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G-covers, 504, 525 G cover of a stable pointed admissible, family of, 527 curve, 556 automorphism of, 526 Beltrami differential, 466 limits of, 526 covers, family of, 526 G-linearization, 340 quasi-diffeomorphism, 468 V -cover, 274 Ahlfors, Lars, 498 Γ-marking, 315 Alexeev, Valery, 879 weak, 314 Algebraic Index Theorem, 422 μ1 map, 808 Algebraic space, 251, 270, 307 ω-coordinate, 463 groupoid presentation of, 306, ω-geodesic, 473 307 ball, 476 normalization of, 308 horizontal, 473 separated, 270 ray, 474 Altman, Allen, 879 vertical, 473 Ample ω-length, 473 cone of Mg, 439 ω-metric, 473 locally free sheaf, 229 2-category, 280 Ampleness Nakai’s criterion of, 424 of bλ − δ + ψ, 425 Abel–Jacobi map, 446 of κ + aλ + b ψ , 435 relative, 790 1 i i of κ + aλ, 425 Abikoff, William, 498 1 of Mumford’s class κ , 398, Abramovich, Dan, 562, 881, 883 1 425 Adapted of the relative dualizing sheaf charts, 56 ω , 424 functions, 56 f Seshadri’s criterion of, 230, 426 metric, 57 Andreotti, Aldo, 248 metric on a Cm family of Aprodu, Marian, 883, 885 vector bundles, 215 Arakelov, Suren Ju., 424, 435, partition of unity, 57 438 relative form, 57 Arbarello, Enrico, 397, 879, 884 section, 57 Arc complex, 609, 613 Additivity property Arrows of a Lie groupoid, 275 of the κ class, 377, 427 1 Arsie, Alessandro, 397 of the Hodge class, 365, 427 Artin, Michael, 323 Adem, Alejandro, 323 Asymptotic expansion, 736 Adjunction isomorphism for in more than one variable, 739 Deligne pairing, 375 of the partition function, 744 Admissible Atiyah, Michael, 769 G-cover, 504, 525, 556

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

Atlas property stable under, 300 and descent data, 329, 337 Beauville, Arnaud, 880, 883 for an algebraic space, 251, 270 Behrend, Kai, 769 for an orbifold, 277 Beilinson, Alexander A., 397 for a Deligne–Mumford stack, Beltrami 300 differential, 466 relative Cm,56 equation, 445, 466 Automorphisms of admissible Benedetti, Riccardo, 665 G-covers, 536 Bernoulli number, 585, 751, 765 Average Bers, Lipman, 497, 498, 665 or expectation value, 734 Bertram, Aaron, 605, 884 stretching of a quasi- Bessis, Daniel, 771 diffeomorphism, 480 Bini, Gilberto, 773, 885 Axis of a hyperbolic Bipartition transformation, 631 of a pair (integer, finite set), 95, 100 stable, 100, 261, 312, 339, 571 Baer, R., 497 Birman, Joan, 497 Baily, Walter L., Jr., 438 Biswas, Indranil, 397 Ballico, Edoardo, 881 Blow-up Barja, Miguel Angel,´ 438 and stable reduction, 106 Barth, Wolf, 161 of CN , real oriented, 488 Base change real oriented, 149 and ampleness, 231 Boggi, Marco, 562 and stable reduction, 105 Borel, Armand, 683 compatibility of Deligne Bost, Jean-Benoˆıt, 438 pairing with, 331, 369, 371 Bott, Raoul, 769 compatibility of Hilbert Boundary scheme with, 46 of moduli, as a determinant, compatibility of Hodge line 361 bundle and points bundles of moduli space, 81, 261, 279 with, 344 of moduli space; irreducible compatibility of Mumford’s components of, 262 class κ with, 377 1 pullback under clutching, 347 compatibility of relative Boundary class, 339, 571, 676, 717 dualizing sheaf with, 98 for the moduli stack of stable compatibility of the boundary hyperelliptic curves, 391 divisor with, 363 Boundary divisor, 261, 262, 312, compatibility of the 313, 339 determinant of the as a determinant 331, 361 cohomology with, 358 contribution from, in Witten’s compatibility of Riemann–Roch conjecture, 724 isomorphism with, 379 in Mumford’s formula for the faithfully flat, 292 canonical class of moduli in cohomology, 1, 8, 12, 13, space, 386 121, 388, 788 Index 947

pullback of, under clutching, for ψ-classes, 694 583, 584 and vanishing theorems for of M 0,P , 599, 601–604, 608 homology of moduli spaces, Boundary strata 671 of moduli space of stable extension to bordification, 614, curves, 312, 321 652 of moduli stack of stable Chang, Mei-Chu, 884 curves, 312 Characteristic pullback of, under clutching, exterior homomorphism, 510 582 homomorphism, 821 Bowditch, Brian H., 665 subgroup, 510 Brézin, Edouard,´ 772 Characteristic linear system, 3, Brill, Alexander von, 779, 883 32, 65, 243 Brill–Noether Characteristic map, 32, 243 matrix, 790 Chasles, Michel, 766 number, 779, 795, 808, 813, Chern character, 382, 586 827, 869 Chern classes subloci of Mg, 793 of the boundary divisors, 339, theory, 779 571, 676, 717 theory, dimension theorem, 835 of the Hodge bundle, 334, 572 varieties, 780, 788–793 of the point bundles, 335, 572, varieties, tangent spaces to, 694, 717 807 of the sheaf of relative Kähler Bruno, Andrea, 884 differentials, 383 Bryan, Jim, 605 Chern, Shiing-Shen, 497 Brylinski, Jean-Luc, 562 Chisini, Oscar, 883 Chow ring Gorenstein conjectures, 597 Canonaco, Alberto, 323 of a moduli stack, 570 Canonical class of a quotient of a smooth of M , 386 g,n variety by a finite group, of M , 386 g,n 570 of M , 332, 344 g,P of M , 565, 570, 605 Caporaso, Lucia, 879, 881 g of M , 599 Cartan, Henri, 209, 257 0,P Chow variety, 70 Cartesian morphism, 280 open, 70 Castelnuovo, Guido, 851, 864, 882 Ciliberto, Ciro, 880 Category fibered in groupoids, Classes 279, 294, 332, 335 boundary, 339, 391, 396, 571, Catenacci, Roberto, 397 602, 676, 678, 710, 713, 717, Cattani, Eduardo, 594 721 Cavalieri, Renzo, 605, 884 Mumford’s, 332, 572, 721 Cellular decomposition of Mumford–Morita–Miller, 572, Teichmüller space, 609, 721 614, 623, 643, 690 point-bundle, 335, 572, 717 and combinatorial expression 948 Index

Classes (cont.) Cukierman, Fernando, 881, 882 tautological, 382, 384, 565, 570, Curvature form of a connection, 572, 573, 581, 596, 604, 669, 224 676, 680, 710, 713, 717, 721 Curve Cleavage, 281 hyperelliptic stable, 101, 192 Clebsch, Alfred, 854, 883 nodal, 83 Clemens, C. Herbert, 161, 880 nodal n-pointed, 94 Clutching, 81, 126, 187, 254, 311– nodal P -pointed, 94 323, 330, 345, 396, 565, 570, nodal, with marked points, 92 581–585, 589, 752 of compact type, 90 Codimension of a regular semistable, 100 embedding, 36 stable, 99 Coherent topology, 615 Cusp, 630 Cohomology Cycle rings of moduli stacks of base change in. See Base curves, 570 change determinant of the. See D’Souza, Cyril, 879 Determinant Date, Etsur¯o, 773 equivariant, 754–759 De Concini, Corrado, 397 of moduli spaces, 445, 485, 565, de Franchis’ theorem, 830 599, 668, 670–689, 708, 710 de Franchis, Michele, 780, 882 of orbifolds, 278 de Jong, Aise Johan, 161, 562 rational, of Γ ,82 g de Jonquières, Ernest, 766, 768 PL, 696 de Rham complex, 591 Collar Lemma, 635 Deformation Colombo, Elisabetta, 883 continuous, of a compact Commutative diagram in complex manifold, 213 a category fibered in differentiable, of a compact groupoids, 280 complex manifold, 213 Composition in a Lie groupoid, first-order, 172 275 first order embedded, 27, 42 Connection, 224 first order, of a morphism, 819, compatible with hermitian 836 product, 225 first order, of a pair (curve, Gauss–Manin, 220, 593 line bundle), 803 hermitian, 225 first order, of an admissible Conormal sheaf, 31 G-cover, 557 Continuous system of plane infinitesimal, 167–171, 197, curves, 847 201, 242, 769, 835 Contraction of a graph, 314 of a morphism, 819 Contraction functor, 125 of an analytic space, 172 Coolidge, Julian L., 65 of a nodal curve, 178 Coppens, Marc, 883 of a scheme, 172 Cornalba, Maurizio, 397, 438 Dehn twist, 82, 145–158, 445, Corti, Alessio, 562, 881, 883 460, 483, 491, 493, 535 Index 949

G2 Dehn, Max, 497 of d , 846 1 ≤ ≤ Dehn–Nielsen of Mg,d for 2 d g/2+1, realization, 443 864 1 theorem, 454, 459 of Mg,d, expected, 813 Deligne pairing, 367, 369 of the Severi variety, 847 as a product of determinants, of the Hilbert scheme, lower 371 bound on, 33, 54 W1 Deligne, Pierre, 323, 396, 397, 562, of d , 811 604, 669, 674, 675, 686, 709 Divisor Deligne–Gysin spectral sequence, admissible, 356–363 669, 685 boundary. See Boundary Descent divisor construction of the stack Cartier, 123, 329, 335, 339, [X/G], 297 356, 366, 422, 783 data, 89, 253, 289, 294 class 365, 391, 599, 606 data defining line bundles on effective, 177, 361, 367, 373, moduli stacks, 336–343 387, 435, 788, 818 data, effective, 295 nef, 426, 433–438 faithfully flat, for quasi- of sections of a family of coherent sheaves, 288–294 curves, 95 theory, 253, 323 relative, 243, 367, 371, 375– Determinant 377, 785, 800 boundary of moduli as, 361 theory of characteristic system Hodge line bundle as, 355, 357, for, 243 359 universal, 243, 784, 789 of a finite complex, 330, 350, 703 universal, effective, 784 of a vector bundle as a Z/2- with normal crossings, 106, graded line bundle, 348 149, 152, 161, 279, 487, 669, of the cohomology, 330, 354, 685, 709 357, 396 zero, 27, 36, 98, 131 of the hypercohomology, 331, 357 Dolgachev, Igor, 437 Determinantal Douglas, Michael R., 772 curve, 75 Dualizing sheaf, 90, 97, 101 variety, generic, 792 logaritmic, relative, 377, 572 Di Francesco, Philippe, 720, 745, relative, 97, 572 771 relative, direct image of, 234, Diaz, Steven, 566, 598, 882, 883 334 Dickey, Leonid A., 773 relative, nefness of, 435 Dijkgraaf, Robbert, 726, 772 relative, positivity properties Dilatation, 469 of, 417–421, 424 minimal, 469 Dilaton equation, 574, 723 Edge Dimension disconnecting, 95 of Brill–Noether varieties, nondisconnecting, 95 expected, 795 of a graph, 93 950 Index

Edidin, Dan, 323, 709, 881, 883 Faber, Carel, 566, 580, 597, 605, Eguchi, Tohru, 770 750, 773 Eisenbud, David, 439, 843, 873, Faithfully flat 880–884 descent, 288–294 Ekedahl, Torsten, 685, 771, 884 algebra, 87 Eliashberg, Yakov, 685 module, 291 Enriques, Federigo, 65, 859, 883 morphism of schemes, 289 Epstein, David B. A., 497, 665 Family Equivalence Cm, of compact complex λ-, of stable curves, 436 manifolds, 62, 213–216 of categories, 280, 282, 284, Cm,ofcurveswithTeichmüller 289, 337 structure, 450 of deformations, 172 Cm, of differentiable of deformations of n-pointed manifolds, 56 curves, 176 Cm, of differentiable vector Equivalence relation bundles, 57, 215 in the context of groupoids, 276 Cm, of projective varieties, 63 quotient of, 270 flat, of subschems of PN ,1,5, quotient of, Grothendieck’s 12, 22, 26 theorem, 784 isotrivial, of curves, 418 relation defining Deligne Mumford’s, of curves in P3, pairing, 367 40–43 schematic, 268 of curves on quadrics, 74 Esteves, Eduardo, 879, 881, 882 of curves with general moduli, Euler sequence, 35, 197, 813, 822 794 Euler–Poincaré characteristic, 63, of curves with level G 361, 382, 527 structure, 508 virtual, 693, 721, 754, 758–766, of curves with level m 773, 777 structure, 503, 538 Exceptional of curves with level ψ chain, 111 structure, 511 divisor, 110, 371, 600, 714, 854, of curves with Teichmüller 876 structure, 444, 449, 471 Excess intersection, 321, 330, 346, of elliptic curves, semistable 396, 582 reduction of, 161 Expectation value, 734 of formally self-adjoint, Expected dimension strongly elliptic differential of Brill–Noether varieties, 795 operators, 215 1 of Mg,d, 813 of Γ-marked stable curves, 315 Exterior of hyperelliptic curves, 418, 606 differentiation, along the fibers, of hypersurfaces in PN ,8 219 of k-planes, 10 r homomorphism, 454, 501, 508, of gd’s, 792 509, 514 of ν-log-canonically embedded isomorphism, 455, 459 curves, 288 Index 951

of nodal curves, 83 Fiorenza, Domenico, 771 of semistable curves, stable Fitting ideal, 196, 788–790 model of, 124 Flag Hilbert scheme, 48 of P -pointed nodal curves, 95, Flat 101 R-module, 4 of quadrics in 3-space, 55 coherent sheaf, 5 of rational normal curves, 73 family of subschemes, 5 of semistable curves, 101 morphism, 5 of smooth cubics in P3, 41, 76 Fogarty, J., 437 of smooth Beltrami Fontanari, Claudio, 883, 885 differentials, 468, 471 Ford, Lester R., 665 of stable n-pointed curves, 81, Formally self-adjoint, strongly 101 elliptic differential operator. of stable curves, isomorphisms See Family of formally self- of, 113–117 adjoint, strongly elliptic of subschemes in the fibers of a differential operators morphism, 43, 53 Fricke, Robert, 443, 461 of subschemes of a scheme, 4 Fuchsian group, 627 of subschemes of an affine Fujiki, Akira, 579 scheme, 66 Fulton, William, 439, 566, 855, of zero-dimensional 883 subschemes, 10 Functor universal, on Hilbert scheme, 25 contraction Contr, 125 universal, on moduli space, deformation, 248 lack of, 266, 267, 283, 286 essentially surjective, 282 universal, on moduli space, fully faithful, 282 surrogate for, 267, 307 p(t) hilbX/S ,43 universal, on moduli stack, 310 Isom, 253, 296 universal, on Teichmüller Hilbert, 2, 6, 25 space, 449 moduli, 285, 504 Fantechi, Barbara, 248, 323, 769, Picard, 782, 879 884 projection Pr, 125 Farkas, Gavril, 439, 881, 883–885 projection, for a category Farkas, Hershel, 812 fibered in groupoids, 279 Fenchel–Nielsen coordinates, 445, representable, 2, 25, 285 485, 487, 494, 497 Gr represented by d(p), 793 Feynman Wr represented by d (p), 789 diagram, 734–744, 776 stable model StMd, 124 move, 621, 692, 701 Teichmüller, 450 Fiber product Fundamental region of stacks, 299, 303 for a Fuchsian group, 629 symmetric, of a family of improper side of, 630 curves, 242, 784, 797 improper vertex of, 629 symmetric, of the universal Funnel, 634 curve, 675 952 Index

Göttsche, Lothar, 323 Grauert, Hans, 248 GAGA, 87, 172 Green operator, 215 Galatius, Søren, 684, 685 Green, Mark, 248, 880, 883, 885 Gardiner, Friederick P., 498 Griffiths, Phillip, 709, 882 Gatto, Letterio, 881, 882 Gromov, Mikhail, 766, 884 Gauss–Bonnet, 476, 478, 628, 644 Gross, David J., 772 Gaussian Grothendieck Riemann–Roch measure, 719, 734–742 formula, 382, 585 map, 880 formula, for the determinant of Gelfand–Dikii form of KdV the cohomology, 379 hierarchy, 726 theorem, 415, 416, 565, 585, Geodesics 588 for the hyperbolic metric. See Grothendieck, Alexander, 64, 248, Geodesics for the Poincaré 323, 396, 498, 580, 668, 784 metric Groupoid, 251 for the metric induced by a complex orbifold, 277 quadratic differential, 473– contravariant functor as a, 283 479 isomorphisms of, 280 for the Poincaré metric, 611, Lie, 275 623, 628, 633, 637, 658 moduli, 286 Geometric realization of a graph, moduli space as a, 281 93 morphisms of, 280 Gervais, Sylvain, 460, 497 orbifold, 276 Getzler, Ezra, 566, 605, 769, 773 presentation of a Deligne– Ghost components, 649 Mumford stack, 304 Gibney, Angela, 439 presentation of an algebraic Gieseker, David, 438, 880 space, 307 Gillet, Henri, 323 proper étale Lie, 276 Givental, Alexander B., 769, 773 quotient, 286 Gorchinskiy, Sergey, 397 represented by a scheme, 283 Gorenstein conjecture, 597 scheme as a, 253 Gorenstein graded algebra, 597 sections of a, 281 Goulden, Ian P., 605, 884 Grzegorczyk, Ivona, 882 Graber, Tom, 323, 396, 604, 605, Gysin homomorphism, 686–689 769, 772 Graph Hain, Richard, 605, 685 P -marked, 93 Half-edge connected, 93 of a graph, 88, 93, 118, 126, dual, 88, 90, 93, 126, 160, 311– 322, 345, 363, 517, 581 323, 545, 548, 555, 582, of a ribbon graph, 616, 700, 648–653, 694 738 numbered, 93 Halpern, Noemi, 665 ribbon. See Ribbon graph Harer, John, 671, 683, 685, 708, semistable, 100 721, 773 stable, 99 Harish-Chandra, Mehrotra, 746 Index 953

Harmonic projector, 215 of hypersurfaces in projective Harris, Joseph, 397, 438, 781, space, 7 843, 850, 873, 880–884 of space conics, 67 Hartshorne, Robin, 27, 64, 90, of twisted cubics, 68 248 of zero-dimensional Hassett, Brendan, 439 subschemes, 10, 33, 72 Hermitian matrix model, 740 restricted, 69 Hilbert the Grassmannian as a, 10 functor. See Functor Hilbert, David, 438 point, 22, 63, 207, 399, 406– Hilbert–Mumford numerical 409, 414–416, 430, 438 criterion, 404 polynomial, 1, 4–26, 41, 43, 48, Hirschowitz, André, 881 67, 72, 112, 195 Hodge bundle, 226, 572, 585, 591 Hilbert scheme, 2, 6, 25, 43, 46 on the moduli stack of stable and base change, 46 curves, 334 of morphisms, 47 semipositivity of, 233, 237 of isomorphisms, 3, 48 Hodge class, 334, 585, 750 flag, 48 additivity of, 365 non-reduced, 40 generalized, 334 of complete intersections, 73 higher, 572 of curves on quadrics, 74 higher generalized, 573 and determinantal curves, 75 nefness of, 433 of k-planes in Pr,10 Hodge line bundle, 334, 344, 359 projectivity of, 26 ampleness on the Satake quasi-complete intersections, 75 compactification of Mg, sections of, 73 435–437 tangent space to, 33, 49–56 Homology lower bound on dimension, 33, equivariant, 755 54 of a group with integral universal property, 25 coefficients, 754 universal property with respect of Mg,n, vanishing of, 671 to analytic families, 26 Hori, Kentaro, 770 universal property with respect Horikawa class of a first-order to Cm families, 63 deformation of a morphism, universal family on, 25 821 variants of, 43 Horikawa, Eiji, 438, 780, 819, 824 of ν-log-canonically embedded Horizontal stable n-pointed genus g trajectory, 480 curves, 196 vector field, 479 of automorphisms of fibers of a Horocycle, 611, 632 standard Kuranishi family, region, 632 209 region, standard, 632 of closed subschemes of Howard, Alan, 882 projective space with given Hubbard, John Hamal, 161, 498 Hilbert polynomial, 7 Hulek, Klaus, 161, 882 954 Index

Humbert, Georges, 882 of a groupoid presentation of a Humphries, Stephen, 497 Deligne–Mumford stack, 304 Hurwitz Itzykson, Claude, 720, 745, 771 numbers, 771, 772 Ivanov, Nikolai V., 683 scheme, 854 Izadi, Elham, 605 space, 857 space, irreducibility of, 857 Jackson, David M., 605, 884 space, unirationality of, 878, Jacobian variety 900 of a nodal curve, 89 Hurwitz, Adolf, 854, 883 relative, 786 Huybrechts, Daniel, 64 Jacobian locus, 461 Hyeon, Donghoon, 439 Jacobson ring, 16 Hyperbolic spine, 611, 623, 640, Jenkins, James A., 771 659, 660, 697, 730 Jimbo, Michio, 773 Jost, J¨urgen, 665

Iarrobino, Anthony, 65 Kähler differentials, 95 Igusa, Kiyoshi, 771 relative, 95, 365 Illusie, Luc, 323 Kac, Victor, 397, 773 Imayoshi, Yôichi, 498, 665 Kaku, Michio, 772 Immersion Kaplan, Aroldo, 594 closed, of Deligne–Mumford Kashiwara, Masaki, 773 stacks, 304, 340 Kawamoto, Noboru, 397 open, of Deligne–Mumford Kazakov, Vladimir A., 772 stacks, 304 Kazarian, Maxim, 772, 883, 884 of algebraic spaces, 307 KdV (Korteweg de Vries) Index hierarchy, 726, 774 of a node of a stable genus Gelfand–Dikii form, 726 zero curve, 392 Keel, Seán, 323, 439, 566, 599 of a ramification point, 839 Keem, Chango, 883 of an admissible cover at a Keen, Linda, 665 node, 505, 527 Kempf, George, 242, 248 Infinitesimal automorphism, 116 Khosla, Deepak, 439 Inverse Kirwan, Frances, 685 function theorem, 57 Kleiman, Steven, 879, 881 in a Lie groupoid, 275, 306, Kleiman, Steven L., 323, 788 323 Kleppe, Jan O., 65 Ionel, Eleny-Nicoleta, 605 Knudsen, Finn Faye, 161, 323, Isomorphism 396, 438, 880, 883 of categories fibered in Knutson, Donald, 323 groupoids, 280 Kodaira, Kunihiko, 32, 65, 167, of deformations, 172 215, 248 Isotrivial family of curves, 418, Kodaira–Spencer 419, 422, 431 class, of a first-order Isotropy group deformation of a manifold, in an orbifold groupoid, 276 173 Index 955

class, of a first-order Lüroth, Jacob, 854, 883 deformation of a nodal Laksov, Dan, 882 curve, 178 Lando, Sergei K., 771, 772, 883, class, of a first-order 884 deformation of an n-pointed Lange, Herbert, 883 nodal curve, 183 Laplace–Beltrami operator, 214 class, of a first-order Laufer, Henry B., 882 deformation of a line Laumon, Gérard, 323 bundle, 201 Lax, Robert F., 882 class, of a first-order Lazarsfeld, Robert, 780, 814, 880, deformation of a pair 883, 885 (curve, line bundle), 804 Lazarsfeld–Mukai bundle, 814 homomorphism, 175, 178 Le Potier, Joseph, 881, 882 homomorphism, in a Kuranishi Lefschetz, Solomon, 161 family, 188 Leg of a graph, 93, 126, 313, 347, homomorphism, in a versal 363, 581, 648 family, 192 Lehn, Manfred, 64 homomorphism, and the Leida, Johann, 323 differential of the period Level map, 217 Jacobi structure of level m, Kollár, János, 64, 248, 323, 438 512 Konno, Kazuhiro, 438 Teichmüller structure of level Kontsevich’s matrix model, 743, G, 508, 511 745–750 structure associated to Kontsevich, Maxim, 397, 612, 702, a surjective exterior 709, 717, 743, 761, 768, 771 homomorphism, 511 Korn, Arthur, 497 structure dominating another Kouvidakis, Alexis, 709 one, 514 Kuranishi family Li, Jun, 769 action of automorphism group Lichtenstein, Leon, 497 on, 189 Lickorish, William B. R., 460, 497 for a morphism, 824 Lie groupoid, 275 for a curve with Teichmüller proper étale, 276 structure, 448 Line bundle for admissible G-covers, 530– even, 348 535, 557 graded, 348 for an n-pointed stable curve, 188 odd, 348 standard, 208 on a nodal curve, 89 standard algebraic, 207 on a Deligne–Mumford stack, standard, of hyperelliptic 333 stable curve, 210, 211 Hodge, 334 universal property with respect nef, 229–231 to continuous deformations, point, 334 212–216 G-equivariant, 343 Kuranishi, Masatake, 248 Poincaré, 781, 782, 785, 786 956 Index

Linear differential operators Martellini, Maurizio, 397 Cm family of, 215 Martens, Gerriet, 883 smooth dependence of solutions Martens, Henrik, 812 on parameters, 216 Martin-Deschamps, Mireille, 65 Linearly reductive linear algebraic Matelski, Peter J., 665 group, 401 Matsmura, Hideyuki, 96 Linearly stable curve in projective Matsuzaki, Katsuhiko, 665 space, 408 Max Noether’s theorem, 223, 241 Liu, Kefeng, 605, 884 Mayer, Alan, 161 Liu, Xiaobo, 769 Melo, Margarida, 879 Local complete intersection (l.c.i.) Mestrano, Nicole, 709 morphism, 86, 97, 578 Metric Local criterion of flatness, 28 conformal, 628 Local Torelli theorem, 223, 420, intrinsic, 633 461 Poincaré, 627, 628 for hyperelliptic curves, 224, Metric topology, 615 420 Migdal, Alexander A., 772 Log-canonical sheaf, 92, 99, 195 Miller, Edward, 604, 684 relative, 377, 572 Miranda, Rick, 880 Looijenga, Eduard, 498, 562, 566, Mirzakhani, Maryam, 772 598, 604, 605, 668, 684, 685, Mishachev, Nikolai M., 685 708, 771, 796, 882 Miwa, Tetsuji, 773 Module with descent data, 292 Moduli map Möbius transformation, 627 finite, onto moduli, 268, 307 dilatation of, 627 of a family of curves, 261 elliptic, 627 Moduli space hyperbolic, 627 of d-gonal curves, irreducibility parabolic, 627 and dimension, 864 translation, 627 coarse, for a stack, 302 Madsen, Ib, 684, 685 for admissible G-covers, 505, Manetti, Marco, 248 535, 556 Manin, Yuri˘ı I., 397, 773 of stable genus g curves, 104 Mapping class group, 144, 450, of elliptic curves, 254–257, 266 451, 454, 458, 459 of stable n-pointed genus g generators of, 460 curves, 257, 259, 260 action on bordification of of stable n-pointed genus zero Teichmüller space, 491 curves, 264, 265, 599 action on the arc complex, 614 of curves with level structure, Marking 508 weak Γ-, 314 of curves with ψ-structure, 510 Γ-, 314 of curves with level structure, of a ribbon graph, 619 compactification of, 522 of a P -pointed stable curve, of stable ribbon graphs, 664 490 of stable maps, 767 Index 957

Moduli space of curves Mumford’s example, 40–43 as an analytic space, 259, 260 Mumford’s formula, 384 boundary of, 261 Mumford’s relations for Hodge completeness, 268 classes, 586–592 as an algebraic space, 271 Mumford, David, 12, 65, 161, 323, as an orbifold, 277 396, 397, 435, 437, 438, 562, as a Deligne–Mumford stack, 565, 566, 591, 604, 605, 665, 300 683, 708, 812, 873, 881, 883, Picard group, 379 884 projectivity, 425 Mumford–Morita–Miller classes, irreducibility, 462, 861 572, 721 unirationality in low genus, Murri, Riccardo, 771 872 Moduli stack Nœther’s theorem, 223, 241, 461 of admissible G-covers, 505, Nag, Subhashis, 397, 498 535 Nagaraj, Donihakkalu S., 883 of stable n-pointed genus g Nagel, Jan, 883 curves, 138, 300 Nakano, Shigeo, 579 Mondello, Gabriele, 665 Namikawa, Yukihiko, 397 Monodromy Narasimhan, Mudumbai S., 881 group, local, 523 Newstead, Peter, 881 representation, 856 Nielsen extension, 634, 658 representation, local, 522 Nielsen kernel, 634 Moret-Bailly, Laurent, 323 Nielsen, Jakob, 497 Morgan, John, 709 Nirenberg, Louis, 248 Mori, Shigefumi, 323, 884 Nitsure, Nitin, 64, 323, 784 Morita, Shigeyuki, 604, 605, 684 Node, 83 Moriwaki, Atsushi, 438 assigned, 853, 877 Morphism nonseparating, 94, 100 of (categories ﬁbered in) nonseparating, on a stable groupoids, 280 hyperelliptic curve, 102, 390 of deformations, 172 separating, 95, 100 of families of nodal curves, 95 separating, on a stable of orbifold groupoids, 276 hyperelliptic curve, 102, 390 of stacks, 296 virtually nonexistent, 853 representable, of stacks, 299 Noether, Max, 779, 880, 883 Morrey, Charles B., 497 Norm map, 366, 375 Morrison, Ian, 439, 880, 883 Normal sheaf, 31 Mukai, Shigeru, 437, 881, 882, of a regular embedding, 38 884, 885 to a clutching morphism, 346 Mulase, Motohico, 771, 773 and Petri’s statement, 824 Multidegree, 89 to a morphism, 345, 819 Mumford class, 572, 721 Normalization κ , 332, 377 1 of a Deligne–Mumford stack, κ , ampleness of, 425 1 305 958 Index

Normalization (cont.) Period matrix, 217 of an algebraic space in an Perrin, Daniel, 65 extension of its function Persson, Ulf, 438 field, 308 Peters, Chris A. M., 161 Petri’s condition, 794, 808, 815 Petri’s statement, 780, 824 Obstructed tangent vector, 53 1 for gd’s, 810 Oda, Tadao, 879 2 Okounkov, Andrei, 769, 771, 884 for gd’s, 845 Oort, Frans, 161, 248, 882 Petri, K., 779, 811, 879, 885 Orbicellular decomposition of Petronio, Carlo, 665 moduli, 614, 623, 661, 690 Pfaffian, 702 extension to compactification, Picard functor, relative, 782, 879 614, 662 Picard group M Orbifold of g,n, 379, 381, 484, 713 local chart, 275 of a Deligne–Mumford stack, groupoid, 276 333 groupoid, complex, 277 of a stack quotient, 343 groupoid, quotient of, 278 of the moduli stack of stable quotient, of a manifold by a hyperelliptic curves, 391, finite group, 277, 323 396 structure, 277 Picard variety. See Relative structure, on the moduli space Picard variety of n-pointed genus g curves, Picard–Lefschetz, 277 representation, 145, 483, 523 Orbit transformation, 144 in an orbifold groupoid, 276 transformation and Dehn in a ribbon graph, 617, 618 twists along vanishing Orientation form on the cycles, 148, 158–160 combinatorial moduli space transformation in the context of curves, 699 of G covers, 539 Outer automorphisms, 454, 540 Pikaart, Martin, 562, 684 Pinkham, Henry, 882 Poincaré Pandharipande, Rahul, 323, 396, duality, on an orbifold, 279 566, 604, 605, 769, 771, 879, line bundle, 781, 782, 785, 786 884 metric, 627, 628 Pants decomposition, 485, 497 Poincaré, Henri, 880 Papadopol, Peter, 161, 498 Point bundles, 334, 344 Pareschi, Giuseppe, 814, 880 nefness of, 434 Partition function, 721 Point bundle classes, 335, 572, asymptotic expansion of, 744 717 Penner, Robert, 665, 771 combinatorial expression for, Period map, 217 697 holomorphicity of, 217, 220 intersection numbers, 721 differential of, 217 Polishchuk, Alexander, 773 Index 959

Polyakov, Alexander M., 397 Quotient Popa, Mihnea, 439, 881, 885 effective, of a schematic Positive component, 649, 653 equivalence relation, 270 Presentation of the mapping class of a schematic equivalence group, 460 relation, 270 Primitive sublattice, 546, 547 groupoid, 286 Procesi, Claudio, 397 orbifold, 277, 323 Projection stack, quasi-coherent sheaves functor, 125 on, 343 morphism, 311, 560 Projectivity Raina, Ashok K., 773 of Hilbert scheme, 26 Ramanan, Sundaraman, 882 of the moduli space of stable Ramification, 835 curves, 425 Ramification divisor, 836 Propagator, 735 Ramification index, 839 Pullback Ran, Ziv, 884 in a category fibered in Ratcliffe, John G., 665 groupoids, 281 Rational functions on an of boundary classes under irreducible algebraic space, projection, 581 271, 308 of boundary classes under Rational tail, 574 clutching, 581–584 moduli space of curves with, of Mumford–Morita–Miller 598 classes under projection, 581 Rauch, H. Ernest, 882 of Mumford–Morita–Miller Reduced degree of a curve in classes under clutching, 581, projective space, 408 582, 584 Rego, C. J., 879 Pushforward of the fundamental Regular embedding, 36, 38, 54, class of a stack, 569 87, 97 Regular sequence, 35–39 Qing Liu, 96 Reina, Cesare, 397 Quadratic differential, 462 Relative Picard variety, 781, 785, Teichmüller deformation 787 associated to a, 463–465 Relative Cm atlas, 56 Teichmüller map associated Relative dualizing sheaf, 97, 572 to a, 463–465 direct image of, 234, 334 canonical, on a Teichmüller nefness of, 435 deformation, 465 positivity properties of, 417– metric attached to a. See 421, 424 ω-metric Relatively minimal fibration, 438 Quadrics through the canonical Reynolds operator, 258, 402 curve, 248 Ribbon graph, 616 Quasi-complete intersection, 75 associated to a proper simplex, Quasi-diffeomorphism, 468 621 admissible, 468 dual of, 619 960 Index

Ribbon graph (cont.) Schechtman, Vadim V., 397 embedded, 620 Schiffer variation, 175–177, 533, genus and boundary 837 components of, 617 Schlessinger, Michael, 248 half-edge of, 616, 700, 738 Schmid, Wilfried, 594, 709 half perimeters of, 619 Schreyer, Frank-Olaf, 883 in connection with Gaussian Schubert, Hermann, 768 integrals, 741 Section of a category fibered in isomorphism, 619 groupoids, 281 marking of, 619 Segal, Graeme, 773 oriented edges of, 617 Segre, Beniamino, 65, 813, 864, stable P -marked, 648 869, 873, 884 stable P -marked, moduli space Seifert, Herbert, 455, 497 of, 664 Semipositive locally free sheaf, topological surface attached to, 229, 230 617 Semistable unital metric on, 619 graph, 100 unital metric on stable P - curve, 100 marked, 648 point, in the sense of Riccati equation, 726 Geometric Invariant Theory, Riemann surface, 151, 167, 216, 401 466, 473, 479 Sernesi, Edoardo, 64, 248, 879, compact n-pointed, 462, 469, 884 559, 672 Serre, Jean-Pierre, 87, 172 hyperbolic, 624 Seshadri’s criterion of ampleness, of finite type, 627, 629, 633 230, 426 parabolic, 624 Seshadri, Conjeeveram S., 879, simply connected, 624 881, 882 with boundary, 489 Severi, Francesco, 65, 811, 812, Riemann’s existence theorem, 850, 882–884 799, 856 Shadrin, Sergey, 772, 773, 884 Riemann’s extension theorem, Shapiro, Michael, 771 257, 260 Sheaf Riemann’s moduli count, 828, 834 G-equivariant quasi-coherent, Rim, Dock Sang, 882 340 Rosenlicht, Maxwell, 397 quasi-coherent, on a Deligne– Ruan, Yongbin, 323, 769 Mumford stack, 333, 337 Shenker, Stephen H., 772 Siegel upper half-space, 217 Samuel, Pierre, 882 Simplex Sard’s lemma for flatness, 18 in the arc complex, 613 Sard’s theorem, 809 proper, 613 Satake compactification of the Singularity moduli space of genus g A , 109 curves, 437 n of a quadratic differential, 463 Satake, Ichir¯o, 323, 439 Index 961

Slope Steenbrink, Joseph, 248 inequality, 417, 438 Stoppino, Lidia, 438 of a divisor in moduli space, Strebel, Kurt, 771 439 Stretching function, 479 Smith, Roy, 248 String equation, 574, 723, 747 Socle, 597 Strongly characteristic Sommese, Andrew J., 882 quotient, 510, 541 Soulé, Christophe, 396 subgroup, 510, 541 Source of an arrow of a Lie Subgraph, 313 groupoid, 275 Substack, 304, 339, 340 Specialization Sullivan, Dennis, 397 of a graph, 314, 319 Symbol map, 804 of an automorphism of a graph, 314, 319 Tangent space Spencer, Donald C., 32, 65, 167, to Gr, 805, 807 215, 248 d to Wr, 807 Springer, George, 665 d to Hilbert scheme, 33, 49–56 Stability of the cohomology of to the Hilbert scheme of ν-log- moduli, 683 canonical stable curves, 198, Stabilization functor, 128 202 as inverse of the contraction to the moduli space of functor, 138 admissible G-covers, 531 Stable to relative Picard variety, 805 graph, 99 Taniguchi, Masahiko, 498, 665 curve, 99 Tannenbaum, Allen, 850, 883 P -marked ribbon graph, 648 Target of an arrow of a Lie point, in the sense of groupoid, 275 Geometric Invariant Theory, Tautological 401 class. See Classes, tautological Stable model relation, 382, 386, 565, 573 of a family of semistable ring, 565, 584, 587, 591, 605, curves, 124 796 of a semistable curve, 118 Teichmüller Stable reduction, 104–113 deformation, 463 for admissible G-covers, 528 map, 462, 463, 465, 469, 470 theorem, 113 marking of a stable curve, 490 uniqueness of, 114, 116 modular group, 144, 441, 450, Stack, 295 453, 454, 459, 483, 683, 757 Artin, 300 space, 441, 446, 453, 454, 459, cycle ring of, 570 471, 483, 509, 614, 757 Deligne–Mumford, 300 space, bordification of, 490– Standard coordinate patch for the 497, 614 Teichmüller space, 448 space, cellular decomposition of. Standard system of parameters, See Cellular decomposition 152 of Teichmüller space 962 Index

Teichmüller (cont.) of a separating node, 95, 261 structure of level G, 508 of node of hyperelliptic stable structure on a pointed curve, curve, 102, 211, 389, 390 445 strucure on a family of pointed Uniformization, 624–626 curves, 447 Unimodular lattice, 551 theory, 167 Unirational variety, 326 uniqueness theorem, 469, 479– Unit of a Lie groupoid, 275 483 Universal deformation of a map, Teichmüller, Oswald, 441 825 Teixidor i Bigas, Montserrat, Universal effective divisor, 881–883 relative of degree d, 784 Thorup, Anders, 882 Vainshtein, Alek, 771, 884 Thurston, William, 665 Vakil, Ravi, 65, 605, 772, 884 Tian, Gang, 769 Van de Ven, Antonius, 161 Tillmann, Ulrike, 684, 685 van der Geer, Gerard, 882 Toda lattice, 772 Vanishing Todd class, 382, 588 cycle, 146, 158, 493, 497, 524, Todorov, Gueorgui T., 605, 884 546, 549, 555 Topology of the homology of M in coherent, 615 g,n high degree, 671 metric, 615 theorem, for the tautological of the bordification of ring, 796 Teichmüller space, 494– Varley, Robert, 248 496, 655 Vector field Topological covering of the stack horizontal, 479 M , 483 g,n vertical, 479 Topological surface Verlinde, Erik Peter, 726, 772 nodal P -pointed, 649 Verlinde, Herman, 726, 772 stable P -pointed, 649 Verra, Alessandro, 884 Torelli group, 460 Versal family, 192 Torelli theorem, 216 Vertical local. See Local Torelli theorem coordinates, 56 Torsion of a complex, 704 derivatives, 56 Total degree of a line bundle, 89 vector field, 479 Totally unimodular Vertices of a graph, 93, 126, 313, lattice, 552 549, 581, 616, 637, 648 matrix, 552 Veselov, Vladimir, 161, 498 Transverse family of stable Viehweg, Eckart, 439 curves, 152, 155, 157, 492 Virasoro Triangle decomposition, 455 algebra, 725 Tromba, Anthony J., 498 equations, 718, 726, 773 Tsuchiya, Akihiro, 397 operators, 718, 722 Type Vistoli, Angelo, 323, 397, 562, of a ramification point, 839 604, 881, 883 Index 963

Vitulli, Marie, 882 Xu, Hao, 605, 884 Viviani, Filippo, 397, 885 Voisin, Claire, 780, 813, 880, Yamada, Yasuhiko, 397 883–885 Yamaki, Kazuhiko, 438 Yoneda lemma, 2-categorical, 284, Wahl, Jonathan, 851, 880, 883 335 Weierstrass point, 262, 388, 798, 881 Weil reciprocity, 366, 396 Zagier, Don, 721, 773 Weil, Andr´e, 248 Zariski’s connectedness theorem, Weiss, Michael S., 684, 685 308 Weyl, Hermann, 438 Zariski’s main theorem, 254, 435 Wick’s lemma, 735 Zariski, Oscar, 65, 882, 883 Wilson, George, 773 Zeuthen, Hieronymus, 766, 768 Witten, Edward, 612, 709, 717, Zuber, Jean Bernard, 720, 745, 766, 771, 884 771 Zucconi, Francesco, 438 Xiao, Gang, 438 Zvonkin, Alexander, 771 Xiong, Chuan-Sheng, 770 Zvonkine, Dimitri, 772, 773, 884