Good moduli spaces for Artin stacks
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2349-2402.

We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford’s geometric invariant theory and tame stacks.

Nous développons une théorie qui associe des espaces de modules ayant de bonnes propriétés géométriques des champs d’Artin arbitraires, généralisant ainsi la théorie géométrique des invariants de Mumford et les « champs modérés ».

DOI: 10.5802/aif.2833
Classification: 14L24,  14L30,  14J15
Keywords: Artin stacks, geometric invariant theory, moduli spaces
Alper, Jarod 1

1 Departmento de Matemáticas Universidad de los Andes Cra No. 18A-10 Bloque H Bogotá, 111711 Colombia
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Alper, Jarod. Good moduli spaces for Artin stacks. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2349-2402. doi : 10.5802/aif.2833. http://www.numdam.org/articles/10.5802/aif.2833/

[1] Abramovich, Dan; Olsson, Martin; Vistoli, Angelo Tame stacks in positive characteristic, Ann. Inst. Fourier, (Grenoble), Volume 58 (2008) no. 4, pp. 1057-1091 | DOI | Numdam | MR | Zbl

[2] Artin, Michael Versal deformations and algebraic stacks, Invent. Math., Volume 27 (1974), pp. 165-189 | DOI | MR | Zbl

[3] Białynicki-Birula, A. On homogeneous affine spaces of linear algebraic groups, Amer. J. Math., Volume 85 (1963), pp. 577-582 | DOI | MR | Zbl

[4] Caporaso, Lucia A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc., Volume 7 (1994) no. 3, pp. 589-660 | DOI | MR | Zbl

[5] Conrad, Brian Keel-mori theorem via stacks, 2005 (http://www.math.stanford.edu/~bdconrad/papers/coarsespace.pdf)

[6] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | DOI | Numdam | MR | Zbl

[7] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 22, Springer-Verlag, Berlin, 1990 (with an appendix by David Mumford) | MR | Zbl

[8] Fogarty, John Geometric quotients are algebraic schemes, Adv. in Math., Volume 48 (1983) no. 2, pp. 166-171 | DOI | MR | Zbl

[9] Fogarty, John Finite generation of certain subrings, Proc. Amer. Math. Soc., Volume 99 (1987) no. 1, pp. 201-204 | MR | Zbl

[10] Gieseker, D. On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Volume 106 (1977) no. 1, pp. 45-60 | DOI | MR | Zbl

[11] Grothendieck, Alexander Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. (1961-1967) no. 4,8,11,17,20,24,28,32 | Numdam | Zbl

[12] Haboush, W. J. Homogeneous vector bundles and reductive subgroups of reductive algebraic groups, Amer. J. Math., Volume 100 (1978) no. 6, pp. 1123-1137 | DOI | MR | Zbl

[13] Hassett, Brendan Classical and minimal models of the moduli space of curves of genus two, Geometric methods in algebra and number theory (Progr. Math.), Volume 235, Birkhäuser Boston, Boston, MA, 2005, pp. 169-192 | MR | Zbl

[14] Hassett, Brendan; Hyeon, Donghoon Log minimal model program for the moduli space of stable curves: The first flip, 2008 (math.AG/0806.3444) | Zbl

[15] Hassett, Brendan; Hyeon, Donghoon Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc., Volume 361 (2009) no. 8, pp. 4471-4489 | DOI | MR | Zbl

[16] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997 | MR | Zbl

[17] Hyeon, Donghoon; Lee, Yongnam Log minimal model program for the moduli space of stable curves of genus three, 2007 (math.AG/0703093) | MR | Zbl

[18] Hyeon, Donghoon; Lee, Yongnam Stability of tri-canonical curves of genus two, Math. Ann., Volume 337 (2007) no. 2, pp. 479-488 | DOI | MR | Zbl

[19] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math., Volume 145 (1997) no. 1, pp. 193-213 | DOI | MR | Zbl

[20] Knop, Friedrich; Kraft, Hanspeter; Vust, Thierry The Picard group of a G-variety, Algebraische Transformationsgruppen und Invariantentheorie (DMV Sem.), Volume 13, Birkhäuser, Basel, 1989, pp. 77-87 | MR | Zbl

[21] Knutson, Donald Algebraic spaces, Lecture Notes in Mathematics, 203, Springer-Verlag, Berlin, 1971 | MR | Zbl

[22] Kraft, Hanspeter G-vector bundles and the linearization problem, Group actions and invariant theory (Montreal, PQ, 1988) (CMS Conf. Proc.), Volume 110, Amer. Math. Soc., Providence, RI, 1989, pp. 111-123 | MR | Zbl

[23] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 39, Springer-Verlag, Berlin, 2000 | Zbl

[24] Lieblich, Max Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007) no. 1, pp. 23-118 | DOI | MR | Zbl

[25] Luna, Domingo Slices étalés, Sur les groupes algébriques (Bull. Soc. Math. France, Mémoire), Volume 33, Soc. Math. France, Paris, 1973, pp. 81-105 | Numdam | MR | Zbl

[26] Maruyama, Masaki Moduli of stable sheaves. I, J. Math. Kyoto Univ., Volume 17 (2007) no. 1, pp. 91-126 MR0450271 (56 #8567) | MR | Zbl

[27] Matsushima, Yozô Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J, Volume 16 (1960), pp. 205-218 | MR | Zbl

[28] Melo, Margarida Compactified picard stacks over the moduli stack of stable curves with marked points, 2008 (math.AG/0811.0763) | Zbl

[29] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR | Zbl

[30] Mumford, David Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, 22, Springer-Verlag, Berlin, 1965 | MR | Zbl

[31] Nagata, Masayoshi On the 14-th problem of Hilbert, Amer. J. Math., Volume 81 (1959), pp. 766-772 | DOI | MR | Zbl

[32] Nagata, Masayoshi Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ., Volume 1 (1961/1962), pp. 87-99 | MR | Zbl

[33] Nagata, Masayoshi Invariants of a group in an affine ring, J. Math. Kyoto Univ., Volume 3 (1963/1964), pp. 369-377 | MR | Zbl

[34] Nironi, Fabio Moduli spaces of semistable sheaves on projective deligne-mumford stacks, 2008 (math.AG/0811.1949)

[35] Olsson, Martin Sheaves on Artin stacks, J. Reine Angew. Math., Volume 603 (2007), pp. 55-112 | MR | Zbl

[36] Raynaud, Michel; Gruson, Laurent Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | MR | Zbl

[37] Richardson, R. W. Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc., Volume 9 (1977) no. 1, pp. 38-41 | DOI | MR | Zbl

[38] Rydh, David Noetherian approximation of algebraic spaces and stacks, 2010 (math.AG/0904.0227v3)

[39] Rydh, David Existence and properties of geometric quotients, J. Algebraic Geom. (2013) (to appear) | DOI | MR | Zbl

[40] Schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, 151,152,153 (1962/1964)

[41] Schubert, David A new compactification of the moduli space of curves, Compositio Math., Volume 78 (1991) no. 3, pp. 297-313 | Numdam | MR | Zbl

[42] Seshadri, C. S. Geometric reductivity over arbitrary base, Advances in Math., Volume 26 (1977) no. 3, pp. 225-274 | DOI | MR | Zbl

[43] Seshadri, C. S. Fibrés vectoriels sur les courbes algébriques, Astérisque, 96, Société Mathématique de France, Paris, 1982 (Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980) | MR | Zbl

[44] Simpson, Carlos T. Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. (1994) no. 79, pp. 47-129 | DOI | Numdam | MR | Zbl

[45] Vistoli, Angelo Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry (Math. Surveys Monogr.), Volume 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1-104 | MR

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