Rocky Mountain Journal of Mathematics

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

Rebecca Goldin, Megumi Harada, David Johannsen, and Derek Krepski

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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

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

First available in Project Euclid: 26 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 57R18: Topology and geometry of orbifolds
Secondary: 14L24: Geometric invariant theory [See also 13A50]

Stacky fan toric Deligne-Mumford stack inertia group weighted projective spaces


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.

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