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?
 A: 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.
A: (I'm not sure this is a real answer, but it was too long for a comment)
I've never seen the log structure you consider.
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).
