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%
\begin{document}
\begin{center} Very basic exercises on logarithmic structures \end{center}

\begin{itemize}

\item{Terrible, Horrible, No Good, Very Bad log structures}

\begin{enumerate}
\item Consider the monoid $P$ generated by 2 and 3 in $\NN$, and let $(\Spec k, M)$ be the log structure associated to the standard map "$0^n:P \to k$". What's wrong with this log structure?
Can it be improved?
\item Consider the same monoid, let $X$ be the cuspidal curve $y^2 = x^3$. Consider $P \to \cO_X$ defined by $2\mapsto x$ and $3\mapsto y$. Want do you think about the associated log structure? Can this one be improved?
\item Compare the above log structure on $X$ with the ones defined by $\NN \to \cO_X$ defined by $1\mapsto x$ or $1 \mapsto y$. (You may want to contemplate the relationship of normalization with log structures, at least in this case).
\item Consider the monoid $M = k^* \sqcup \NN_{>0}$ and the map $\alpha:M \to k$ defined by the standard embedding on $k^*$ and 0 on $\NN_{>0}$. Show that this is a log structure on $\Spec k$, that $(\Spec k, M)$ maps to the standard log point associated to $0^n:\NN\to k$, but is incoherent.
\item Recall that if $U\subset X$ is open then we can define a log structure by $M =$ local sections of $\cO_X$ invertible when restricted to $U$. Consider $X$ defined by $xy=uv$. Show that the log structure corresponding to the principal open $D(xyuv)$  is log smooth (over a point with trivial log structure), but if you take away any of the divisor it gets worse. Describe these log structures explicitly.
\item (some exercises with reducible schemes....)
\end{enumerate}
\item{Comparing some log structures}

\begin{enumerate}
\item The map $0 \to \NN$ induces a map from the log point to the point. Does the analogue hold for the left inverse $\NN \to 0$? Compare log differential when possible.
\item On $\AA^1$ consider the log structures associated to $\NN\to k[x]$ defined by  $1\mapsto x$ and $ 1\mapsto x^a$ for $ a $ a positive integer. Which way do you get a map? compare log differentials.
\item Fix $a,b$ nonnegative integers. On $\AA^1$ consider the log structures associated to $\NN^2\to k[x]$ defined by $(n,m) \mapsto x^{an+bm}$. Compare the log differentials, especially noting the cases where $ a$ or $b$ are 0.
\item Investigate when there are maps between the log structures of the previous two exercises and how the log differentials behave.
\item Consider the log structure on $\AA^2$ associated to $(m,n) \mapsto x^m y^n$. If $P\subset \NN^2$ is a submonoid of  finite index, how does the associated log structure change? How about log differentials?
\end{enumerate}


\end{itemize}






\end{document}