These are interesting times to look over the algebraic geometry arxiv postings. Just over a week ago, there was a posting by Tanaka which claimed the minimal model program was false in characteristic two. Then yesterday at the top of the page was a paper by Castravet and Tevelev claiming that the Mori Dream Space conjecture for was false. Then today, there is a paper by Fontanari claiming instead that the Mori Dream Space conjecture is TRUE for the same space, but modded out by the finite group .
I’ll keep my remarks brief here, but roughly a Mori Dream Space is an algebraic variety which “looks like” a toric variety, and so its birational geometry is strongly determined by some combinatorial invariants. There’s obviously much more to be said here, but for that I will defer to either the papers listed above or to a survey on Cox rings such as this one. The Mori Dream Space conjecture for a family of varieties is simply the statement that for all , is a Mori Dream Space. What Castravet and Tevelev showed was that for , there are Birational maps between and spaces which are not Mori Dream Spaces. I haven’t yet had a chance to do more than skim Fontanari’s paper.
One thing I can note here is that the object you most often study to get a handle on the birational geometry of is the cone of nef divisors. I myself had a paper with Arap, Gibney and Swinarski studying a certain selection of these divisors called Conformal Blocks Divisors, which can be defined either on or . One of the things that made this conjecture difficult experimentally was that the number of generators of this nef cone gets very big very quickly, and the work with Conformal Blocks was in part an attempt to focus on some subcone where we could actually do experiments. It might be interesting to do some more experiements and see what exactly goes wrong with the simple quotient under .
Finally, even though a fly may drop, it may also start buzzing again, as noted by Ellenberg. Still, interesting stuff!