# Fortnightly links (56)

2 days late, but jet lag and work on the new Stacks project website is to be blamed for that. More on that later this week.

Alexander Kuznetsov, Embedding derived category of an Enriques surface into derived category of a Fano variety is a really cute result: certain Enriques surfaces can be embedded into the Grassmannian $\mathrm{Gr}(2,4)$, which is a Fano 4-fold, and blowing up the surface

**preserves**the ampleness of the anticanonical bundle! So that means you've embedded the derived category of the Enriques surface into the derived category of a Fano 4-fold, which moreover has its Hodge numbers concentrated on the diagonal. For a general Enriques surface a similar construction with a Fano 6-fold can be done.Can something similar be done for a fake projective plane?

Dmitri Kaledin, How to glue derived categories is an expository long article (almost book-length) about enhancements of triangulated categories, with a view towards gluing questions. It was also the topic of his talk at the Antwerp conference 2 years ago.

S. Paul Smith: Simple modules over the 4-dimensional Sklyanin Algebras at points of finite order is a preprint which was written in 1993 but never published, on 4-dimensional Sklyanin algebras associated to translations of finite orders. Such algebras are necessarily finite over their center, and they have an interesting fat point geometry.

In Chelsea Walton, Xingting Wang, Milen Yakimov: Poisson geometry and representations of PI 4-dimensional Sklyanin algebras the Poisson geometry of these algebras is studied, which is closely related to their fat point geometry (which Smith studied using algebro-geometric techniques, avoiding all representation theory). It should be remarked that they discuss some issues with the 1993 preprint, e.g. in remark 2.13.