Log-smooth deformation functor over a toric Artin base

Let $k$ be a field, $Q$ a sharp toric monoid, $\mathrm{\Lambda }=k\mathrm{⟦}Q\mathrm{⟧}$ the completion of $k[Q]$ at the ideal generated by all monomials $z^q$ with $q\mathrm{\ne }0$, and $\mathfrak{m}$ its maximal ideal. The category ${\mathbf{A}\mathbf{r}\mathbf{t}}_{\mathrm{\Lambda }}$ has local Artinian $\mathrm{\Lambda }$-algebras $A$ with identified residue field $A\mathrm{/}{\mathfrak{m}}_A\cong k$ as objects and local $\mathrm{\Lambda }$-algebra maps as arrows. The log base $\mathrm{Spec}(Q\to A)$ has underlying scheme $\mathrm{Spec}A$ and the log structure associated to the monoid chart induced by $\mathrm{\Lambda }\to A$.

Fix $S_0=\mathrm{Spec}(Q\to k)$ and a proper log-smooth saturated morphism $f_0:X_0\to S_0$ of relative dimension $d$. For $A\in {\mathbf{A}\mathbf{r}\mathbf{t}}_{\mathrm{\Lambda }}$, write $S_A=\mathrm{Spec}(Q\to A)$ and let $S_0\to S_A$ be the base change induced by $A\to k$. A log-smooth deformation of $f_0$ over $A$ is a proper log-smooth morphism $f_A:X_A\to S_A$ of relative dimension $d$, together with a specified isomorphism of log spaces over $S_0$ ${\iota }_A:X_A{\times }_{S_A}{S}_0\stackrel{\sim }{\to }{X}_0\mathrm{.}$ Two such deformations $(X_A,f_A,{\iota }_A)$ and $(X_A^{\mathrm{\prime }},f_A^{\mathrm{\prime }},{\iota }_A^{\mathrm{\prime }})$ are isomorphic when an isomorphism $h:X_A\to X_A^{\mathrm{\prime }}$ over $S_A$ preserves the log structures and satisfies ${\iota }_A^{\mathrm{\prime }}\circ (h{\times }_{S_A}{S}_0)={\iota }_A$. Thus the reduction includes a chosen identification, and isomorphisms preserve it. Its log-smooth deformation functor ${\mathrm{L}\mathrm{D}}_{X_0\mathrm{/}{S}_0}:{\mathbf{A}\mathbf{r}\mathbf{t}}_{\mathrm{\Lambda }}⟶\mathbf{S}\mathbf{e}\mathbf{t}$ assigns to $A$ the set of isomorphism classes of log-smooth deformations $f:X\to \mathrm{Spec}(Q\to A)$ reducing to $f_0$. Smoothness of this functor means the lifting property: for every surjection $B\to A$ in ${\mathbf{A}\mathbf{r}\mathbf{t}}_{\mathrm{\Lambda }}$, every deformation over $A$ lifts to a deformation over $B$. These definitions assert neither that lifts exist nor that the functor is smooth.

Fine saturated log schemes and toric monoids

Log-smooth saturated morphism and relative log-canonical line