# Fortnightly links (119)

Introduction to stacks and moduli is a course that Jarod Alper will be running next term (starting in 2 weeks). There's also a working draft of the course notes available, which suggests it will be an absolute banger of a course!

Long Wang: On automorphisms of Hilbert squares of smooth hypersurfaces shows that automorphisms of Hilbert squares of low-degree hypersurfaces are always induced from the hypersurface. Cool stuff! I really enjoy this type of result, and in a joint work with Oberdieck and Rennemo we tackled the Ur-case of $\operatorname{Hilb}^2\mathbb{P}^n$, so it's nice to see this question gaining some traction.

Shizhuo Zhang: Bridgeland moduli spaces and Kuznetsov's Fano threefold conjecture disproves one of my favourite conjectures, phrased by Kuznetsov. This conjecture predicts a surprising connection between the Kuznetsov components (the interesting piece of the derived category) of Fano 3-folds in different deformation families, giving a categorical twist on what is a priori only a coincidental equality of Hodge numbers. The proof goes by describing moduli spaces of Bridgeland-stable objects, which if the components are equivalent should be identified. But in one family the moduli space is always reducible, whilst in the other it is always irreducible. Hence the Kuznetsov components cannot be identified in any way, let alone in the more precise way predicted by Kuznetsov's conjecture (this latter fact was already known).

The different work-in-progress's (by different groups of authors) cited in this preprint are also very promising, with a categorical Torelli theorem for Gushel–Mukai 3-folds; a description of the Kuznetsov components in the $d=1$ case of Kuznetsov's conjecture; a description of moduli spaces of Bridgeland stable objects in Enriques categories; and a different disproof of Kuznetsov's conjecture. I'm looking forward to all of these works!