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Toric stacks I: The theory of stacky fans

Geraschenko, Anton and Satriano, Matthew (2014) Toric stacks I: The theory of stacky fans. Transactions of the American Mathematical Society, 367 (2). pp. 1033-1071. ISSN 0002-9947.

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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 P^n and [A^1/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:Article copyright: © Copyright 2014 Anton Geraschenko and Matthew Satriano. Received by editor(s): December 12, 2012. Article electronically published on July 25, 2014. The second author was partially supported by NSF grant DMS-0943832. We thank Jesse Kass and Martin Olsson for conversations which helped get this project started, and Vera Serganova and the MathOverflow community (especially Torsten Ekedahl, Jim Humphreys, Peter McNamara, David Speyer, and Angelo Vistoli) for their help with several technical points. We also thank Smiley for helping to track down many references. Finally, we would like to thank the anonymous referee for helpful suggestions and interesting questions.
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Issue or Number:2
Classification Code:MSC (2010): Primary 14D23, 14M25
Record Number:CaltechAUTHORS:20150122-114139342
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Official Citation:Anton Geraschenko and Matthew Satriano Journal: Trans. Amer. Math. Soc. 367 (2015), 1033-1071 Published electronically: July 25, 2014
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:53994
Deposited By: Ruth Sustaita
Deposited On:22 Jan 2015 20:27
Last Modified:03 Oct 2019 07:54

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