Definition
A morphism \(f:X\to Y\) of fine log schemes is said to be log smooth if etale locally on \(X\) and \(Y\) it fits into a commutative diagram
such that
- The horizontal maps induce charts \(\underline P \to \mathcal O_X\) and \(\underline Q\to \mathcal O_Y\) for \(X\) and \(Y\), i.e. the two rows above are strict morphisms
- The induced morphism
\[X\to Y\times_{\Spec \mathbb Z[Q]}\Spec \mathbb Z[P]\] is a smooth morphism of schemes
- The righthand vertical arrow is induced by a monoid homomorphism \(Q\to P\) so that \(\ker(Q^{gp}\to P^{gp})\) and the torsion portion of \(\coker(Q^{gp}\to P^{gp})\) are both finite groups of orders invertible on \(X\).
This last requirement is significant only in positive and mixed characteristic, so I usually can ignore it.
A note from Dhruv’s “an invitation to enumerative geometry of degenerations” is that a morphism \((X, \mathcal M_X)\to (Y,\mathcal M_Y)\) is log smooth if an only if, locally on the source and the target, it is the pullback of a dominant equivariant morphism of toric varieties. This may only be the case for fine and saturated log schemes, in which case the above definition basically gives it to us.