So, I was talking to Mikhail Khovanov at tea today (there are advantages to being at IAS), when the following question “How is it that one could take a ‘nice’ variety (where for now, we’ll let ‘nice’ mean ‘rational’), take a quotient by a free finite group action, and end up with a variety which is ‘bad’ (i.e. not rational)? Or is such even possible?”
I insist it is, having come across examples in my research, but Mikhail seemed to be unconvinced, and I couldn’t summon the energy to come up with a good example (also, let’s honest, sometimes when one thinks something is true, but has a seed of doubt that one might somehow be wrong, it’s hard to argue forcefully. It’s so much less embarrassing when you turn out to wrong if you were tentative).
Thinking about this while making dinner, it occurred to me that this is just another instance of mathematicians refusing to believe that higher dimensional algebraic geometry is really is bad as it is. It’s like the femme fatale (or handsome bad boy) who breaks our heart over and over, but who we can’t bring ourselves to believe the fundamental badness of.
So let’s be honest with ourselves, higher dimensional algebraic geometry is a terrifying place, nothing like the world of curves (which, let’s be honest, is what most of us having mind when thinking about algebraic geometry). It even has a Murphy’s law!
Anyone out there have good tidbits illuminating the horribleness of algebraic varieties that are explainable (if not in full detail) over tea?