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?