TITLE: The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification
QUESTION [12 upvotes]: Let $g \geq 2$, and consider the moduli space $\bar M_{g,n}$ of stable n-pointed curves of genus g. There is a natural forgetful map to $\bar M_g$, which forgets the markings and contracts any resulting unstable component. I am thinking about what the fibers of this map look like. Consider first the n-fold fibered product of the universal curve over $\bar M_g$ -- this would be a moduli space of curves of genus g with n not necessarily distinct markings. We know that this product is singular at all points where more than one of the markings coincide, or when a marking is placed on a node of a singular base curve. Moreover, we know that $\bar M_{g,n}$ is a desingularization. 
Of course, the difference between the two spaces is that on $\bar M_{g,n}$, whenever two markings or a marking and a node come together, one ''bubbles off'' a rational curve, and this can be seen as a blow-up of an ambient space in which the curve is embedded. My question is: can one in a similar way explicitly realize the total space $\bar M_{g,n}$ as a sequence of blow-ups of the n-fold fibered product of the universal curve, or a similar construction?
And a related naive question, which may or may not be equivalent to the one above: given a point [C] in the interior of $\bar M_g$, the fiber over it in $\bar M_{g,n} \to \bar M_g$ is a compactification of the configuration space of n distinct ordered points on C. Is this fiber isomorphic to the Fulton-MacPherson compactification of this configuration space?

REPLY [2 votes]: It may be worth mentioning briefly that the special case of $\overline{M_{0,n}}$ can be constructed fairly explicitly by blow-ups. The Fulton-MacPherson construction works. There are at least two additional constructions: an inductive one by Sean Keel in "Intersection theory on moduli space of stable N-pointed curves of genus zero" and the construction of Kapranov using what he calls Veronese curves towards the end of this paper and in an earlier paper not on the arXiv that he sites. Kapranov's construction expresses $\overline{M_{0,n}}$ as a composition of blow-ups of $\mathbb{P}^{n-3}$ at an explicit sequence of (strict transforms of) linear subspaces.