:of a genus
3795:unirational
2423:J-invariant
1591:compactness
187:. Over the
107:August 2019
7506:Categories
7286:Dual curve
6914:Topics in
6652:: 75–109.
6577:0142.33503
6124:1904.08081
5894:: 75–109.
5855:References
5487:dual graph
5332:. Level 1
4139:and hence
3974:and found
1587:Properness
1582:Properness
77:newspapers
7399:Morphisms
7147:Bitangent
6654:CiteSeerX
6530:cite book
6522:681426064
6359:120642775
6351:0020-9910
6289:cite book
6281:881814709
6242:2214-2584
6225:1307.6614
6097:¯
6062:0003-486X
6016:cite book
6008:861706007
5935:1101.5489
5573:¯
5556:∏
5517:¯
5418:¯
5376:¯
5346:structure
5302:¯
5262:−
5222:¯
5137:¯
5121:∗
5107:Chow ring
5050:¯
5039:×
5028:¯
5014:∐
4973:¯
4950:∗
4941:Δ
4937:∐
4932:∗
4923:Δ
4904:¯
4893:∂
4760:whenever
4705:¯
4694:×
4675:¯
4661:≅
4644:Δ
4586:−
4577:¯
4566:≅
4561:∗
4552:Δ
4512:≤
4484:−
4475:¯
4464:×
4453:¯
4442:≅
4437:∗
4428:Δ
4398:∗
4389:Δ
4366:≤
4360:≤
4353:∐
4338:¯
4327:∂
4276:¯
4265:∂
4237:¯
4101:
4089:−
4071:κ
4018:≥
3983:κ
3945:¯
3911:κ
3880:≥
3820:→
3740:∐
3647:−
3615:≅
3607:−
3533:∈
3439:Γ
3413:…
3184:≠
3149:Δ
3107:Δ
3103:∪
3088:Δ
3084:∪
3069:Δ
3062:∖
3015:∈
2971:−
2956:−
2941:−
2926:−
2914:−
2801:∈
2575:×
2475:→
2110:⊗
2101:ω
2055:−
2030:⊗
2021:ω
1997:−
1967:ω
1931:⊗
1922:ω
1899:≅
1884:∨
1871:⊗
1862:ω
1839:≅
1595:orbifolds
1506:∗
1452:∗
1358:−
1261:∖
1243:∗
1133:∐
1130:⋯
1127:∐
1108:∐
906:−
842:−
827:−
768:⊗
759:ω
710:−
704:−
533:−
494:
488:−
464:
458:−
444:
354:−
325:−
288:¯
139:algebraic
6826:(2013).
6725:(1985).
6676:16482150
6635:(1969).
6621:29184987
6593:(1994).
6379:(1982),
6258:54876684
6192:15638167
5908:16482150
5880:(1969).
5833:See also
5466:′
5439:′
5428:′
4806:′
4780:is even.
3211:and the
3207:Using a
2748:Rapoport
2370:to mean
1004:′
958:. Using
181:complete
7459:Tacnode
7444:Crunode
6853:3184167
6815:2952773
6613:1304906
6482:8281102
6331:Bibcode
6250:3322200
6184:2282935
6154:Bibcode
6070:1970233
5972:, URL:
3563:Genus 3
2872:Genus 2
2744:Deligne
2411:Genus 1
2268:is the
1652:Genus 0
664:in the
91:scholar
7439:Acnode
7363:Moduli
6851:
6841:
6813:
6803:
6764:
6737:
6705:
6674:
6656:
6619:
6611:
6601:
6575:
6520:
6510:
6480:
6470:
6438:
6402:
6357:
6349:
6279:
6256:
6248:
6240:
6190:
6182:
6172:
6068:
6060:
6006:
5996:
5965:
5906:
5342:level
4416:where
3399:where
3140:where
2509:where
1376:(as a
1165:where
270:stable
178:smooth
149:or an
147:scheme
143:curves
93:
86:
79:
72:
64:
6831:(PDF)
6789:(PDF)
6672:S2CID
6640:(PDF)
6561:(PDF)
6504:(PDF)
6478:S2CID
6384:(PDF)
6355:S2CID
6254:S2CID
6220:arXiv
6188:S2CID
6144:arXiv
6119:arXiv
6066:JSTOR
5930:arXiv
5904:S2CID
5491:graph
1784:With
1589:, or
189:field
185:genus
98:JSTOR
84:books
7449:Cusp
6839:ISBN
6801:ISBN
6762:ISBN
6735:ISBN
6703:ISBN
6617:OCLC
6599:ISBN
6536:link
6518:OCLC
6508:ISBN
6468:ISBN
6436:ISBN
6400:ISBN
6347:ISSN
6299:link
6295:link
6277:OCLC
6238:ISSN
6170:ISBN
6058:ISSN
6022:link
6004:OCLC
5994:ISBN
5963:ISBN
5489:, a
5470:<
5206:(or
4822:The
4518:<
4501:for
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4007:for
3992:>
3583:Hilb
2746:and
2718:Spec
2598:Spec
2580:Spec
1628:>
1593:for
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598:and
380:>
253:>
169:and
167:fine
133:, a
70:news
6793:doi
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6664:doi
6573:Zbl
6460:doi
6428:doi
6392:doi
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6230:doi
6162:doi
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6050:doi
5955:doi
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4098:dim
2280:PGL
2047:is
660:of
491:dim
461:dim
441:dim
191:of
129:In
53:by
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3636:4
3633:6
3628:(
3620:P
3610:4
3604:t
3601:8
3594:2
3589:P
3545:2
3539:M
3530:]
3527:C
3524:[
3504:C
3484:)
3481:)
3478:g
3475:(
3470:O
3465:,
3462:)
3459:1
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3453:3
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3446:P
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3407:a
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3379:6
3375:y
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3352:f
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3340:y
3334:2
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3326:e
3323:+
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3259:x
3255:b
3252:+
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3236:=
3231:2
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3187:j
3181:i
3159:j
3156:,
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3125:,
3122:)
3117:c
3114:,
3111:b
3098:c
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3057:3
3052:A
3025:1
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3012:c
3009:,
3006:b
3003:,
3000:a
2977:)
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2968:x
2965:(
2962:)
2959:b
2953:x
2950:(
2947:)
2944:a
2938:x
2935:(
2932:)
2929:1
2923:x
2920:(
2917:x
2909:2
2905:y
2850:1
2847:,
2844:1
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2813:1
2807:M
2798:]
2795:C
2792:[
2770:1
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2730:)
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2722:(
2695:1
2689:C
2683:P
2658:C
2652:|
2645:1
2642:,
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2610:)
2606:C
2602:(
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2551:=
2545:C
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2450:1
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2438::
2435:j
2393:3
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2387:0
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2325:1
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2290:,
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2284:(
2254:1
2249:P
2225:1
2220:P
2198:0
2176:0
2170:M
2147:0
2124:0
2121:=
2118:)
2113:2
2105:C
2097:,
2094:C
2091:(
2086:0
2082:H
2058:4
2033:2
2025:C
2000:2
1971:C
1939:)
1934:2
1926:C
1918:,
1915:C
1912:(
1907:0
1903:H
1889:)
1879:C
1875:T
1866:C
1858:,
1855:C
1852:(
1847:0
1843:H
1832:)
1827:C
1823:T
1819:,
1816:C
1813:(
1808:1
1804:H
1769:)
1764:C
1760:T
1756:,
1753:C
1750:(
1745:1
1741:H
1715:1
1710:P
1688:0
1664:0
1631:1
1625:g
1564:g
1558:M
1533:g
1529:H
1502:S
1479:g
1475:H
1448:S
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1421:H
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1364:)
1361:6
1355:g
1352:5
1349:(
1345:L
1342:G
1339:P
1334:/
1328:0
1323:g
1319:H
1315:=
1310:0
1305:g
1299:M
1274:o
1269:g
1265:H
1256:g
1252:H
1248:=
1239:S
1216:i
1213:,
1210:g
1206:H
1183:o
1178:g
1174:H
1161:,
1147:n
1144:,
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1137:H
1122:1
1119:,
1116:g
1112:H
1103:o
1098:g
1094:H
1068:g
1064:H
1037:g
1031:M
1008:)
1001:C
997:,
994:C
991:(
986:S
981:m
978:o
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915:]
912:)
909:6
903:g
900:5
897:(
893:L
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887:P
882:/
876:g
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868:[
848:)
845:1
839:g
836:(
833:)
830:1
824:n
821:6
818:(
815:=
812:)
809:n
806:(
801:g
797:P
771:3
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736:)
733:n
730:(
725:g
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713:1
707:5
701:g
698:5
693:P
686:b
683:l
680:i
677:H
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642:H
619:g
613:M
572:1
566:M
530:=
520:)
517:)
514:2
511:(
507:L
504:G
501:P
497:(
485:0
482:=
475:)
467:(
455:)
447:(
412:0
406:M
383:1
377:g
357:3
351:g
348:3
328:3
322:g
319:3
294:g
284:M
256:1
250:g
228:g
222:M
137:(
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
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