(1) Let $X$ be a scheme and $Y\subseteq X$ a closed subset of codimension one. Denote by $j:X\setminus Y\rightarrow X$ the inclusion. Then the inclusion

of the sheaf of regular functions invertible off of $Y$ is a log structure on $X$. This is called a divisorial log structure on $X$.

(2) A prelog structure, i.e., an arbitrary homomorphism of sheaves of monoids $\varphi:\mathcal{P}\rightarrow\mathcal{O}_{X}$, defines an associated log structure $\mathcal{M}_{X}$ by

and $\alpha_{X}(p,h)=h\cdot\varphi(p)$.

(3) If $f:X\rightarrow Y$ is a morphism of schemes and $\alpha_{Y}:\mathcal{M}_{Y}\rightarrow\mathcal{O}_{Y}$ is a log structure on $Y$, then the prelog structure $f^{-1}(\mathcal{M}_{Y})\rightarrow\mathcal{O}_{X}$ given as the composition of $\alpha_{Y}:f^{-1}(\mathcal{M}_{Y})\rightarrow f^{-1}\mathcal{O}_{Y}$ and $f^{*}:f^{-1}\mathcal{O}_{Y}\rightarrow\mathcal{O}_{X}$ defines an associated log structure on $X$, the pull-back log structure.

(4) In (1) we can pull back the log structure on $X$ to $Y$ using (3). Thus in particular, if $\mathcal{X}\rightarrow D$ is a toric degeneration, the inclusion $\mathcal{X}_{0}\subseteq\mathcal{X}$ gives a log structure on $\mathcal{X}$ and an induced log structure on $\mathcal{X}_{0}$. Similarly the inclusion $0\in D$ gives a log structure on $D$ and an induced one on $0$. Here $\mathcal{M}_{0}=\mathbb{C}^{\times}\oplus\mathbb{N}$, where $\mathbb{N}$ is the (additive) monoid of natural (non-negative) numbers, and

$0^{\dagger}$ is usually called the standard log point.

We then have log morphisms $\mathcal{X}^{\dagger}\rightarrow D^{\dagger}$ and $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$.

(5) If $\sigma\subseteq M_{\mathbb{R}}=\mathbb{R}^{n}$ is a strictly convex rational polyhedral cone, ${\sigma}^{\scriptscriptstyle\vee}\subseteq N_{\mathbb{R}}$ the dual cone, let $P={\sigma}^{\scriptscriptstyle\vee}\cap N$: this is a monoid under addition. The affine toric variety defined by $\sigma$ can be written as $X=\operatorname{Spec}\mathbb{C}[P]$. We then have a pre-log structure induced by the homomorphism of monoids

given by $p\mapsto z^{p}$. There is then an associated log structure on $X$. This is in fact the same as the log structure induced by $\partial X\subseteq X$, where $\partial X$ is the toric boundary of $X$, i.e., the union of toric divisors of $X$.

If $p\in P$, then the monomial $z^{p}$ defines a map $f:X\rightarrow\operatorname{Spec}\mathbb{C}[\mathbb{N}]=\mathbb{A}^{1}$ which is a log morphism with the log structure on $\operatorname{Spec}\mathbb{C}[\mathbb{N}]$ induced similarly by $\mathbb{N}\rightarrow\mathbb{C}[\mathbb{N}]$. The fibre $X_{0}=\operatorname{Spec}\mathbb{C}[P]/(z^{p})$ is a subscheme of $X$, there is an induced log structure on $X_{0}$, and a map $X_{0}^{\dagger}\rightarrow 0^{\dagger}$ as in (4). The log morphism $f$ is an example of a log smooth morphism. Essentially all log smooth morphisms are étale locally of this form (if $\mathbb{N}$ is replaced by a more general monoid). See [48] for details.

Condition (4) of Definition 7.1 in fact implies that locally, away from $Z$, $\mathcal{X}^{\dagger}$ and $\mathcal{X}_{0}^{\dagger}$ are of the above form. So we should view $\mathcal{X}^{\dagger}\rightarrow D^{\dagger}$ as log smooth away from $Z$, and from the log point of view, $\mathcal{X}_{0}^{\dagger}$ can be treated much like a non-singular scheme away from $Z$.

(6) Given a monoid $P$ as in (5) and a morphism $X\rightarrow\operatorname{Spec}\mathbb{C}[P]$, we can pull back the log structure defined above on $\operatorname{Spec}\mathbb{C}[P]$ to $X$. If $X^{\dagger}$ is a log scheme which étale locally can be described in this way, we say $X^{\dagger}$ is a fine saturated log scheme. The adjective “fine” tells us it is locally described via maps to schemes of the form $\operatorname{Spec}\mathbb{C}[P]$ where $P$ is a finitely generated integral monoid, i.e., the canonical homomorphism $P\rightarrow P^{{\operatorname{gp}}}$ is an injection. The adjective “saturated” tells us the monoid $P$ is saturated. This means that $P$ is integral and whenever $p\in P^{{\operatorname{gp}}}$ satisfies $mp\in P$ for some $m>0$, $p\in P$. Such monoids arise, e.g., as the intersection of a rational polyhedral cone with a lattice.

Most of the literature on log geometry tends to apply only to fine log structures. In the key example of $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$, the log structure is fine saturated away from the set $Z$. However, it is not in general fine along $Z$, and this tends to cause many technical problems as new techniques have to be developed to deal properly with the log structure along $Z$. ∎