Toric orbifolds associated to Cartan matrices
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 863-901.

We investigate moduli stacks of pointed chains of 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.

Nous étudions les champs de modules des chaînes de 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.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.2946
Classification: 14M25,  14D23,  14H10
Keywords: Losev-Manin moduli spaces, toric stacks, root systems, Cartan matrices, permutohedron.
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Blume, Mark. Toric orbifolds associated to Cartan matrices. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 863-901. doi : 10.5802/aif.2946. https://aif.centre-mersenne.org/articles/10.5802/aif.2946/

[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, Grundlehren math. Wiss., Tome 179, Springer-Verlag, Berlin - Heidelberg - New York, 1971 | MR: 344253 | Zbl: 0226.14011

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

[10] Grothendieck, A. Séminaire de géométrie algébrique, Théorie des topos et cohomologie étale des schémas, Tome 3, Lecture Notes in Mathematics, Tome 305, Springer-Verlag, Berlin - Heidelberg - New York, 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 (Math. Surveys Monogr.) Tome 123, Amer. Math. Soc., Providence, RI, 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 (Math. Surveys Monogr.) Tome 123, Amer. Math. Soc., Providence, RI, 2005, p. 1-104, arXiv:math/0412512 | MR: 2223406

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