Good moduli spaces for Artin stacks  [ Bons espaces de modules pour les champs d’Artin ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 6, p. 2349-2402
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 ».
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.
DOI : https://doi.org/10.5802/aif.2833
Classification:  14L24,  14L30,  14J15
Mots clés: champs d’Artin, théorie géométrique des invariants, espaces de modules
@article{AIF_2013__63_6_2349_0,
     author = {Alper, Jarod},
     title = {Good moduli spaces for Artin stacks},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     pages = {2349-2402},
     doi = {10.5802/aif.2833},
     mrnumber = {3237451},
     zbl = {06325437},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_6_2349_0}
}
Alper, Jarod. Good moduli spaces for Artin stacks. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2349-2402. doi : 10.5802/aif.2833. http://www.numdam.org/item/AIF_2013__63_6_2349_0/

[1] Abramovich, Dan; Olsson, Martin; Vistoli, Angelo Tame stacks in positive characteristic, Ann. Inst. Fourier, (Grenoble), Tome 58 (2008) no. 4, pp. 1057-1091 | Article | Numdam | MR 2427954 | Zbl 1222.14004

[2] Artin, Michael Versal deformations and algebraic stacks, Invent. Math., Tome 27 (1974), pp. 165-189 | Article | MR 399094 | Zbl 0317.14001

[3] Białynicki-Birula, A. On homogeneous affine spaces of linear algebraic groups, Amer. J. Math., Tome 85 (1963), pp. 577-582 | Article | MR 186674 | Zbl 0116.38202

[4] Caporaso, Lucia A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc., Tome 7 (1994) no. 3, pp. 589-660 | Article | MR 1254134 | Zbl 0827.14014

[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 | Article | Numdam | MR 262240 | Zbl 0181.48803

[7] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 22 (1990) (with an appendix by David Mumford) | MR 1083353 | Zbl 0744.14031

[8] Fogarty, John Geometric quotients are algebraic schemes, Adv. in Math., Tome 48 (1983) no. 2, pp. 166-171 | Article | MR 700982 | Zbl 0556.14023

[9] Fogarty, John Finite generation of certain subrings, Proc. Amer. Math. Soc., Tome 99 (1987) no. 1, pp. 201-204 | MR 866454 | Zbl 0627.13006

[10] Gieseker, D. On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Tome 106 (1977) no. 1, pp. 45-60 | Article | MR 466475 | Zbl 0381.14003

[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 0122.16102

[12] Haboush, W. J. Homogeneous vector bundles and reductive subgroups of reductive algebraic groups, Amer. J. Math., Tome 100 (1978) no. 6, pp. 1123-1137 | Article | MR 522693 | Zbl 0432.14029

[13] Hassett, Brendan Classical and minimal models of the moduli space of curves of genus two, Geometric methods in algebra and number theory, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 235 (2005), pp. 169-192 | MR 2166084 | Zbl 1094.14017

[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 1273.14034

[15] Hassett, Brendan; Hyeon, Donghoon Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc., Tome 361 (2009) no. 8, pp. 4471-4489 | Article | MR 2500894 | Zbl 1172.14018

[16] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, E31 (1997) | MR 1450870 | Zbl 0872.14002

[17] Hyeon, Donghoon; Lee, Yongnam Log minimal model program for the moduli space of stable curves of genus three (2007) (math.AG/0703093) | MR 2661168 | Zbl 1230.14035

[18] Hyeon, Donghoon; Lee, Yongnam Stability of tri-canonical curves of genus two, Math. Ann., Tome 337 (2007) no. 2, pp. 479-488 | Article | MR 2262795 | Zbl 1111.14017

[19] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math., Tome 145 (1997) no. 1, pp. 193-213 | Article | MR 1432041 | Zbl 0881.14018

[20] Knop, Friedrich; Kraft, Hanspeter; Vust, Thierry The Picard group of a G-variety, Algebraische Transformationsgruppen und Invariantentheorie, Birkhäuser, Basel (DMV Sem.) Tome 13 (1989), pp. 77-87 | MR 1044586 | Zbl 0705.14005

[21] Knutson, Donald Algebraic spaces, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 203 (1971) | MR 302647 | Zbl 0221.14001

[22] Kraft, Hanspeter G-vector bundles and the linearization problem, Group actions and invariant theory (Montreal, PQ, 1988), Amer. Math. Soc., Providence, RI (CMS Conf. Proc.) Tome 110 (1989), pp. 111-123 | MR 1021283 | Zbl 0703.14009

[23] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Springer-Verlag, Berlin, 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], Tome 39 (2000) | Zbl 0945.14005

[24] Lieblich, Max Moduli of twisted sheaves, Duke Math. J., Tome 138 (2007) no. 1, pp. 23-118 | Article | MR 2309155 | Zbl 1122.14012

[25] Luna, Domingo Slices étalés, Sur les groupes algébriques, Soc. Math. France, Paris (Bull. Soc. Math. France, Mémoire) Tome 33 (1973), pp. 81-105 | Numdam | MR 342523 | Zbl 0286.14014

[26] Maruyama, Masaki Moduli of stable sheaves. I, J. Math. Kyoto Univ., Tome 17 (2007) no. 1, pp. 91-126 (MR0450271 (56 #8567)) | MR 450271 | Zbl 0374.14002

[27] Matsushima, Yozô Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J, Tome 16 (1960), pp. 205-218 | MR 109854 | Zbl 0094.28201

[28] Melo, Margarida Compactified picard stacks over the moduli stack of stable curves with marked points (2008) (math.AG/0811.0763) | Zbl 1208.14010

[29] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Tome 34 (1994) | MR 1304906 | Zbl 0797.14004

[30] Mumford, David Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Tome 22 (1965) | MR 214602 | Zbl 0147.39304

[31] Nagata, Masayoshi On the 14-th problem of Hilbert, Amer. J. Math., Tome 81 (1959), pp. 766-772 | Article | MR 105409 | Zbl 0192.13801

[32] Nagata, Masayoshi Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ., Tome 1 (1961/1962), pp. 87-99 | MR 142667 | Zbl 0106.25201

[33] Nagata, Masayoshi Invariants of a group in an affine ring, J. Math. Kyoto Univ., Tome 3 (1963/1964), pp. 369-377 | MR 179268 | Zbl 0146.04501

[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., Tome 603 (2007), pp. 55-112 | MR 2312554 | Zbl 1137.14004

[36] Raynaud, Michel; Gruson, Laurent Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Tome 13 (1971), pp. 1-89 | Article | MR 308104 | Zbl 0227.14010

[37] Richardson, R. W. Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc., Tome 9 (1977) no. 1, pp. 38-41 | Article | MR 437549 | Zbl 0355.14020

[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) | Article | MR 3084720 | Zbl 1278.14003

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

[41] Schubert, David A new compactification of the moduli space of curves, Compositio Math., Tome 78 (1991) no. 3, pp. 297-313 | Numdam | MR 1106299 | Zbl 0735.14022

[42] Seshadri, C. S. Geometric reductivity over arbitrary base, Advances in Math., Tome 26 (1977) no. 3, pp. 225-274 | Article | MR 466154 | Zbl 0371.14009

[43] Seshadri, C. S. Fibrés vectoriels sur les courbes algébriques, Société Mathématique de France, Paris, Astérisque, Tome 96 (1982) (Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980) | MR 699278 | Zbl 0517.14008

[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 | Article | Numdam | MR 1307297 | Zbl 0891.14005

[45] Vistoli, Angelo Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Amer. Math. Soc., Providence, RI (Math. Surveys Monogr.) Tome 123 (2005), pp. 1-104 | MR 2223406