There are two main definitions.
Definition
(Illusie) A log structure \((\mathcal M, \alpha)\) on a scheme \(X\) is said to be fine if it is etale locally the log structure associated to the pre-log structure \((P_X, \beta)\) where \(P_X\) is the constant sheaf of monoids of value \(P \), and \(P \) is finitely generated and integral.
We then say that a log scheme \(X\) is a fine log scheme if its log structure is fine.
Equivalently, \((X,\mathcal M_X)\) is a fine log scheme if etale locally there is a chart \(P\to \mathcal M_X\) with \(P\) a finitely generated integral monoid. If \(P\) can moreover be chosen to be saturated, then \(X\) is a fine and saturated log structure.
Definition
(Ogus) A log structure is fine if it is coherent (locally finitely generated as a monoid) and integral. A log scheme is a fine log scheme if its log structure is fine.
These two definitions are equivalent.
- Illusie \(\implies\)Ogus: If \(\mathcal M\) is etale locally a constant finitely generated integral sheaf of monoids, then it is exactly a coherent sheaf of integral monoids.
- Ogus \(\implies\)Illusie: This is slightly more subtle, not quite sure how to do it. Probably some quirk of the etale topology that I don’t understand.