Compositio Mathematica

Table of Contents - 2009 - Volume 145, Issue 03  

Research Article

The category of toric stacks

Isamu Iwanaria1

a1 Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan (email: iwanari@math.kyoto-u.ac.jp)

Abstract

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.

(Received October 25 2007)

(Accepted September 15 2008)

(Online publication May 13 2009)

2000 Mathematics Subject Classification

  • 14M25;
  • 14A20

Keywords

  • toric geometry;
  • logarithmic geometry;
  • algebraic stacks