Toric orbifolds associated to Cartan matrices
[Orbifolds toriques associés aux matrices de Cartan]
Blume, Mark
Annales de l'Institut Fourier, Tome 65 (2015), p. 863-901 / Harvested from Numdam
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Nous étudions les champs de modules des chaînes de ℙ 1 marquées, reliés aux espaces de modules de Losev-Manin, et montrons que ces champs de modules coïncident avec certains champs toriques qui peuvent être décrits en termes de matrices de Cartan de systèmes de racines de type A. Nous considérons également les variantes de ces champs liés aux systèmes de racines de type B et C.

We investigate moduli stacks of pointed chains of ℙ 1 related to the Losev-Manin moduli spaces and show that these moduli stacks coincide with certain toric stacks which can be described in terms of the Cartan matrices of root systems of type A. We also consider variants of these stacks related to root systems of type B and C.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2946
Classification:  14M25,  14D23,  14H10
Mots clés: Espaces de modules de Losev-Manin, champs toriques, systèmes de racines, matrices de Cartan, permutoèdre. 
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     author = {Blume, Mark},
     title = {Toric orbifolds associated to Cartan matrices},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {863-901},
     doi = {10.5802/aif.2946},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_2_863_0}
}

Blume, Mark. Toric orbifolds associated to Cartan matrices. Annales de l'Institut Fourier, Tome 65 (2015) pp. 863-901. doi : 10.5802/aif.2946. http://gdmltest.u-ga.fr/item/AIF_2015__65_2_863_0/

[1] Abramovich, D.; Olsson, M.; Vistoli, A. Tame stacks in positive characteristic, Ann. Inst. Fourier, Tome 58 (2008), pp. 1057-1091 (arXiv:math/0703310) | Article | Numdam | MR 2427954 | Zbl 1222.14004

[2] Batyrev, V.; Blume, M. The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces, Tohoku Math. J., Tome 63 (2011), pp. 581-604 (arXiv:0911.3607) | Article | MR 2872957 | Zbl 1255.14041

[3] Batyrev, V.; Blume, M. On generalisations of Losev-Manin moduli spaces for classical root systems, Pure and Applied Mathematics Quarterly, Tome 7 (2011, (Special Issue: In memory of Eckart Viehweg)), pp. 1053-1084 (arXiv:0912.2898) | Article | MR 2918154

[4] Borisov, L.; Chen, L.; Smith, G. The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc., Tome 18 (2005), pp. 193-213 (arXiv:math/0309229) | Article | MR 2114820 | Zbl 1178.14057

[5] Cox, D. The functor of a smooth toric variety, Tohoku Math. J., Tome 47 (1995), pp. 251-262 (arXiv:alg-geom/9312001) | Article | MR 1329523 | Zbl 0828.14035

[6] Cox, D. The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., Tome 4 (1995), pp. 17-50 (arXiv:alg-geom/9210008) | MR 1299003 | Zbl 0846.14032

[7] Fantechi, B.; Mann, E.; Nironi, F. Smooth toric Deligne-Mumford stacks, J. reine angew. Math., Tome 648 (2010), pp. 201-244 (arXiv:0708.1254) | MR 2774310 | Zbl 1211.14009

[8] Giraud, J. Cohomologie non abélienne, Springer-Verlag, Berlin - Heidelberg - New York, Grundlehren math. Wiss., Tome 179 (1971) | MR 344253 | Zbl 0226.14011

[9] Grothendieck, A.; Dieudonné, J. Éléments de Géométrie Algébrique, Publ. Math. IHES Tome 4,8,11,17,20,24,28,32 (1960-1967)

[10] Grothendieck, A. Séminaire de géométrie algébrique, Théorie des topos et cohomologie étale des schémas, Tome 3, Springer-Verlag, Berlin - Heidelberg - New York, Lecture Notes in Mathematics, Tome 305 (1973)

[11] Iwanari, I. Integral Chow rings of toric stacks (arXiv:0705.3524)

[12] Kapranov, M. Chow quotients of Grassmannians I, Adv. Soviet Math., Tome 16 (1993), pp. 29-110 (arXiv:alg-geom/9210002) | MR 1237834 | Zbl 0811.14043

[13] Kleiman, S. The Picard scheme, Fundamental Algebraic Geometry, Amer. Math. Soc., Providence, RI (Math. Surveys Monogr.) Tome 123 (2005), p. 237-321, arXiv:math/0504020 | MR 2223410

[14] Knudsen, F. The projectivity of the moduli space of stable curves II: The stacks M g,n , Math. Scand., Tome 52 (1983), pp. 161-199 | MR 702953 | Zbl 0544.14020

[15] Laumon, G.; Moret-Bailly, L. Champs algébriques, Springer-Verlag, Berlin - Heidelberg - New York (2000) | MR 1771927 | Zbl 0945.14005

[16] Losev, A.; Manin, Yu New Moduli Spaces of Pointed Curves and Pencils of Flat Connections, Michigan Math. J., Tome 48 (2000), pp. 443-472 (arXiv:math/0001003) | Article | MR 1786500 | Zbl 1078.14536

[17] Perroni, F. A note on toric Deligne-Mumford stacks, Tohoku Math. J., Tome 60 (2008), pp. 441-458 (arXiv:0705.3823) | Article | MR 2453733 | Zbl 1174.14044

[18] Vistoli, A. Grothendieck topologies, fibered categories and descent theory, Fundamental Algebraic Geometry, Amer. Math. Soc., Providence, RI (Math. Surveys Monogr.) Tome 123 (2005), p. 1-104, arXiv:math/0412512 | MR 2223406