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1. Abikoff, William, The real analytic theory of Teichmuller space,Lecture Notes in Mathematics, 820, Springer-Verlag, Berlin, 1980.
2. Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles andadmissible covers, Comm. Algebra 31 (2003), 3547–3618.
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Index
G-covers, 504, 525admissible, family of, 527automorphism of, 526limits of, 526
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
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
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,
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,
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,