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, pp. 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},
     pages = {2349--2402},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     doi = {10.5802/aif.2833},
     mrnumber = {3237451},
     zbl = {06325437},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2833/}
}
TY  - JOUR
AU  - Alper, Jarod
TI  - Good moduli spaces for Artin stacks
JO  - Annales de l'Institut Fourier
PY  - 2013
DA  - 2013///
SP  - 2349
EP  - 2402
VL  - 63
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2833/
UR  - https://www.ams.org/mathscinet-getitem?mr=3237451
UR  - https://zbmath.org/?q=an%3A06325437
UR  - https://doi.org/10.5802/aif.2833
DO  - 10.5802/aif.2833
LA  - en
ID  - AIF_2013__63_6_2349_0
ER  - 
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://archive.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 | Article | Numdam | MR 2427954 | Zbl 1222.14004

[2] Artin, Michael Versal deformations and algebraic stacks, Invent. Math., Volume 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., Volume 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., Volume 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, 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 1083353 | Zbl 0744.14031

[8] Fogarty, John Geometric quotients are algebraic schemes, Adv. in Math., Volume 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., Volume 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), Volume 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., Volume 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 (Progr. Math.), Volume 235, Birkhäuser Boston, Boston, MA, 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., Volume 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, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 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., Volume 337 (2007) no. 2, pp. 479-488 | Article | MR 2262795 | Zbl 1111.14017

[19] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math., Volume 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 (DMV Sem.), Volume 13, Birkhäuser, Basel, 1989, pp. 77-87 | MR 1044586 | Zbl 0705.14005

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

[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 1021283 | Zbl 0703.14009

[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 0945.14005

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

[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 342523 | Zbl 0286.14014

[26] Maruyama, Masaki Moduli of stable sheaves. I, J. Math. Kyoto Univ., Volume 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, Volume 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, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR 1304906 | Zbl 0797.14004

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

[31] Nagata, Masayoshi On the 14-th problem of Hilbert, Amer. J. Math., Volume 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., Volume 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., Volume 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., Volume 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., Volume 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., Volume 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, 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 1106299 | Zbl 0735.14022

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

[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 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 (Math. Surveys Monogr.), Volume 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1-104 | MR 2223406

Cité par Sources :