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## The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification

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?

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You don't mean that the forgetful map contracts any rational component, only those touching less than three nodes. Also, the n-fold fibered power is not singular where several markings coincide at a smooth point. After all, a fibered product of smooth morphisms is smooth. It will, however, disagree with $\overline{M}_{g,n}$ at the locus where three or more markings coincide, so blow-ups will indeed be required.

However, what you wish for in the last paragraph is indeed true. One has to assume that the automorphism group of $C$ is trivial, for otherwise, even the fiber of $\overline{M}_{g,1}$ over $\overline{M}_g$ will be not $C$ but $C/ Aut C$. (Working with moduli stacks rather than moduli spaces would cure this problem.) But assuming this, the fiber of $\overline{M}_{g,n} \to \overline{M}_g$ is indeed the Fulton-MacPherson configuration space. Indeed, that space is described in the original F-Mac paper as a moduli space of stable configurations of distinct smooth points on a fixed curve $C$ with nodal trees of projective lines attached, modulo projective equivalence on the lines. Here "stable" means that the pointed nodal curve has finite automorphism group, that is, each line carries at least three distinguished points. Those are exactly the configurations that you see in your fiber.

As for your earlier question, it is true in some sense, but it would be quite subtle in practice. Fulton-MacPherson explain how to obtain their space from the product $C^n$ by an explicit sequence of blow-ups. One blows up first the "small diagonal" where all the points come together, and then proper transforms of other diagonals. You could try to blow up all the corresponding loci in the fibered power of $\overline{M}_{g,n}$ over $\overline{M}_g$.

But you would have to resolve other loci over the boundary too: if two markings coincide with a node, for example, you want to pull apart the node, glue in an extra line, and draw the two markings on that line. But since there is a 1-parameter family of ways to do this (due to the cross-ratio of the two nodes and the two markings), you want to replace the corresponding points in the fibered power with a $P^1$. Even worse, this would not be a blow-up in the obvious sense. The fibered product of two nodes, in local analytic coordinates, looks like $xy = t = wz$, that is, $xy-wz=0$, the cone on a smooth quadric surface. You can resolve this singularity by blowing up the origin, but then the exceptional divisor is $P^1 \times P^1$, not $P^1$ as you wish. Instead, you should perform one of the "small resolutions" obtained by blowing up one of the Weil divisors $x=w=0$ or $x=z=0$.

Since any birational morphism is the blow-up at some sheaf of ideals, in some abstract sense you are guaranteed that $\overline{M}_{g,n}$ is obtained from the fibered power by blowing up. But that doesn't mean that you can give an explicit list of smooth centers to be blown up in turn. The example with the two markings at the node suggests that this will be a thorny problem.

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 In the process of contraction we only contract unstable components and components with less than three nodes are allowed: Rational components with less than three special points are contracted. Special points consist of nodes and markings. – Passenger Jun 3 2010 at 9:11 Thanks a lot for your comments, you've given me a lot to mull over. I edited my post to reflect your absolutely correct remarks in the beginning. You are right that you really have to take care to distinguish between working with the moduli stack and the coarse moduli scheme. I now see that you only get the Fulton-MacPherson compactification when you work with the coarse space, but I really want to work with the stack... for instance, I don't feel comfortable stating that the universal family over $M_2$ is a bunch of rational curves. – Dan Petersen Jun 3 2010 at 9:54 ...which raises the question whether one can construct the moduli stack by a sequence of blow-ups at closed substacks in a natural way. One way might be to not try to do the blow-ups directly on the nth fibered power, but to take first the second fibered power, make all necessary blow-ups, take the fibered product with a third copy of the universal curve, make more blow-ups, and so on... this might solve the problems you get from your example of two markings on the same node, and similar issues. – Dan Petersen Jun 3 2010 at 9:59

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.

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 Thanks for the pointers! – Dan Petersen Aug 11 2010 at 8:19