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Toric stacks I: The theory of stacky fans

Authors: Anton Geraschenko and Matthew Satriano
Journal: Trans. Amer. Math. Soc. 367 (2015), 1033-1071
MSC (2010): Primary 14D23, 14M25
Published electronically: July 25, 2014
MathSciNet review: 3280036
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Abstract: The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties.

In this paper, we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a stacky fan. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stressing stacky phenomena such as canonical stacks and good moduli space morphisms.

We also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of $ \mathbb{P}^n$ and $ [\mathbb{A}^1/\mathbb{G}_m]$. Indeed, smooth toric stacks precisely solve moduli problems specified by (generalized) effective Cartier divisors with given linear relations and given intersection relations. Smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric DM stacks in papers by Cox and Perroni, respectively.

We include a plethora of examples to illustrate the general theory. We hope that this theory of toric stacks can serve as a companion to an introduction to stacks, in much the same way that toric varieties can serve as a companion to an introduction to schemes.

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Additional Information

Anton Geraschenko
Affiliation: Department of Mathematics 253-37, California Institute of Technology, 1200 California Boulevard, Pasadena, California 91125

Matthew Satriano
Affiliation: Department of Mathematics, 2074 East Hall, University of Michigan, Ann Arbor, Michigan 48109-1043

Received by editor(s): December 12, 2012
Published electronically: July 25, 2014
Additional Notes: The second author was partially supported by NSF grant DMS-0943832
Article copyright: © Copyright 2014 Anton Geraschenko and Matthew Satriano

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