# A log structure on the moduli space of curves

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each marked point. There is a map $i:M_{g, n}\to M_{g, \vec{n}}$ (the zero section), and there is a log structure on $M_{g, \vec{n}}$ associated to the normal crossings divisor which is the complement to the locus where all specified tangent vectors are nonzero. Let $M_{g, n}^{log}$ be the pullback of this log structure on $M_{g, \vec{n}}$ to $M_{g, n}$ along $i.$

Question: what is $M_{g, n}^{log}$ called? Who has studied it? At one point I heard the term "log moduli space of curves with log structure" floating around, though I've never seen it explicitly defined. Is my $M_{g, n}^{log}$ the (restriction from $\bar{M}_{g,n}$ to $M_{g,n}$) of this log moduli space?

• Could you please explain what it means to pull back a log structure along a morphism of varieties? I seems to me that every time you have an open subscheme U⊂X with D:=X\U a normal crossing divisor, you can take the standard log structure on X associated to D and pull it back to the deepest stratum of D (whatever that means). What do you get in the case when X is affine n-space and D is a bunch of coordinate hyperplanes? – André Henriques Mar 23 '18 at 23:44
• Right, the log structure will be given by a local system of semigroups locally isomorphic to some power of the natural numbers. I don't think the local system will in general be trivial. – Dmitry Vaintrob Mar 23 '18 at 23:52
• Have you tried looking at this paper of Olsson? – Ben Lim Mar 24 '18 at 5:21
• – Piotr Achinger Mar 24 '18 at 8:07
• @DmitryVaintrob sorry, after reading Mattia's answer I realized I didn't read your question too carefully. What makes you think that the local system is not constant? Your marked points are ordered, and it seems that the $i$-th copy of the natural numbers corresponds to the $i$-th marked point. So it seems that your log structure is constant, i.e., the pull-back of the log structure on the log point ${\rm Spec}(\mathbf{N}^n\to \mathbf{C})$. – Piotr Achinger Mar 24 '18 at 19:30

As Piotr Achinger suggested in a comment, your log moduli space is the direct product of $M_{g,n}$ with the log point $\operatorname{Spec}(\mathbb{N}^n \to \mathbb{C})$ given by the monoid map $(x_1,\ldots,x_n) \mapsto 0$. The main reason is that the monoid $\mathbb{N}^n$ only has automorphisms given by reordering basis elements, and in this case, the basis elements are attached to marked points that have a specified order. That is, the local system is trivial for rather elementary reasons.
• Sorry, after a little more thinking I ended up un-accepting this answer (based on Piotr's comment). The issue is that a family of constant $\mathbb{N}$-motives over a variety $X$ may not be (e.g. étale) locally constant, since it may be twisted by invertible functions on the base. As an example, if $L$ is a line bundle on $X$ and $X_L$ is the normal log structure on $X$ induced on $L$, its Kato-Nakayama space is the circle bundle associated to $L$, which remembers at least the Chern class (in Betti homology): something that can be nontrivial if $X$ is e.g. simply connected. – Dmitry Vaintrob Mar 25 '18 at 22:49
By "log moduli space of (stable) log curves" usually one refers to the natural log structure on $\overline{\mathcal M}_{g,n}$ given by the boundary normal crossings divisor $\Delta=\overline{\mathcal M}_{g,n}\setminus \mathcal M_{g,n}$. It turns out that the resulting log stack $(\overline{\mathcal M}_{g,n}, \Delta)$ represents the moduli functor of families of log smooth (stable) curves (with the right genus and "marked points") over the category of log schemes (I'm omitting some words, like fine saturated). This is in the paper by F. Kato linked by Piotr, and is also explained in this survey paper http://arxiv.org/abs/1006.5870 .
Of course if you restrict this log structure to $\mathcal M_{g,n}$ you obtain the trivial one. The log structure you describe is very different, as it's "supported everywhere" (as you point out in your comment).