# Fortnightly links (126)

Michael Wemyss: A lockdown survey on cDV singularities is a beautiful survey about the beautiful mathematics Michael and collaborators have been developing in the past decade or so. The density of ideas might make it less of an easy read if you have never seen some of the material before, but this is a great read in any case.

Thorsten Beckmann: Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure is about generalising results for K3 surfaces to hyperkähler varieties, understanding what lattice theory has to say about derived categories. One highlight for me is Proposition 9.4, which almost, but not quite, gives finiteness of Fourier–Mukai partners of hyperkählers of K3

^{[n]}-type: the problem is that it's not known that a variety derived equivalent to a hyperkähler of such a type is itself of this type. It's an intriguing thought that, even though hyperkählers seem to be scarce!Corollary 9.7 is also fun, showing that if the derived categories of Hilbert schemes of points on K3s are equivalent, then the derived categories of the surfaces are equivalent to begin with. In an email exchange Thorsten and I discussed how this generalises to (anti-)Fano surfaces. It'd be interesting to figure out whether it holds universally, or whether it can fail for some surfaces.

Barbara Bolognese, Domenico Fiorenza: Fullness of exceptional collections via stability conditions–A case study: the quadric threefold shows how stability conditions can be used to study admissible subcategories. So if you have some way of constructing a stability condition, you can study the fullness of a collection or a geometric description of a subcategory. This intriguingly reverses the usual approach to these questions!