Let \(\mathsf{Log}\) denote the stack of logarithmic structures introduced in [Ols].
Some facts
- A log scheme \(X\) is smooth if and only if the map \(\underline X \to \mathsf{Log}\) is smooth
- This map factors as a map \(\underline X \to \underline{\mathscr{X}} \to \mathsf{Log}\) where \(X\to \mathscr{X}\) is a strict smooth map and \(\mathscr{X}\) is a locally toric stack. This last bit means \(\mathscr{X}\) has an etale cover by finitely many stacks of the form \([V/T]\) where \(V\) is a toric variety and \(T\) is its dense torsu.
- We call \(\mathscr{X}\) the Artin fan of \(X\).