# Fortnightly links (97)

Alexander Perry, Laura Pertusi, Xiaolei Zhao: Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties develops the machinery on stability conditions in families in the case of Gushel–Mukai varieties. These are (at least in dimension 4) somewhat parallel to cubic 4-folds, with very precise conjectures about their geometry and rationality. For cubics there are two conjectures concerning their rationality (one in terms of a Hodge-theoretically associated K3 surface, due to Hassett, and one in terms of a categorically associated K3 surface, due to Kuznetsov) which are recently proven to be equivalent.

The intriguing result (for me) is that for Gushel–Mukai 4-folds the analogous conjectures are

*not*equivalent!Yves André, Luisa Fiorot: On the canonical, fpqc, and finite topologies on affine schemes. The state of the art discusses how some very classical questions regarding Grothendieck topologies on affine schemes, and relates them to very recent techniques (involving splinters, and prisms). The geographer in me likes these types of results.

Chunyi Li, Howard Nuer, Paolo Stellari, Xiaolei Zhao: A refined derived Torelli Theorem for Enriques surfaces discusses how for all but a codimension-2 subset of the moduli space of Enriques surfaces, the "Kuznetsov component" of the derived category of the Enriques surface (orthogonal to 10 completely orthogonal line bundles) determines the surface uniquely. I have seen talks about this result, and I quite like the approach of studying 3-spherical objects!