Question

Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces singular objects in our moduli. The question is: What is the use of knowing the singularities parametrized by the boundary of the compactified moduli space i.e $\overline{M} \setminus M$. Usually $M$ has diferent compactifications, and so different " limit singular objects". Does this difference mean something?

For example: the smooth genus $g=3$ have the $\overline{M_3}$ compactification with only stable curves in the boundary, but is possible to find another compactification by the GIT analysis of degree four plane curves. The singular curves present in the boundaries are quite different. What is the use of having an explicit descriptions of them?

Answer 1

here's a try: suppose you have a quartic surface in P^3 and ask whether the isomorphism type of plane sections varies or not. If the planes pass through a general common line, the general singularity of the curve section is an odp. If these curve sections are also irreducible, it seems that the conclusion is that the holomorphic type of the sections varies. I.e. the fact that the compactified moduli space of smooth genus 3 curves contains all irreducible nodal curves of genus 3, and is hausdorff, implies that the plane sections of a quartic surface are not all isomorphic.

Answer 2

Here's an example: Suppose you'd like to know about the divisors on $X = \overline{M_{g,n}}$. Say for instance that you have a divisor $D$ and you'd like to know whether $D$ is ample or nef, that is, if for all curves $C$ on $X$, we have $D\cdot C > 0$ or $\ge 0$.

There's a conjecture out there called the $F$ conjecture which says that if we want to show $D$ is nef, it suffices to check $D\cdot C \ge 0$ for a smaller set of curves in the boundary strata, called the $F$-curves. Because that's a definition for which pictures help, I refer you over to http://www-irm.mathematik.hu-berlin.de/~larsen/talkM2Goettingen.pdf

Of course that's a conjecture to be proven, but it's related to a big circle of ideas surrounding the minimal model program, including the question of Hu and Keel on whether or not $\overline{M_{0,n}}$ is a Mori Dream Space ( http://arxiv.org/PS_cache/math/pdf/0004/0004017v1.pdf )