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The category of toric stacks

Part of: Special varieties Algebraic geometry: Foundations

Published online by Cambridge University Press:  01 May 2009

Isamu Iwanari
Isamu Iwanari*
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan (email: [email protected])
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Abstract

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In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

Keywords

toric geometrylogarithmic geometryalgebraic stacks

MSC classification

Secondary: 14M25: Toric varieties, Newton polyhedra 14A20: Generalizations (algebraic spaces, stacks)
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 718 - 746
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Abramovich, D. and Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27–75.CrossRefGoogle Scholar
[2]Abramovich, D., Olsson, M. and Vistoli, A., Tame stacks in positive characteristic, Ann. Inst. Fourier 58 (2008), 1057–1091.CrossRefGoogle Scholar
[3]Ash, A., Mumford, D., Rapoport, M. and Tai, Y.-S., Smooth compactifications of locally symmetric varieties (Mathematical Science Press, Brookline, MA, 1975).Google Scholar
[4]Borisov, L., Chen, L. and Smith, G., The orbifold Chow rings of toric Deligne–Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193–215.CrossRefGoogle Scholar
[5]Cadman, C., Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405–427.CrossRefGoogle Scholar
[6]Conrad, B., Arithmetic moduli of generalized elliptic curves, J. Math. Inst. Jussieu 6 (2007), 209–278.Google Scholar
[7]Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.Google Scholar
[8]Cox, D., The functor of a smooth toric variety, Tôhoku Math. J. 47 (1995), 251–262.Google Scholar
[9]Dieudonné, J. and Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).Google Scholar
[10]Fantechi, B., Mann, E. and Nironi, F., Smooth toric DM stacks, Preprint (2007), arXiv:0708.1254.Google Scholar
[11]Hoshi, Y., The exactness of log homotopy sequence, Preprint (2006), (RIMS-1558), Hiroshima Math. J., to appear.Google Scholar
[12]Iwanari, I., Toroidal geometry and Deligne–Mumford stacks, Preprint (2006), the first version of [Iwa07a], http://www.math.kyoto-u.ac.jp/preprint/preprint2006.html.Google Scholar
[13]Iwanari, I., Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007).Google Scholar
[14]Iwanari, I., Integral Chow rings of toric stacks, Preprint (2007), arXiv:0705.3524.Google Scholar
[15]Jiang, Y., A note on finite abelian gerbes over toric DM stacks, Proc. Amer. Math. Soc. 136 (2008), 4151–4156.Google Scholar
[16]Jiang, Y. and Tsen, H-H., The integral (orbifold) Chow rings of toric DM stacks, Preprint (2007), arXiv:0707.2972.Google Scholar
[17]Kato, K., Logarithmic structure of Fontaine-Illusie, in Algebraic analysis, geometry and number theory (Baltimore, MD, 1988) (Johns Hopkins University Press, Baltimore, MD, 1989), 191–224.Google Scholar
[18]Kato, K., Toric singularities, Amer. J. Math. 116 (1994), 1073–1099.CrossRefGoogle Scholar
[19]Keel, S. and Mori, S., Quotients by groupoids, Ann. of Math. (2) 145 (1997), 193–213.Google Scholar
[20]Kemp, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973).Google Scholar
[21]Laumon, G. and Moret-Bailly, L., Champs algébriques (Springer, Berlin, 2000).CrossRefGoogle Scholar
[22]Mochizuki, S., Extending families of curves over log regular schemes, J. Reine Angew. Math. 511 (1999), 43–71.Google Scholar
[23]Olsson, M., Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. 36 (2003), 747–791.CrossRefGoogle Scholar
[24]Olsson, M., Hom-stacks and restriction of scalars, Duke Math. J. 134 (2006), 139–164.CrossRefGoogle Scholar
[25]Perroni, F., A note on toric Deligne–Mumford stacks, Tohoku Math. J. (2) 60 (2008), 441–458.CrossRefGoogle Scholar
[26]Reid, M., Decomposition of toric morphisms, in Arithmetic and Geometry II, Progress in Mathematics, vol. 36 (Birkhäuser, Basel, 1983), 395–418.CrossRefGoogle Scholar
[27]Sumihiro, H., Eqiuvariant completion II, J. Math. Kyoto Univ. 15 (1975), 573–605.Google Scholar
[28]Stix, J., Projective anabelian curves in positive characteristic and descent theory for log-étale covers, PhD dissertation, University of Bonn (2002).Google Scholar