Today I’m going to talk a bit about an important paper from 1969. This one. It’s a bit hard to read at some points, but it was revolutionary. In it, Pierre Deligne and David Mumford prove that the moduli space of curves is always irreducible. They proved this fact in two ways: first by using the Stable Reduction Theorem, which is a hard theorem on Abelian varieties. The second way was by using stacks.
Now, I’m mostly going to talk about the impact of the paper, rather than going in detail through the proof (though I can do that in another post, if people are interested). For some intuition on stacks, check out this post at the Secret Blogging Seminar. Though the first proof, with stable reduction, was sufficient to establish the result that the moduli space of genus curves is irreducible, the second proof was the more important one.
Before, the general method of solving classification problems was to attempt to construct a variety or, if needed (and it often was) a scheme , such that the families of objects over a scheme was the same thing as . This is what we call a fine moduli space. Problem is, they almost never exist. Sure, there’s the Hilbert Scheme, and its special cases, things like Grassmannians, and there’s Chow Varieties. But most moduli problems don’t behave well. For instance, smooth curves of a given genus don’t have a moduli scheme.
However, stacks are the ultimate in classifying spaces. You can always construct a category classifying things: let be the category of schemes and let be a functor such that the fiber over any given scheme is the set of families of object parameterized by that scheme. There’s a real sense in which this category classifies all of your objects. However, a category isn’t geometric enough.
So what do Deligne and Mumford do? They introduce the notion of an algebraic stack (stacks themselves had been introduced a few years earlier) which first starts with a category and insists it be a stack, which mostly says that we can take open covers of schemes, define families on them that agree on overlaps, and then glue to get a family over the whole scheme. The algebraic condition, that’s the big one. The condition here amounts to saying “Well, the stack is ALMOST a scheme” and in fact, the failure to be a scheme is geometric too, so algebraic stacks are geometric objects. This gives us a real, honest, classifying space (for suitable definitions of words like “real” and “honest”…but mostly “space.”)
They then go to show that this thing is, in fact, irreducible when we’re looking at the moduli stack of genus curves. The irreducibility was known over much earlier, through some nice topological methods, and this proof was brought into positive characteristic with some work, for sufficiently large genus and prime. Deligne and Mumford pushed the proof for all primes and all genuses (is this the right plural for genus?) greater than . Zero and one needed to be done by hand, but that’s not too unreasonable.
Aside from the great achievement of solving this long standing open problem, the real value here is that Deligne and Mumford made algebraic geometers start taking the notion of stacks seriously, because they could be used to solve actual mathematical problems. Nowadays, stacks are essential to large branches of mathematics research.
For instance, in my previous series on Gromov-Witten invariants and Quantum Cohomology, I just the moduli space of stable maps, and could only really do things in the genus zero case, becuase that was a fine moduli space. Well, if you use stacks, you can get fine moduli spaces in all genuses, though with the downside of the Gromov-Witten invariants being, a priori, only rational numbers. But still, there are many good questions about these numbers, and we need stacks to even define them!