Definition
A toric stack is a quotient stack \([X/G]\) where \(X\) is a normal toric variety with torus \(T_0\) and let \(G\subset T_0\) is a subgroup, together with the action of the torus \(T = T_0/G\). It is an Artin stack.
There is also a toric stack associated to a stacky fan:
Definition
Let \((\Sigma, \beta)\) be a stacky fan and \(X_\Sigma\) be the toric variety associated to \(\Sigma\).
- The map \(\beta^*:N^*\to L^*\) induces a homomorphism of tori \(T_\beta:T_L\to T_N\) which naturally identifies \(\beta\) with the induced map on lattices of 1-parameter subgroups.
- Since \(\coker \beta\) is finite, \(\beta^*\) is injective and thus \(T_\beta\) is surjective.
- Set \(G_\beta = \ker(T_\beta)\), noting that \(G_\beta \subset T_L\) is a subgroup of the torus of \(X_\Sigma\).
We then define the toric stack \(\mathcal X_{\Sigma, \beta}\) to be \([X_\Sigma/G_\beta]\) with the torus \(T = T_L/G_\beta\).