Smooth toric Deligne-Mumford stacks

Abstract

Abstract. We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a"torus". We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

On Generalisations of Losev-Manin Moduli Spaces for Classical Root Systems

“…The stack X (C n ) is a toric Deligne-Mumford stack as introduced in [BCS04] (see also [FMN10]): we define the stacky fan Σ(C n ) as the fan Σ(C n ) in the lattice N (C n ) with the difference that we choose on the rays generated by X(C n ) as fine moduli space. We give a characterisation of X(C n ) as a fine moduli space L ± n of 2n-pointed chains of projective lines.…”

Stack structures on GIT quotients parametrizing hypersurfaces

“…We refer the reader to Lieblich [13] and Fantechi et al [9] for systematic expositions, but mention briefly that a μ n -gerbe over a stack X corresponds to an element of H 2 (X, μ n ), and is called essentially trivial if the push forward to H 2 (X, G m ) vanishes. This is equivalent [13, Proposition 2.3.4.4] and [9, Remark 6.4] to the statement that the gerbe is the n'th root stack of a line bundle on X.…”

The integral (orbifold) Chow ring of toric Deligne–Mumford stacks

“…Remark 2.10 In the paper [11] by Fantechi, Mann and Nironi, the authors studies smooth toric Deligne-Mumford stacks. Their definition of toric Deligne-Mumford stacks generalized the general definition of toric varieties.…”