Fine saturated log schemes and toric monoids

A commutative monoid is written additively, with identity $0$. It is integral if its map to its group completion $P^{\mathrm{g}\mathrm{p}}$ is injective, fine if it is integral and finitely generated, saturated if $x\in P^{\mathrm{g}\mathrm{p}}$ and $rx\in P$ for an integer $r>0$ imply $x\in P$, and sharp if its only invertible element is $0$. Here a toric monoid means $Q=\sigma \cap M$ for a pointed rational polyhedral cone $\sigma \subset M_{\mathbb{R}}$ and a lattice $M$; it is fine, saturated and sharp. This fixes the convention for this page.

A log structure on a scheme $X$ is a sheaf of commutative monoids $M_X$ on the étale site, together with a monoid map $\alpha :M_X\to {\mathcal{O}}_X$ (where multiplication is used on ${\mathcal{O}}_X$), for which ${\alpha }^{-1}({\mathcal{O}}_X^{\times })\to {\mathcal{O}}_X^{\times }$ is an isomorphism. Its characteristic or ghost monoid at a geometric point $\bar{x}$ is ${\bar{M}}_{X,\bar{x}}=M_{X,\bar{x}}\mathrm{/}{\mathcal{O}}_{X,\bar{x}}^{\times }$. A chart by a monoid $P$ is a monoid map $P\to \mathrm{\Gamma }(U,{\mathcal{O}}_U)$ whose associated log structure is $M_X{\mathrm{\mid }}_U$. A log scheme is fine and saturated (fs) if étale locally it has a chart by an fs monoid.

A morphism of log schemes $f:(X,M_X,{\alpha }_X)\to (Y,M_Y,{\alpha }_Y)$ consists of a scheme morphism $f:X\to Y$ and a homomorphism of sheaves of monoids $f^{\mathrm{♭}}:f^{-1}{M}_Y\to M_X$ satisfying ${\alpha }_X\circ f^{\mathrm{♭}}=f^{\mathrm{♯}}\circ f^{-1}{\alpha }_Y$. Composition uses the composed scheme maps and the induced composite monoid maps. An isomorphism is a morphism with a two-sided inverse in this category.

For a sharp toric monoid $Q$ and field $k$, the log point $\mathrm{Spec}(Q\to k)$ has underlying scheme $\mathrm{Spec}k$; its chart sends $0$ to $1$ and every nonzero $q\in Q$ to $0\in k$. These definitions do not assert that any given scheme admits a log structure or that a morphism is log smooth.

Log-smooth saturated morphism and relative log-canonical line