A note on toric Deligne-Mumford stacks

Abstract

We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.

Version2: typos corrected; we add sections 6 and 7 where we relate the present work with the paper of Iwanari: arXiv:math/0610548 and that of Fantechi-Mann-Nironi: arXiv:0708.1254. Accepted for publication in the Tohoku Math. J