# Fortnightly links (77)

Olivier Debarre, Alexander Kuznetsov: Gushel-Mukai varieties: moduli continues the study of Gushel–Mukai varieties (when they are 3-dimensional they are the Fano 3-fold 1–5), with 2 earlier installments by the same authors. They show how two moduli stacks (one for the varieties, one for the linear data describing them) are related.

The thing that intrigues me the most is the generalised root stack construction they introduce, I look forward to understanding it better.

Alex Chirvasitu, Ryo Kanda, S. Paul Smith: Feigin and Odesskii's elliptic algebras rather is the

*first*paper in a series discussing Feigin–Odesskii algebras $Q_{n,k}(E,\tau)$, which generalise Sklyanin algebras (where $k=1$). In the original papers by Feigin–Odesskii, dating almost 30 years ago, quite a few statements were left as exercises to the reader, or as conjectures, and they provide careful proofs for these.Exercise for the reader of this edition of fortnightly links: what can be said about $Q_{4,2}(E,\tau)$? That is the first interesting case it seems (as $Q_{3,2}(E,\tau)$ is just the polynomial algebra in 3 variables). Presumably it must fit somewhere in Pym's classification, but what is $X_{4/2}$ supposed to be in this case? The definition is left for a later paper unfortunately...

Hayato Morimura: Derived equivalence for Mukai flop via mutation of semiorthogonal decomposition is an interesting (and short) paper which I found a very interesting read. In this paper a new and streamlined proof is given of the theorem that varieties connected by either a standard flop, an Abuaf flop, or a Mukai flop are derived equivalent. Sometimes a paper comes along at precisely the right time, and for me this paper came along at precisely the right time for understanding what higher-dimensional flops are.