trivialities on log-structures

Question

I would like to understand some trivialities about log-structures. Given a log-scheme $(X,M_X)$ the log-structure $M_X$ is defined via push-out. Are there stupid examples in which this push-out is actually a direct sum $\mathbb{G}_m\oplus P$ for some monoid $P$?

Furthermore assume we have a ring $R$, an element $t\in R$ defining a log-structure on $Spec(R)$ via $\mathbb{N}\rightarrow t^n$. Consider $X=Spec(R[x,y]/(xy-t))$ with the standard log-structure on the two brances $x$ and $y$. Let $p:Y\rightarrow X$ be the blow-up of $X$ at $(x,y,t)$ and put a log-structure on $Y$ over the preimage $\tilde{x},\tilde{y}$ of the two branches $x,y$ but not on the exceptional divisor. Namely locally on $Y$ we take $\mathbb{N}\rightarrow \mathcal{O}_Y$ where we send $1$ to $\tilde{x}$ (resp. to $\tilde{y}$). Call this log-structure $M_Y$. Is it true that $p_{*}M_Y$ splits as a direct sum? If so how do the summands look like?

Answer 1

For the first question, given a sharp monoid $P$ (i.e., the group of units of $P$ is the zero group), there is a log structure on any scheme $X$ given by $M_X={\mathcal O}_X^*\oplus P$ with the structure map given by $(f,p)\mapsto f$ if $p=0$ and $(f,p)\mapsto 0$ if $p\not =0$. This comes from the prelog structure $P\rightarrow {\mathcal O}_X$ given by $p\mapsto 1$ if $p=0$, $p\mapsto 0$ if $p\not=0$. I wouldn't call this a stupid log structure: it in fact can be quite useful. For example, the standard log point is defined as the log structure on the scheme $Spec(k)$ with $P$ the natural numbers as above.

For the second question, I'm not quite sure what direct sum decomposition you are expecting, but I can't think of a natural direct sum decomposition here. If you are thinking that $p_*M_Y$ will decompose as a direct sum of the sheaf ${\mathcal O}_X^*$ and a sheaf supported on the branches $x$ and $y$, this is not the case. To take a simpler example, taking the divisorial log structure associated to $0 \in \mathbb{A}^1$, the log structure does not decompose as $O_{A^1}\oplus {\mathbb N}_0$. Indeed, there is no section of $M_{\mathbb{A}^1}$ with support at $0$. In the example you describe, there will be an exact sequence $$ 1 \rightarrow {\mathcal O}_Y^*\rightarrow M_Y \rightarrow {\mathbb{N}}_x \oplus {\mathbb{N}}_y\rightarrow 0 $$ where the subscripts $x$ and $y$ denote which branch the constant sheaf with coefficients ${\mathbb N}$ is sitting on. I believe when pushing down, one obtains an exact sequence $$ 1\rightarrow {\mathcal O}_X^*\rightarrow p_*M_Y\rightarrow {\mathcal F} \rightarrow 0 $$ where ${\mathcal F}$ is the sheaf supported on $D$, the vanishing locus of $t$, which is obtained by taking the constant sheaf ${\mathbb N}$ on $D-\{0\}$ and extending by zero across $0$. This is because there are no functions on the inverse image of a open neighbourhood of $0$ which do not vanish off the two branches, but do vanish one or both of the branches.