I posted a new paper on the arXiv this week.
As I’ve discussed before, I’m interested in the birational geometry of minimal resolutions of 3-fold Gorenstein quotient singularities; i.e. quotients of 
by finite subgroups of 
.
I’ve had some ideas floating around since my time at Nagoya University in 2016 and, with a lot of great input from Alastair Craw, I distilled a selection of these ideas into a decent theorem with some promising consequences.
To summarise, as I’ve written about before, resolutions of 
correspond (not bijectively) to chambers in a big vector space 
, at least when 
is abelian. There is a particular resolution that has been studied more than any of the others, called the -Hilbert scheme 
. In general, the chamber structure inside is quite mysterious – for instance, as the singularities here are not symplectic there is no reason to expect that the walls define full hyperplanes instead of just cones. Using the mechanics of Reid’s recipe – which is a clever, representation-theoretic way of labelling the exceptional fibre inside – I produced explicit inequalities defining the chamber for and determined which of them are necessary, thus which define walls. The work of Craw-Ishii allows us to say exactly what wall-crossing behaviour happens at each of these walls, again using Reid’s recipe.
Hope it’s interesting to you! I’m hoping to publish some applications of this technology soon in joint work with Yukari Ito.
lisnbaa 