(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
I will introduce N-resolutions, which are the negative analog of the Kollár–Shepherd-Barron (1988) P-resolutions of a 2-dimensional cyclic quotient singularity. (We instead work with the corresponding M-resolutions of Benkhe-Christophersen (1994).) I will start by describing an algorithm to find all of them based on the explicit algorithm for P-resolutions in Park-Park-Shin-Urzúa (2018) (that geometrically recovers Christophersen-Stevens’ zero continued fractions correspondence (1991)), which in turn is based on the explicit MMP described by Hacking-Tevelev-Urzúa (HTU 2017). I will also describe another way to find N-resolutions via antiflips (HTU 2017) starting with an M-resolution, showing an action of the braid group on all its associated Wahl resolutions. This will bring us to Hacking exceptional collections (2013-2016) on surfaces that are Q-Gorenstein smoothings of particular singular surfaces, where Karmazyn-Kuznetsov-Shinder (2022) have described their derived categories via derived categories of the Kalck-Karmazyn algebras (2017). This can be put together through Kawamata’s bundles (2018-2022), and I will describe our main theorem on semi-orthogonal decompositions defined by these M- and N-resolutions. I will end with applications to all simply-connected Dolgachev surfaces. I will mention open problems. This is on the joint recent work with Jenia Tevelev. This computer program finds all M- and N-resolutions.