## Rocky Mountain Journal of Mathematics

### Inertia groups of a toric Deligne-Mumford stack, fake weighted projective stacks, and labeled sheared simplices

#### Abstract

This paper determines the inertia groups (isotropy groups) of the points of a toric Deligne-Mumford stack $[Z/G]$ (considered over the category of smooth manifolds) that is realized from a quotient construction using a stacky fan or stacky polytope. The computation provides an explicit correspondence between certain geometric and combinatorial data. In particular, we obtain a computation of the connected component of the identity element $G_0 \subset G$ and the component group $G/G_0$ in terms of the underlying stacky fan, enabling us to characterize the toric DM stacks which are global quotients. As another application, we obtain a characterization of those stacky polytopes that yield stacks equivalent to weighted projective stacks and, more generally, to \textit {`fake' weighted projective stacks}. Finally, we illustrate our results in detail in the special case of \textit {labeled sheared simplices}, where explicit computations can be made in terms of the facet labels.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 481-517.

Dates
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1469537474

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-481

Mathematical Reviews number (MathSciNet)
MR3529080

Zentralblatt MATH identifier
1362.14053

#### Citation

Goldin, Rebecca; Harada, Megumi; Johannsen, David; Krepski, Derek. Inertia groups of a toric Deligne-Mumford stack, fake weighted projective stacks, and labeled sheared simplices. Rocky Mountain J. Math. 46 (2016), no. 2, 481--517. doi:10.1216/RMJ-2016-46-2-481. https://projecteuclid.org/euclid.rmjm/1469537474

#### References

• J. Alper, A guide to the literature on algebraic stacks, https://maths-people.anu.edu.au/$\sim$alperj/papers/stacks-guide.pdf, available from the author's webpage.
• V. Batyrev and D. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293–338.
• K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. Göttssche and A. Kresch, Algebraic stacks, http://www.math.uzh.ch/index.php?pr_vo_det&key1 =1287&key2=580&no_cache=1, in progress.
• S. Boissière, É. Mann and F. Perroni, A model for the orbifold Chow ring of weighted projective spaces, Comm. Alg. 37 (2009), 503–514.
• L.A. Borisov, L. Chen and G.G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193–215 (electronic).
• W. Buczyńska, Fake weighted projective space, M.S. thesis, Warsaw University, arXiv:0805.1211 [math.AG].
• D. Edidin, What is a stack?, Not. Amer. Math. Soc. 50 (2003), 458–459.
• B. Fantechi, Stacks for everybody, in European congress of mathematics, Vol. I, Progr. Math. 201, Birkhäuser, Basel, 2001.
• B. Fantechi, E. Mann and F. Nironi, Smooth toric Deligne-Mumford stacks, J. reine angew. Math. 648 (2010), 201–244.
• W. Fulton, Introduction to toric varieties, Ann. Math. Stud. 131, Princeton University Press, Princeton, NJ, 1993.
• A. Geraschenko and M. Satriano, Toric stacks I: The theory of stacky fans, Trans. Amer. Math. Soc. 367 (2015), 1033–1071.
• ––––, Toric stacks II: Intrinsic characterization of toric stacks, Trans. Amer. Math. Soc. 367 (2015), 1073–1094.
• R. Goldin, T.S. Holm and A. Knutson, Orbifold cohomology of torus quotients, Duke Math. J. 139 (2007), 89–139.
• M. Harada and D. Krepski, Global quotients among toric Deligne-Mumford stacks, Osaka J. Math. 52 (2014), 236–269.
• I. Iwanari, Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Publ. Res. Inst. Math. Sci. 45 (2009), 1095–1140.
• Y. Jiang, The Chen-Ruan cohomology of weighted projective spaces, Canad. J. Math. 59 (2007), 981–1007.
• A. Kasprzyk, Bounds on fake weighted projective spaces, Kodai Math. J. 32 (2009), 197–208.
• E. Lerman, Orbifolds as stacks?, Enseign. Math. 56 (2010), 315–363.
• E. Lerman and A. Malkin, Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds, Adv. Math. 229 (2012), 984–1000.
• E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201–4230.
• E. Mann, Orbifold quantum cohomology of weighted projective spaces, J. Alg. Geom. 17 (2008), 137–166.
• D. Metzler, Topological and smooth stacks. [math.DG].
• B. Noohi, Foundations of topological stacks I. [math.AG].
• F. Perroni, A note on toric Deligne-Mumford stacks, Tohoku Math. J. 60 (2008), 441–458.
• M. Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), 209–236.
• H. Sakai, The symplectic Deligne-Mumford stack associated to a stacky polytope, Results Math. 63 (2013), 903–922.
• C.A. Weibel, An introduction to homological algebra, Cambr. Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994.