# Properties of log smooth schemes

Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the structure morphism from $X$ to $\eta_M$ is log-smooth.( We can assume $X$ has a global chart $a:X \to \operatorname{Spec}(\mathbb{Z}[P]))$. Does that imply that $X$ is reduced? What if $M= \mathbb{N}$ or identity monoid (i.e the log structure on $\operatorname{Spec}(k)$ is the trivial one)? If no, what is the simplest counterexample?

Notation: $\mathbf{A}_P = {\rm Spec}(k[P])$ with the standard log structure.
Basic counterexample. Let $n$ be an integer invertible in $k$, and consider the map $$f_n\colon \mathbf{A}_P\to \mathbf{A}_P$$ induced by the multiplication by $n$ map $P\to P$. The map $f_n$ is log smooth (or even log etale), and hence so is its base change to the log point $\eta_P = {\rm Spec}(P\to k)$. But the fiber over $\eta_P$ is $${\rm Spec}(P\to k[P]/I)$$ where $I$ is the ideal generated by $n\cdot (P\setminus\{0\})$, which is non-reduced if $n>1$ and $P$ is nontrivial.
In the simplest example $P=\mathbf{N}$ you get the standard tamely ramified map $t\mapsto t^n\colon \mathbf{A}^1\to \mathbf{A}^1$.
• Thank you very much for the answer. What if the log structure on $k$ is trivial? Shouldn't it be true then? – ABC Jun 9 '18 at 18:10
• Yes. In this case $X$ looks locally like $\mathbf{A}_P$ for a fine and saturated $P$, which is reduced. – Piotr Achinger Jun 11 '18 at 9:54