# Fortnightly links (168)

Christian Böhning, Hans-Christian Graf von Bothmer, Yuri Tschinkel: Equivariant birational geometry of cubic fourfolds and derived categories explains how an equivariant version of the conjecture that rationality of a cubic fourfold corresponds to geometricity of its Kuznetsov category (or the Hodge-theoretic version, or the Fano variety of lines being birational to a Hilbert square)

*fails*! Check out Section 3 for the details. Cool!Johannes Krah: A Phantom on a Rational Surface is an even bigger shocker than the previous preprint. Johannes construct a phantom category in $\mathrm{Bl}_{10}\mathbb{P}^2$, and thus there is also a non-transitive action of the braid group by mutation. As explained in Remark 5.2, the Hochschild cohomology of this phantom is big, and its second Hochschild cohomology being 12 corresponds to the blowup having 12 deformation directions, thus the deformation theories agree. Cool! This was a phenomenon of phantoms for surfaces of general type, but it now also happens for this rational surface.

Jenia Tevelev: Braid and Phantom does the opposite of the previous preprint, namely show that there is

*no*phantom in the complement of a natural semiorthogonal decomposition for the moduli space of rank 2 bundles. This settles a conjecture which is very dear to my heart, and I'd love to see the techniques extended to other moduli spaces!