I’m finally back in Ithaca after about a month of travelling about. Hopefully this will alleviate the lull the blog has been in the last few weeks. Though, Jim Pivarski filled some of the otherwise dead-space with some excellent physics posts. Kudos to him!
The most mathy thing I did while I was gone was the aforementioned mini-course at MSRI on Deformation Theory. It was very well put together, and the lectures were a cut above what I went in expecting. However, I couldn’t help but feel a little dissappointed. The conference was mostly on the nitty-gritty of making stacks work, and very few concepts I hadn’t seen before were introduced. It was good for me to work out lots of examples and discover computational tricks, but I was hoping to have my imagination sparked by some nifty new ideas.
There were some great moments, though. Ravi Vakil, one of the organizers, gave a fun talk on Murphy’s Law for moduli spaces. It was on a recent paper of his which gave a concrete meaning and proof to an old folklore meta-theorem called Murphy’s Law for the Hilbert scheme. I first read it in Morrison and Harris’s wonderful book “Moduli of Curves“, as follows:
There is no geometric possibility so horrible that it cannot be found generically on some component of the Hilbert scheme.
Apparently, this idea goes back to Mumford and his paper “Further Pathologies in Algebraic Geometry”, where he showed that there is a component of the Hilbert scheme that is everywhere non-reduced. The meta-theorem was meant to stop mathematicians from wasting their time trying to find a line the Hilbert scheme wouldn’t cross.
In his paper, Vakil considers a few possibilities for a concrete meaning to ‘satisfying Murphy’s Law’, and settles on ‘every singularity occurs’. This isn’t precisely true, to get a workable statement we need to restrict to smooth equivalence classes of singularities of finite type. Once that is settled, he shows that the Hilbert scheme, and a number of other moduli spaces, satisfy this terribleness condition.
The best part is how simple the proof is, conceptually. The first step is to connect all the moduli spaces in question by showing that if one satisfies Murphy’s Law, they all do. The second step is to find a moduli space amongst your collection that you can show it for.
The moduli space used in this second step is an ‘incidence scheme of lines and points in ‘. This is the scheme that parametrizes all possible lines and points in with a given incidence, that is, with each point being required to lie on some of the lines. Effectively, we are just requiring that some sets of lines each have a common intersection point.
The fact that the collection of all incidence schemes satisfies Murphy’s Law is older than Vakil’s paper. It goes by the name Mnev’s Theorem, and the cuteness of its proof is the motivation for this post. The trick is, using incidences, one can encode all basic arithmetic operations, and therefore write out equations that define singularities in terms of incidence data.
Start with a line in . Internally, it is just a copy of , so to rigidify it we need to fix three points. Pick three points on and call them 0, 1 and . Note that this canonically identifies with . Next, pick a line going through , and call it ‘the line at infinity’. Now, we can think of as plus the line at infinity. Thus, the problem is slowly turning into one about plane geometry (if we only think about the real part).
Now, suppose I give you two more points on , called and (thought of as numbers in ). Can you use incidence data to describe a line that must intersect at the point corresponding to ? How about ? I won’t tell you the answer here, since I think it is too much fun to figure it out on your own. I will advise you to do it first for just the real parts, where the problem is just one about lines in plane geometry.
I should mention that Mnev’s theorem is true schemes over as well as over , and its this version that is necessary for Vakil’s result. However, this obfuscates the classical plane geometry I was trying to highlight above.
Anyway, now that we can add and multiply points, we can do quite alot more. Given three points , and on , I can find a line which hits at the point , and I can require that it pass through 0. Therefore, the points and must collectively satisfy . If we did things right, this is the only condition they must satisfy, and so the incidence scheme is then isomorphic to the affine scheme determined by that equation, which is the Kleinian singularity . Clearly, given enough points, we can carve out the defining equation for any singularity (up to smooth equivalance).