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The category of toric stacks

Part of: Algebraic geometry: Foundations Special varieties

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: iwanari@math.kyoto-u.ac.jp)
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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

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