# Fortnightly links (153), part 1

The reader of this blog might have noticed an unusual silence. I had some eye problems recently, which made me have to focus on the urgent things only, and blogging is not one of these. But all is well now, so let's get back to fortnightly links. A lot of interesting things were published, so I will split things into two posts.

Federico Barbacovi, Ed Segal: Serre functors of residual categories via hybrid models gives a beautiful and short argument for a description of the Serre functor of the residual category (formerly known as Kuznetsov category) of a complete intersection, recovering results of Kuznetsov and Perry. They do this by describing the residual category as a category of matrix factorisations, bringing back the geometry in what is from what is a priori just defined as an admissible subcategory. Cool!

Raymond Cheng: Geometry of $q$-bic Hypersurfaces is Raymond's PhD thesis, on a certain type of hypersurface in positive characteristic. They are curious (dare I say $q$-rious?) objects indeed, having many properties similar to quadric hypersurfaces but having much higher degree. I strongly advise you to read his introduction, for an infinitely better introduction to these objects than I could ever provide.

Studying homological projective duality could also be interesting in this context: Beauville has shown that these are exactly the hypersurfaces for which smooth hyperplane sections are all isomorphic. Hence just like for quadric hypersurfaces, their homological projective dual is a trivial family away for the dual hypersurface. Except that the dual hypersurface is much more complicated than the initial smooth hypersurface, the ambient dimension has to be larger than the degree $q+1$ for any interesting decompositions to exist, etc. So not sure whether it is

*that*meaningful. But it might be fun to continue this train of thought.Alexander Kuznetsov, Yuri Prokhorov: On higher-dimensional del Pezzo varieties is a great preprint on mildly singular analogues of del Pezzo varieties: Fano varieties of coindex 2 (so of complexity just above that of projective space and quadric hypersurfaces).

The story about homological projective duality Thorsten and I explained for the Segre cubic is in fact an instance of a natural duality that exists for certain almost del Pezzo varieties. There is also something I find really interesting regarding homological projective self-duality contained implicitly within the paper, which I might discuss in a blogpost at some point (or maybe even write up as a paper).

Valery Alexeev: Root systems and hyperkahler varieties is a curious little preprint. Who is the first to provide an interpretation of OG

_{6}and OG_{10}in this setup? What makes smooth hyperkählers so special from this point of view really?

There are also some more interesting ICM-related preprints:

- Michel Van den Bergh: Non-commutative crepant resolutions, an overview
- Catharina Stroppel: Categorification: tangle invariants and TQFTs

Lastly, Okounkov has written 4 beautiful expository articles on the work of the Fields medalists: