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Index

G-covers, 504, 525admissible, family of, 527automorphism of, 526limits of, 526

G-linearization, 340V -cover, 274Γ-marking, 315

weak, 314μ1 map, 808ω-coordinate, 463ω-geodesic, 473

ball, 476horizontal, 473ray, 474vertical, 473

ω-length, 473ω-metric, 4732-category, 280

Abel–Jacobi map, 446relative, 790

Abikoff, William, 498Abramovich, Dan, 562, 881, 883Adapted

charts, 56functions, 56metric, 57metric on a Cm family of

vector bundles, 215partition of unity, 57relative form, 57section, 57

Additivity propertyof the κ1 class, 377, 427of the Hodge class, 365, 427

Adem, Alejandro, 323Adjunction isomorphism for

Deligne pairing, 375Admissible

G-cover, 504, 525, 556

G cover of a stable pointedcurve, 556

Beltrami differential, 466covers, family of, 526quasi-diffeomorphism, 468

Ahlfors, Lars, 498Alexeev, Valery, 879Algebraic Index Theorem, 422Algebraic space, 251, 270, 307

groupoid presentation of, 306,307

normalization of, 308separated, 270

Altman, Allen, 879Ample

cone of Mg, 439locally free sheaf, 229

AmplenessNakai’s criterion of, 424of bλ − δ + ψ, 425of κ1 + aλ +

∑biψi, 435

of κ1 + aλ, 425of Mumford’s class κ1, 398,

425of the relative dualizing sheaf

ωf , 424Seshadri’s criterion of, 230, 426

Andreotti, Aldo, 248Aprodu, Marian, 883, 885Arakelov, Suren Ju., 424, 435,

438Arbarello, Enrico, 397, 879, 884Arc complex, 609, 613Arrows of a Lie groupoid, 275Arsie, Alessandro, 397Artin, Michael, 323Asymptotic expansion, 736

in more than one variable, 739of the partition function, 744

Atiyah, Michael, 769

E. Arbarello et al., Geometry of Algebraic Curves,Grundlehren der mathematischen Wissenschaften 268,DOI 10.1007/978-3-540-69392-5, c© Springer-Verlag Berlin Heidelberg 2011

946 Index

Atlasand descent data, 329, 337for an algebraic space, 251, 270for an orbifold, 277for a Deligne–Mumford stack,

300relative Cm, 56

Automorphisms of admissibleG-covers, 536

Averageor expectation value, 734stretching of a quasi-

diffeomorphism, 480Axis of a hyperbolic

transformation, 631

Baer, R., 497Baily, Walter L., Jr., 438Ballico, Edoardo, 881Barja, Miguel Angel, 438Barth, Wolf, 161Base change

and ampleness, 231and stable reduction, 105compatibility of Deligne

pairing with, 331, 369, 371compatibility of Hilbert

scheme with, 46compatibility of Hodge line

bundle and points bundleswith, 344

compatibility of Mumford’sclass κ1 with, 377

compatibility of relativedualizing sheaf with, 98

compatibility of the boundarydivisor with, 363

compatibility of thedeterminant of thecohomology with, 358

compatibility of Riemann–Rochisomorphism with, 379

faithfully flat, 292in cohomology, 1, 8, 12, 13,

121, 388, 788

property stable under, 300Beauville, Arnaud, 880, 883Behrend, Kai, 769Beilinson, Alexander A., 397Beltrami

differential, 466equation, 445, 466

Benedetti, Riccardo, 665Bernoulli number, 585, 751, 765Bers, Lipman, 497, 498, 665Bertram, Aaron, 605, 884Bessis, Daniel, 771Bini, Gilberto, 773, 885Bipartition

of a pair (integer, finite set),95, 100

stable, 100, 261, 312, 339, 571Birman, Joan, 497Biswas, Indranil, 397Blow-up

and stable reduction, 106of C

N , real oriented, 488real oriented, 149

Boggi, Marco, 562Borel, Armand, 683Bost, Jean-Benoıt, 438Bott, Raoul, 769Boundary

of moduli, as a determinant,361

of moduli space, 81, 261, 279of moduli space; irreducible

components of, 262pullback under clutching, 347

Boundary class, 339, 571, 676, 717for the moduli stack of stable

hyperelliptic curves, 391Boundary divisor, 261, 262, 312,

313, 339as a determinant 331, 361contribution from, in Witten’s

conjecture, 724in Mumford’s formula for the

canonical class of modulispace, 386

Index 947

pullback of, under clutching,583, 584

of M0,P , 599, 601–604, 608Boundary strata

of moduli space of stablecurves, 312, 321

of moduli stack of stablecurves, 312

pullback of, under clutching,582

Bowditch, Brian H., 665Brezin, Edouard, 772Brill, Alexander von, 779, 883Brill–Noether

matrix, 790number, 779, 795, 808, 813,

827, 869subloci of Mg, 793theory, 779theory, dimension theorem, 835varieties, 780, 788–793varieties, tangent spaces to,

807Bruno, Andrea, 884Bryan, Jim, 605Brylinski, Jean-Luc, 562

Canonaco, Alberto, 323Canonical class

of Mg,n, 386of Mg,n, 386of Mg,P , 332, 344

Caporaso, Lucia, 879, 881Cartan, Henri, 209, 257Cartesian morphism, 280Castelnuovo, Guido, 851, 864, 882Category fibered in groupoids,

279, 294, 332, 335Catenacci, Roberto, 397Cattani, Eduardo, 594Cavalieri, Renzo, 605, 884Cellular decomposition of

Teichmuller space, 609,614, 623, 643, 690

and combinatorial expression

for ψ-classes, 694and vanishing theorems for

homology of moduli spaces,671

extension to bordification, 614,652

Chang, Mei-Chu, 884Characteristic

exterior homomorphism, 510homomorphism, 821subgroup, 510

Characteristic linear system, 3,32, 65, 243

Characteristic map, 32, 243Chasles, Michel, 766Chern character, 382, 586Chern classes

of the boundary divisors, 339,571, 676, 717

of the Hodge bundle, 334, 572of the point bundles, 335, 572,

694, 717of the sheaf of relative Kahler

differentials, 383Chern, Shiing-Shen, 497Chisini, Oscar, 883Chow ring

Gorenstein conjectures, 597of a moduli stack, 570of a quotient of a smooth

variety by a finite group,570

of Mg, 565, 570, 605of M0,P , 599

Chow variety, 70open, 70

Ciliberto, Ciro, 880Classes

boundary, 339, 391, 396, 571,602, 676, 678, 710, 713, 717,721

Mumford’s, 332, 572, 721Mumford–Morita–Miller, 572,

721point-bundle, 335, 572, 717

948 Index

Classes (cont.)tautological, 382, 384, 565, 570,

572, 573, 581, 596, 604, 669,676, 680, 710, 713, 717, 721

Cleavage, 281Clebsch, Alfred, 854, 883Clemens, C. Herbert, 161, 880Clutching, 81, 126, 187, 254, 311–

323, 330, 345, 396, 565, 570,581–585, 589, 752

Codimension of a regularembedding, 36

Coherent topology, 615Cohomology

base change in. See Basechange

determinant of the. SeeDeterminant

equivariant, 754–759of moduli spaces, 445, 485, 565,

599, 668, 670–689, 708, 710of orbifolds, 278rational, of Γg, 82PL, 696

Collar Lemma, 635Colombo, Elisabetta, 883Commutative diagram in

a category fibered ingroupoids, 280

Composition in a Lie groupoid,275

Connection, 224compatible with hermitian

product, 225Gauss–Manin, 220, 593hermitian, 225

Conormal sheaf, 31Continuous system of plane

curves, 847Contraction of a graph, 314Contraction functor, 125Coolidge, Julian L., 65Coppens, Marc, 883Cornalba, Maurizio, 397, 438Corti, Alessio, 562, 881, 883

Cukierman, Fernando, 881, 882Curvature form of a connection,

224Curve

hyperelliptic stable, 101, 192nodal, 83nodal n-pointed, 94nodal P -pointed, 94nodal, with marked points, 92of compact type, 90semistable, 100stable, 99

Cusp, 630Cycle rings of moduli stacks of

curves, 570

D’Souza, Cyril, 879Date, Etsuro, 773De Concini, Corrado, 397de Franchis’ theorem, 830de Franchis, Michele, 780, 882de Jong, Aise Johan, 161, 562de Jonquieres, Ernest, 766, 768de Rham complex, 591Deformation

continuous, of a compactcomplex manifold, 213

differentiable, of a compactcomplex manifold, 213

first-order, 172first order embedded, 27, 42first order, of a morphism, 819,

836first order, of a pair (curve,

line bundle), 803first order, of an admissible

G-cover, 557infinitesimal, 167–171, 197,

201, 242, 769, 835of a morphism, 819of an analytic space, 172of a nodal curve, 178of a scheme, 172

Dehn twist, 82, 145–158, 445,460, 483, 491, 493, 535

Index 949

Dehn, Max, 497Dehn–Nielsen

realization, 443theorem, 454, 459

Deligne pairing, 367, 369as a product of determinants,

371Deligne, Pierre, 323, 396, 397, 562,

604, 669, 674, 675, 686, 709Deligne–Gysin spectral sequence,

669, 685Descent

construction of the stack[X/G], 297

data, 89, 253, 289, 294data defining line bundles on

moduli stacks, 336–343data, effective, 295faithfully flat, for quasi-

coherent sheaves, 288–294theory, 253, 323

Determinantboundary of moduli as, 361Hodge line bundle as, 355, 357,

359of a finite complex, 330, 350, 703of a vector bundle as a Z/2-

graded line bundle, 348of the cohomology, 330, 354,

357, 396of the hypercohomology, 331, 357

Determinantalcurve, 75variety, generic, 792

Di Francesco, Philippe, 720, 745,771

Diaz, Steven, 566, 598, 882, 883Dickey, Leonid A., 773Dijkgraaf, Robbert, 726, 772Dilatation, 469

minimal, 469Dilaton equation, 574, 723Dimension

of Brill–Noether varieties,expected, 795

of G2d , 846

of M1g,d for 2 ≤ d ≤ g/2 + 1,

864of M1

g,d, expected, 813of the Severi variety, 847of the Hilbert scheme, lower

bound on, 33, 54of W 1

d , 811Divisor

admissible, 356–363boundary. See Boundary

divisorCartier, 123, 329, 335, 339,

356, 366, 422, 783class 365, 391, 599, 606effective, 177, 361, 367, 373,

387, 435, 788, 818nef, 426, 433–438of sections of a family of

curves, 95relative, 243, 367, 371, 375–

377, 785, 800theory of characteristic system

for, 243universal, 243, 784, 789universal, effective, 784with normal crossings, 106,

149, 152, 161, 279, 487, 669,685, 709

zero, 27, 36, 98, 131Dolgachev, Igor, 437Douglas, Michael R., 772Dualizing sheaf, 90, 97, 101

logaritmic, relative, 377, 572relative, 97, 572relative, direct image of, 234,

334relative, nefness of, 435relative, positivity properties

of, 417–421, 424

Edgedisconnecting, 95nondisconnecting, 95of a graph, 93

950 Index

Edidin, Dan, 323, 709, 881, 883Eguchi, Tohru, 770Eisenbud, David, 439, 843, 873,

880–884Ekedahl, Torsten, 685, 771, 884Eliashberg, Yakov, 685Enriques, Federigo, 65, 859, 883Epstein, David B. A., 497, 665Equivalence

λ-, of stable curves, 436of categories, 280, 282, 284,

289, 337of deformations, 172of deformations of n-pointed

curves, 176Equivalence relation

in the context of groupoids, 276quotient of, 270quotient of, Grothendieck’s

theorem, 784relation defining Deligne

pairing, 367schematic, 268

Esteves, Eduardo, 879, 881, 882Euler sequence, 35, 197, 813, 822Euler–Poincare characteristic, 63,

361, 382, 527virtual, 693, 721, 754, 758–766,

773, 777Exceptional

chain, 111divisor, 110, 371, 600, 714, 854,

876Excess intersection, 321, 330, 346,

396, 582Expectation value, 734Expected dimension

of Brill–Noether varieties, 795of M1

g,d, 813Exterior

differentiation, along the fibers,219

homomorphism, 454, 501, 508,509, 514

isomorphism, 455, 459

Faber, Carel, 566, 580, 597, 605,750, 773

Faithfully flatdescent, 288–294algebra, 87module, 291morphism of schemes, 289

FamilyCm, of compact complex

manifolds, 62, 213–216Cm, of curves with Teichmuller

structure, 450Cm, of differentiable

manifolds, 56Cm, of differentiable vector

bundles, 57, 215Cm, of projective varieties, 63flat, of subschems of P

N , 1, 5,12, 22, 26

isotrivial, of curves, 418Mumford’s, of curves in P

3,40–43

of curves on quadrics, 74of curves with general moduli,

794of curves with level G

structure, 508of curves with level m

structure, 503, 538of curves with level ψ

structure, 511of curves with Teichmuller

structure, 444, 449, 471of elliptic curves, semistable

reduction of, 161of formally self-adjoint,

strongly elliptic differentialoperators, 215

of Γ-marked stable curves, 315of hyperelliptic curves, 418, 606of hypersurfaces in P

N , 8of k-planes, 10of gr

d’s, 792of ν-log-canonically embedded

curves, 288

Index 951

of nodal curves, 83of semistable curves, stable

model of, 124of P -pointed nodal curves, 95,

101of quadrics in 3-space, 55of rational normal curves, 73of semistable curves, 101of smooth cubics in P

3, 41, 76of smooth Beltrami

differentials, 468, 471of stable n-pointed curves, 81,

101of stable curves, isomorphisms

of, 113–117of subschemes in the fibers of a

morphism, 43, 53of subschemes of a scheme, 4of subschemes of an affine

scheme, 66of zero-dimensional

subschemes, 10universal, on Hilbert scheme, 25universal, on moduli space,

lack of, 266, 267, 283, 286universal, on moduli space,

surrogate for, 267, 307universal, on moduli stack, 310universal, on Teichmuller

space, 449Fantechi, Barbara, 248, 323, 769,

884Farkas, Gavril, 439, 881, 883–885Farkas, Hershel, 812Fenchel–Nielsen coordinates, 445,

485, 487, 494, 497Feynman

diagram, 734–744, 776move, 621, 692, 701

Fiber productof stacks, 299, 303symmetric, of a family of

curves, 242, 784, 797symmetric, of the universal

curve, 675

Fiorenza, Domenico, 771Fitting ideal, 196, 788–790Flag Hilbert scheme, 48Flat

R-module, 4coherent sheaf, 5family of subschemes, 5morphism, 5

Fogarty, J., 437Fontanari, Claudio, 883, 885Ford, Lester R., 665Formally self-adjoint, strongly

elliptic differential operator.See Family of formally self-adjoint, strongly ellipticdifferential operators

Fricke, Robert, 443, 461Fuchsian group, 627Fujiki, Akira, 579Fulton, William, 439, 566, 855,

883Functor

contraction Contr, 125deformation, 248essentially surjective, 282fully faithful, 282hilbp(t)

X/S , 43Isom, 253, 296Hilbert, 2, 6, 25moduli, 285, 504Picard, 782, 879projection Pr, 125projection, for a category

fibered in groupoids, 279representable, 2, 25, 285represented by G r

d(p), 793represented by W r

d(p), 789stable model StMd, 124Teichmuller, 450

Fundamental regionfor a Fuchsian group, 629improper side of, 630improper vertex of, 629

Funnel, 634

952 Index

Gottsche, Lothar, 323GAGA, 87, 172Galatius, Søren, 684, 685Gardiner, Friederick P., 498Gatto, Letterio, 881, 882Gauss–Bonnet, 476, 478, 628, 644Gaussian

measure, 719, 734–742map, 880

Gelfand–Dikii form of KdVhierarchy, 726

Geodesicsfor the hyperbolic metric. See

Geodesics for the Poincaremetric

for the metric induced by aquadratic differential, 473–479

for the Poincare metric, 611,623, 628, 633, 637, 658

Geometric realization of a graph,93

Gervais, Sylvain, 460, 497Getzler, Ezra, 566, 605, 769, 773Ghost components, 649Gibney, Angela, 439Gieseker, David, 438, 880Gillet, Henri, 323Givental, Alexander B., 769, 773Gorchinskiy, Sergey, 397Gorenstein conjecture, 597Gorenstein graded algebra, 597Goulden, Ian P., 605, 884Graber, Tom, 323, 396, 604, 605,

769, 772Graph

P -marked, 93connected, 93dual, 88, 90, 93, 126, 160, 311–

323, 545, 548, 555, 582,648–653, 694

numbered, 93ribbon. See Ribbon graphsemistable, 100stable, 99

Grauert, Hans, 248Green operator, 215Green, Mark, 248, 880, 883, 885Griffiths, Phillip, 709, 882Gromov, Mikhail, 766, 884Gross, David J., 772Grothendieck Riemann–Roch

formula, 382, 585formula, for the determinant of

the cohomology, 379theorem, 415, 416, 565, 585,

588Grothendieck, Alexander, 64, 248,

323, 396, 498, 580, 668, 784Groupoid, 251

complex orbifold, 277contravariant functor as a, 283isomorphisms of, 280Lie, 275moduli, 286moduli space as a, 281morphisms of, 280orbifold, 276presentation of a Deligne–

Mumford stack, 304presentation of an algebraic

space, 307proper etale Lie, 276quotient, 286represented by a scheme, 283scheme as a, 253sections of a, 281

Grzegorczyk, Ivona, 882Gysin homomorphism, 686–689

Hain, Richard, 605, 685Half-edge

of a graph, 88, 93, 118, 126,322, 345, 363, 517, 581

of a ribbon graph, 616, 700,738

Halpern, Noemi, 665Harer, John, 671, 683, 685, 708,

721, 773Harish-Chandra, Mehrotra, 746

Index 953

Harmonic projector, 215Harris, Joseph, 397, 438, 781,

843, 850, 873, 880–884Hartshorne, Robin, 27, 64, 90,

248Hassett, Brendan, 439Hermitian matrix model, 740Hilbert

functor. See Functorpoint, 22, 63, 207, 399, 406–

409, 414–416, 430, 438polynomial, 1, 4–26, 41, 43, 48,

67, 72, 112, 195Hilbert scheme, 2, 6, 25, 43, 46

and base change, 46of morphisms, 47of isomorphisms, 3, 48flag, 48non-reduced, 40of complete intersections, 73of curves on quadrics, 74and determinantal curves, 75of k-planes in P

r, 10projectivity of, 26quasi-complete intersections, 75sections of, 73tangent space to, 33, 49–56lower bound on dimension, 33,

54universal property, 25universal property with respect

to analytic families, 26universal property with respect

to Cm families, 63universal family on, 25variants of, 43of ν-log-canonically embedded

stable n-pointed genus gcurves, 196

of automorphisms of fibers of astandard Kuranishi family,209

of closed subschemes ofprojective space with givenHilbert polynomial, 7

of hypersurfaces in projectivespace, 7

of space conics, 67of twisted cubics, 68of zero-dimensional

subschemes, 10, 33, 72restricted, 69the Grassmannian as a, 10

Hilbert, David, 438Hilbert–Mumford numerical

criterion, 404Hirschowitz, Andre, 881Hodge bundle, 226, 572, 585, 591

on the moduli stack of stablecurves, 334

semipositivity of, 233, 237Hodge class, 334, 585, 750

additivity of, 365generalized, 334higher, 572higher generalized, 573nefness of, 433

Hodge line bundle, 334, 344, 359ampleness on the Satake

compactification of Mg,435–437

Homologyequivariant, 755of a group with integral

coefficients, 754of Mg,n, vanishing of, 671

Hori, Kentaro, 770Horikawa class of a first-order

deformation of a morphism,821

Horikawa, Eiji, 438, 780, 819, 824Horizontal

trajectory, 480vector field, 479

Horocycle, 611, 632region, 632region, standard, 632

Howard, Alan, 882Hubbard, John Hamal, 161, 498Hulek, Klaus, 161, 882

954 Index

Humbert, Georges, 882Humphries, Stephen, 497Hurwitz

numbers, 771, 772scheme, 854space, 857space, irreducibility of, 857space, unirationality of, 878,

900Hurwitz, Adolf, 854, 883Huybrechts, Daniel, 64Hyeon, Donghoon, 439Hyperbolic spine, 611, 623, 640,

659, 660, 697, 730

Iarrobino, Anthony, 65Igusa, Kiyoshi, 771Illusie, Luc, 323Imayoshi, Yoichi, 498, 665Immersion

closed, of Deligne–Mumfordstacks, 304, 340

open, of Deligne–Mumfordstacks, 304

of algebraic spaces, 307Index

of a node of a stable genuszero curve, 392

of a ramification point, 839of an admissible cover at a

node, 505, 527Infinitesimal automorphism, 116Inverse

function theorem, 57in a Lie groupoid, 275, 306,

323Ionel, Eleny-Nicoleta, 605Isomorphism

of categories fibered ingroupoids, 280

of deformations, 172Isotrivial family of curves, 418,

419, 422, 431Isotropy group

in an orbifold groupoid, 276

of a groupoid presentation of aDeligne–Mumford stack, 304

Itzykson, Claude, 720, 745, 771Ivanov, Nikolai V., 683Izadi, Elham, 605

Jackson, David M., 605, 884Jacobian variety

of a nodal curve, 89relative, 786

Jacobian locus, 461Jacobson ring, 16Jenkins, James A., 771Jimbo, Michio, 773Jost, Jurgen, 665

Kahler differentials, 95relative, 95, 365

Kac, Victor, 397, 773Kaku, Michio, 772Kaplan, Aroldo, 594Kashiwara, Masaki, 773Kawamoto, Noboru, 397Kazakov, Vladimir A., 772Kazarian, Maxim, 772, 883, 884KdV (Korteweg de Vries)

hierarchy, 726, 774Gelfand–Dikii form, 726

Keel, Sean, 323, 439, 566, 599Keem, Chango, 883Keen, Linda, 665Kempf, George, 242, 248Khosla, Deepak, 439Kirwan, Frances, 685Kleiman, Steven, 879, 881Kleiman, Steven L., 323, 788Kleppe, Jan O., 65Knudsen, Finn Faye, 161, 323,

396, 438, 880, 883Knutson, Donald, 323Kodaira, Kunihiko, 32, 65, 167,

215, 248Kodaira–Spencer

class, of a first-orderdeformation of a manifold,173

Index 955

class, of a first-orderdeformation of a nodalcurve, 178

class, of a first-orderdeformation of an n-pointednodal curve, 183

class, of a first-orderdeformation of a linebundle, 201

class, of a first-orderdeformation of a pair(curve, line bundle), 804

homomorphism, 175, 178homomorphism, in a Kuranishi

family, 188homomorphism, in a versal

family, 192homomorphism, and the

differential of the periodmap, 217

Kollar, Janos, 64, 248, 323, 438Konno, Kazuhiro, 438Kontsevich’s matrix model, 743,

745–750Kontsevich, Maxim, 397, 612, 702,

709, 717, 743, 761, 768, 771Korn, Arthur, 497Kouvidakis, Alexis, 709Kuranishi family

action of automorphism groupon, 189

for a morphism, 824for a curve with Teichmuller

structure, 448for admissible G-covers, 530–

535, 557for an n-pointed stable curve, 188standard, 208standard algebraic, 207standard, of hyperelliptic

stable curve, 210, 211universal property with respect

to continuous deformations,212–216

Kuranishi, Masatake, 248

Luroth, Jacob, 854, 883Laksov, Dan, 882Lando, Sergei K., 771, 772, 883,

884Lange, Herbert, 883Laplace–Beltrami operator, 214Laufer, Henry B., 882Laumon, Gerard, 323Lax, Robert F., 882Lazarsfeld, Robert, 780, 814, 880,

883, 885Lazarsfeld–Mukai bundle, 814Le Potier, Joseph, 881, 882Lefschetz, Solomon, 161Leg of a graph, 93, 126, 313, 347,

363, 581, 648Lehn, Manfred, 64Leida, Johann, 323Level

Jacobi structure of level m,512

Teichmuller structure of levelG, 508, 511

structure associated toa surjective exteriorhomomorphism, 511

structure dominating anotherone, 514

Li, Jun, 769Lichtenstein, Leon, 497Lickorish, William B. R., 460, 497Lie groupoid, 275

proper etale, 276Line bundle

even, 348graded, 348odd, 348on a nodal curve, 89on a Deligne–Mumford stack,

333Hodge, 334nef, 229–231point, 334G-equivariant, 343Poincare, 781, 782, 785, 786

956 Index

Linear differential operatorsCm family of, 215smooth dependence of solutions

on parameters, 216Linearly reductive linear algebraic

group, 401Linearly stable curve in projective

space, 408Liu, Kefeng, 605, 884Liu, Xiaobo, 769Local complete intersection (l.c.i.)

morphism, 86, 97, 578Local criterion of flatness, 28Local Torelli theorem, 223, 420,

461for hyperelliptic curves, 224,

420Log-canonical sheaf, 92, 99, 195

relative, 377, 572Looijenga, Eduard, 498, 562, 566,

598, 604, 605, 668, 684, 685,708, 771, 796, 882

Mobius transformation, 627dilatation of, 627elliptic, 627hyperbolic, 627parabolic, 627translation, 627

Madsen, Ib, 684, 685Manetti, Marco, 248Manin, Yuriı I., 397, 773Mapping class group, 144, 450,

451, 454, 458, 459generators of, 460action on bordification of

Teichmuller space, 491action on the arc complex, 614

Markingweak Γ-, 314Γ-, 314of a ribbon graph, 619of a P -pointed stable curve,

490

Martellini, Maurizio, 397Martens, Gerriet, 883Martens, Henrik, 812Martin-Deschamps, Mireille, 65Matelski, Peter J., 665Matsmura, Hideyuki, 96Matsuzaki, Katsuhiko, 665Max Noether’s theorem, 223, 241Mayer, Alan, 161Melo, Margarida, 879Mestrano, Nicole, 709Metric

conformal, 628intrinsic, 633Poincare, 627, 628

Metric topology, 615Migdal, Alexander A., 772Miller, Edward, 604, 684Miranda, Rick, 880Mirzakhani, Maryam, 772Mishachev, Nikolai M., 685Miwa, Tetsuji, 773Module with descent data, 292Moduli map

finite, onto moduli, 268, 307of a family of curves, 261

Moduli spaceof d-gonal curves, irreducibility

and dimension, 864coarse, for a stack, 302for admissible G-covers, 505,

535, 556of stable genus g curves, 104of elliptic curves, 254–257, 266of stable n-pointed genus g

curves, 257, 259, 260of stable n-pointed genus zero

curves, 264, 265, 599of curves with level structure,

508of curves with ψ-structure, 510of curves with level structure,

compactification of, 522of stable ribbon graphs, 664of stable maps, 767

Index 957

Moduli space of curvesas an analytic space, 259, 260boundary of, 261completeness, 268as an algebraic space, 271as an orbifold, 277as a Deligne–Mumford stack,

300Picard group, 379projectivity, 425irreducibility, 462, 861unirationality in low genus,

872Moduli stack

of admissible G-covers, 505,535

of stable n-pointed genus gcurves, 138, 300

Mondello, Gabriele, 665Monodromy

group, local, 523representation, 856representation, local, 522

Moret-Bailly, Laurent, 323Morgan, John, 709Mori, Shigefumi, 323, 884Morita, Shigeyuki, 604, 605, 684Moriwaki, Atsushi, 438Morphism

of (categories fibered in)groupoids, 280

of deformations, 172of families of nodal curves, 95of orbifold groupoids, 276of stacks, 296representable, of stacks, 299

Morrey, Charles B., 497Morrison, Ian, 439, 880, 883Mukai, Shigeru, 437, 881, 882,

884, 885Mulase, Motohico, 771, 773Multidegree, 89Mumford class, 572, 721

κ1, 332, 377κ1, ampleness of, 425

Mumford’s example, 40–43Mumford’s formula, 384Mumford’s relations for Hodge

classes, 586–592Mumford, David, 12, 65, 161, 323,

396, 397, 435, 437, 438, 562,565, 566, 591, 604, 605, 665,683, 708, 812, 873, 881, 883,884

Mumford–Morita–Miller classes,572, 721

Murri, Riccardo, 771

Nœther’s theorem, 223, 241, 461Nag, Subhashis, 397, 498Nagaraj, Donihakkalu S., 883Nagel, Jan, 883Nakano, Shigeo, 579Namikawa, Yukihiko, 397Narasimhan, Mudumbai S., 881Newstead, Peter, 881Nielsen extension, 634, 658Nielsen kernel, 634Nielsen, Jakob, 497Nirenberg, Louis, 248Nitsure, Nitin, 64, 323, 784Node, 83

assigned, 853, 877nonseparating, 94, 100nonseparating, on a stable

hyperelliptic curve, 102, 390separating, 95, 100separating, on a stable

hyperelliptic curve, 102, 390virtually nonexistent, 853

Noether, Max, 779, 880, 883Norm map, 366, 375Normal sheaf, 31

of a regular embedding, 38to a clutching morphism, 346and Petri’s statement, 824to a morphism, 345, 819

Normalizationof a Deligne–Mumford stack,

305

958 Index

Normalization (cont.)of an algebraic space in an

extension of its functionfield, 308

Obstructed tangent vector, 53Oda, Tadao, 879Okounkov, Andrei, 769, 771, 884Oort, Frans, 161, 248, 882Orbicellular decomposition of

moduli, 614, 623, 661, 690extension to compactification,

614, 662Orbifold

local chart, 275groupoid, 276groupoid, complex, 277groupoid, quotient of, 278quotient, of a manifold by a

finite group, 277, 323structure, 277structure, on the moduli space

of n-pointed genus g curves,277

Orbitin an orbifold groupoid, 276in a ribbon graph, 617, 618

Orientation form on thecombinatorial moduli spaceof curves, 699

Outer automorphisms, 454, 540

Pandharipande, Rahul, 323, 396,566, 604, 605, 769, 771, 879,884

Pants decomposition, 485, 497Papadopol, Peter, 161, 498Pareschi, Giuseppe, 814, 880Partition function, 721

asymptotic expansion of, 744Penner, Robert, 665, 771Period map, 217

holomorphicity of, 217, 220differential of, 217

Period matrix, 217Perrin, Daniel, 65Persson, Ulf, 438Peters, Chris A. M., 161Petri’s condition, 794, 808, 815Petri’s statement, 780, 824

for g1d’s, 810

for g2d’s, 845

Petri, K., 779, 811, 879, 885Petronio, Carlo, 665Pfaffian, 702Picard functor, relative, 782, 879Picard group

of Mg,n, 379, 381, 484, 713of a Deligne–Mumford stack,

333of a stack quotient, 343of the moduli stack of stable

hyperelliptic curves, 391,396

Picard variety. See RelativePicard variety

Picard–Lefschetz,representation, 145, 483, 523transformation, 144transformation and Dehn

twists along vanishingcycles, 148, 158–160

transformation in the contextof G covers, 539

Pikaart, Martin, 562, 684Pinkham, Henry, 882Poincare

duality, on an orbifold, 279line bundle, 781, 782, 785, 786metric, 627, 628

Poincare, Henri, 880Point bundles, 334, 344

nefness of, 434Point bundle classes, 335, 572,

717combinatorial expression for,

697intersection numbers, 721

Polishchuk, Alexander, 773

Index 959

Polyakov, Alexander M., 397Popa, Mihnea, 439, 881, 885Positive component, 649, 653Presentation of the mapping class

group, 460Primitive sublattice, 546, 547Procesi, Claudio, 397Projection

functor, 125morphism, 311, 560

Projectivityof Hilbert scheme, 26of the moduli space of stable

curves, 425Propagator, 735Pullback

in a category fibered ingroupoids, 281

of boundary classes underprojection, 581

of boundary classes underclutching, 581–584

of Mumford–Morita–Millerclasses under projection, 581

of Mumford–Morita–Millerclasses under clutching, 581,582, 584

Pushforward of the fundamentalclass of a stack, 569

Qing Liu, 96Quadratic differential, 462

Teichmuller deformationassociated to a, 463–465

Teichmuller map associatedto a, 463–465

canonical, on a Teichmullerdeformation, 465

metric attached to a. Seeω-metric

Quadrics through the canonicalcurve, 248

Quasi-complete intersection, 75Quasi-diffeomorphism, 468

admissible, 468

Quotienteffective, of a schematic

equivalence relation, 270of a schematic equivalence

relation, 270groupoid, 286orbifold, 277, 323stack, quasi-coherent sheaves

on, 343

Raina, Ashok K., 773Ramanan, Sundaraman, 882Ramification, 835Ramification divisor, 836Ramification index, 839Ran, Ziv, 884Ratcliffe, John G., 665Rational functions on an

irreducible algebraic space,271, 308

Rational tail, 574moduli space of curves with,

598Rauch, H. Ernest, 882Reduced degree of a curve in

projective space, 408Rego, C. J., 879Regular embedding, 36, 38, 54,

87, 97Regular sequence, 35–39Reina, Cesare, 397Relative Picard variety, 781, 785,

787Relative Cm atlas, 56Relative dualizing sheaf, 97, 572

direct image of, 234, 334nefness of, 435positivity properties of, 417–

421, 424Relatively minimal fibration, 438Reynolds operator, 258, 402Ribbon graph, 616

associated to a proper simplex,621

dual of, 619

960 Index

Ribbon graph (cont.)embedded, 620genus and boundary

components of, 617half-edge of, 616, 700, 738half perimeters of, 619in connection with Gaussian

integrals, 741isomorphism, 619marking of, 619oriented edges of, 617stable P -marked, 648stable P -marked, moduli space

of, 664topological surface attached to,

617unital metric on, 619unital metric on stable P -

marked, 648Riccati equation, 726Riemann surface, 151, 167, 216,

466, 473, 479compact n-pointed, 462, 469,

559, 672hyperbolic, 624of finite type, 627, 629, 633parabolic, 624simply connected, 624with boundary, 489

Riemann’s existence theorem,799, 856

Riemann’s extension theorem,257, 260

Riemann’s moduli count, 828, 834Rim, Dock Sang, 882Rosenlicht, Maxwell, 397Ruan, Yongbin, 323, 769

Samuel, Pierre, 882Sard’s lemma for flatness, 18Sard’s theorem, 809Satake compactification of the

moduli space of genus gcurves, 437

Satake, Ichiro, 323, 439

Schechtman, Vadim V., 397Schiffer variation, 175–177, 533,

837Schlessinger, Michael, 248Schmid, Wilfried, 594, 709Schreyer, Frank-Olaf, 883Schubert, Hermann, 768Section of a category fibered in

groupoids, 281Segal, Graeme, 773Segre, Beniamino, 65, 813, 864,

869, 873, 884Seifert, Herbert, 455, 497Semipositive locally free sheaf,

229, 230Semistable

graph, 100curve, 100point, in the sense of

Geometric Invariant Theory,401

Sernesi, Edoardo, 64, 248, 879,884

Serre, Jean-Pierre, 87, 172Seshadri’s criterion of ampleness,

230, 426Seshadri, Conjeeveram S., 879,

881, 882Severi, Francesco, 65, 811, 812,

850, 882–884Shadrin, Sergey, 772, 773, 884Shapiro, Michael, 771Sheaf

G-equivariant quasi-coherent,340

quasi-coherent, on a Deligne–Mumford stack, 333, 337

Shenker, Stephen H., 772Siegel upper half-space, 217Simplex

in the arc complex, 613proper, 613

SingularityAn, 109of a quadratic differential, 463

Index 961

Slopeinequality, 417, 438of a divisor in moduli space,

439Smith, Roy, 248Socle, 597Sommese, Andrew J., 882Soule, Christophe, 396Source of an arrow of a Lie

groupoid, 275Specialization

of a graph, 314, 319of an automorphism of a

graph, 314, 319Spencer, Donald C., 32, 65, 167,

215, 248Springer, George, 665Stability of the cohomology of

moduli, 683Stabilization functor, 128

as inverse of the contractionfunctor, 138

Stablegraph, 99curve, 99P -marked ribbon graph, 648point, in the sense of

Geometric Invariant Theory,401

Stable modelof a family of semistable

curves, 124of a semistable curve, 118

Stable reduction, 104–113for admissible G-covers, 528theorem, 113uniqueness of, 114, 116

Stack, 295Artin, 300cycle ring of, 570Deligne–Mumford, 300

Standard coordinate patch for theTeichmuller space, 448

Standard system of parameters,152

Steenbrink, Joseph, 248Stoppino, Lidia, 438Strebel, Kurt, 771Stretching function, 479String equation, 574, 723, 747Strongly characteristic

quotient, 510, 541subgroup, 510, 541

Subgraph, 313Substack, 304, 339, 340Sullivan, Dennis, 397Symbol map, 804

Tangent spaceto G r

d , 805, 807to W r

d , 807to Hilbert scheme, 33, 49–56to the Hilbert scheme of ν-log-

canonical stable curves, 198,202

to the moduli space ofadmissible G-covers, 531

to relative Picard variety, 805Taniguchi, Masahiko, 498, 665Tannenbaum, Allen, 850, 883Target of an arrow of a Lie

groupoid, 275Tautological

class. See Classes, tautologicalrelation, 382, 386, 565, 573ring, 565, 584, 587, 591, 605,

796Teichmuller

deformation, 463map, 462, 463, 465, 469, 470marking of a stable curve, 490modular group, 144, 441, 450,

453, 454, 459, 483, 683, 757space, 441, 446, 453, 454, 459,

471, 483, 509, 614, 757space, bordification of, 490–

497, 614space, cellular decomposition of.

See Cellular decompositionof Teichmuller space

962 Index

Teichmuller (cont.)structure of level G, 508structure on a pointed curve,

445strucure on a family of pointed

curves, 447theory, 167uniqueness theorem, 469, 479–

483Teichmuller, Oswald, 441Teixidor i Bigas, Montserrat,

881–883Thorup, Anders, 882Thurston, William, 665Tian, Gang, 769Tillmann, Ulrike, 684, 685Toda lattice, 772Todd class, 382, 588Todorov, Gueorgui T., 605, 884Topology

coherent, 615metric, 615of the bordification of

Teichmuller space, 494–496, 655

Topological covering of the stackMg,n, 483

Topological surfacenodal P -pointed, 649stable P -pointed, 649

Torelli group, 460Torelli theorem, 216

local. See Local Torelli theoremTorsion of a complex, 704Total degree of a line bundle, 89Totally unimodular

lattice, 552matrix, 552

Transverse family of stablecurves, 152, 155, 157, 492

Triangle decomposition, 455Tromba, Anthony J., 498Tsuchiya, Akihiro, 397Type

of a ramification point, 839

of a separating node, 95, 261of node of hyperelliptic stable

curve, 102, 211, 389, 390

Uniformization, 624–626Unimodular lattice, 551Unirational variety, 326Unit of a Lie groupoid, 275Universal deformation of a map,

825Universal effective divisor,

relative of degree d, 784

Vainshtein, Alek, 771, 884Vakil, Ravi, 65, 605, 772, 884Van de Ven, Antonius, 161van der Geer, Gerard, 882Vanishing

cycle, 146, 158, 493, 497, 524,546, 549, 555

of the homology of Mg,n inhigh degree, 671

theorem, for the tautologicalring, 796

Varley, Robert, 248Vector field

horizontal, 479vertical, 479

Verlinde, Erik Peter, 726, 772Verlinde, Herman, 726, 772Verra, Alessandro, 884Versal family, 192Vertical

coordinates, 56derivatives, 56vector field, 479

Vertices of a graph, 93, 126, 313,549, 581, 616, 637, 648

Veselov, Vladimir, 161, 498Viehweg, Eckart, 439Virasoro

algebra, 725equations, 718, 726, 773operators, 718, 722

Vistoli, Angelo, 323, 397, 562,604, 881, 883

Index 963

Vitulli, Marie, 882Viviani, Filippo, 397, 885Voisin, Claire, 780, 813, 880,

883–885

Wahl, Jonathan, 851, 880, 883Weierstrass point, 262, 388, 798,

881Weil reciprocity, 366, 396Weil, Andre, 248Weiss, Michael S., 684, 685Weyl, Hermann, 438Wick’s lemma, 735Wilson, George, 773Witten, Edward, 612, 709, 717,

766, 771, 884

Xiao, Gang, 438Xiong, Chuan-Sheng, 770

Xu, Hao, 605, 884

Yamada, Yasuhiko, 397Yamaki, Kazuhiko, 438Yoneda lemma, 2-categorical, 284,

335

Zagier, Don, 721, 773Zariski’s connectedness theorem,

308Zariski’s main theorem, 254, 435Zariski, Oscar, 65, 882, 883Zeuthen, Hieronymus, 766, 768Zuber, Jean Bernard, 720, 745,

771Zucconi, Francesco, 438Zvonkin, Alexander, 771Zvonkine, Dimitri, 772, 773, 884