Applications for knowing the singularities parametrized by the boundary of a moduli space

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 )

Answer 3

This example is about moduli of weighted pointed stable curves. In this case the knowledge of the boundary objects is fundamental to work out the birational geometry (for instance in relation to the log minimal model program) of the spaces themselves.

Let $S$ be a Noetherian scheme and $g,n$ two non-negative integers. A family of nodal curves of genus $g$ with $n$ marked points over $S$ consists of a flat proper morphism $\pi:C\rightarrow S$ whose geometric fibers are nodal connected curves of arithmetic genus $g$, and sections $s_{1},...,s_{n}$ of $\pi$. A collection of input data $(g,A) := (g, a_{1},...,a_{n})$ consists of an integer $g\geq 0$ and the weight data: an element $(a_{1},...,a_{n})\in\mathbb{Q}^{n}$ such that $0<a_{i}\leq 1$ for $i = 1,...,n$, and $$2g-2 + \sum_{i = 1}^{n}a_{i} > 0.$$ A family of nodal curves with marked points $\pi:(C,s_{1},...,s_{n})\rightarrow S$ is stable of type $(g,A)$ if

• the sections $s_{1},...,s_{n}$ lie in the smooth locus of $\pi$, and for any subset $\{s_{i_{1}}, . . . , s_{i_{r}}\}$ with non-empty intersection we have $a_{i_{1}} +...+ a_{i_{r}} \leq 1$,
• $K_{\pi}+\sum_{i=1}^{n}a_{i}s_{i}$ is $\pi$-relatively ample.

Now, given a collection $(g,A)$ of input data, there exists a connected Deligne-Mumford stack $\overline{\mathcal{M}}_{g,A[n]}$, smooth and proper over $\mathbb{Z}$, representing the moduli problem of pointed stable curves of type $(g,A)$. The corresponding coarse moduli scheme $\overline{M}_{g,A[n]}$ is projective over $\mathbb{Z}$.

B. Hassett, "Moduli spaces of weighted pointed stable curves", Advances in Mathematics 173 (2003), Issue 2, 316-352.

Fixed $g,n$, consider two collections of weight data $A[n],B[n]$ such that $a_i\geq b_i$ for any $i = 1,...,n$. Then there exists a birational \textit{reduction morphism} $$\rho_{B[n],A[n]}:\overline{M}_{g,A[n]}\rightarrow\overline{M}_{g,B[n]}$$ associating to a curve $[C,s_1,...,s_n]\in\overline{M}_{g,A[n]}$ the curve $\rho_{B[n],A[n]}([C,s_1,...,s_n])$ obtained by collapsing components of $C$ along which $K_C+b_1s_1+...+b_ns_n$ fails to be ample.

The reduction morphisms are defined in terms of the parametrized objects. Some of the spaces $\overline{M}_{0,A[n]}$ endowed with these reduction morphisms appear as intermediate steps of Kapranov's blow-up construction of $\overline{M}_{0,n}$,

M. Kapranov, "Veronese curves and Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$", Jour. Alg. Geom. 2 (1993), 239-262.

In higher genus $\overline{M}_{g,A[n]}$ may be related to the log minimal model program on $\overline{M}_{g,n}$,

H. Moon, "Log canonical models for $\overline{M}_{g,n}$", https://archive.org/details/arxiv-1111.5354.