TITLE: A log structure on the moduli space of curves
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: 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.