I’m currently in Maryland at a conference to celebrate the 62nd birthday of John Millson (why 62? Beats me. I guess people like him so much they couldn’t wait for 65).
Ravi Vakil is talking about “Murphy’s Law” in algebraic geometry, which was ably summed up by Harris and Morrison (not our Morrison) as “There is no geometric configuration so horrible that it does not appear in a Hilbert scheme.” A Hilbert scheme is a moduli space of subvarieties of , so it’s a very natural object to look at in algebraic geometry, and thus it’s very upsetting that it’s as singular as you can possibly imagine. This means all kinds of horrifying objects must exist, like surfaces in characteristic 31 which deform to characteristic 0 to 43rd order, but no further. Yuck!
Now, there’s lots more to say about this from a high level perspective (like a long list of moduli spaces satisfying Murphy’s law), but I wanted to talk about an example of a singular moduli space you already know.
So. Grab the nearest Rubik’s cube (or vivid mental image thereof). Now look at it in it while all the sides are aligned (when it’s actually cube shaped). What is the space of “deformations” at the point (the Zariski tangent space)? There’s 9 of them, since in each of the planes of rotation, there are 3 freely rotating pieces. But if you rotation a little bit in one of the planes, suddenly, the other directions of rotation lock up, and you have a lot less symmetry, 3 dimensions from the direction you rotated, and one each from rotating the whole cube, which gives us 5. As you may have noticed, 5 is not 9, so by the usual definition of a singular point, the aligned configuration must be a singular point.
Congratulations! You’ve shown that the moduli space of configurations of the Rubik’s cube is singular. As you can see, no moduli space is safe.
In fact, this just the tip of the iceberg. It’s a well-known principle that if an object has a deformation with less symmetry than the original object, it will be a singular point of the moduli space, which is basically what we are applying here. What Ravi’s theorem says is there are lots other, much less obvious singularities out there in the world.