# Fortnightly links (79)

Andrea Petracci: An example of mirror symmetry for Fano threefolds is a careful elaboration of an example illustrating the role that toric degenerations play in the study of Fano varieties via methods from mirror symmetry. It shows how to study the Fano threefold $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ (also known as 3-27) and the Fano threefold $\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$ (also known as 2-32). They are related via a deformation which goes through a singular toric Fano threefold, and it is explained how the different smoothings lead to different properties from the point of view of mirror symmetry.

Daniel Bergh, Olaf Schnürer: Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties is a cool application of the notion of conservative descent, introduced earlier by the same authors. They explain how the derived categories of gerbes and Brauer–Severi stacks decompose, which was known over schemes, but in this greater generality it is a really neat application of their machinery.

A special case was proven a few days earlier in Michael Brown, Tasos Moulinos: Topological K-theory of twisted equivariant perfect complexes.

Andreas Hochenegger: Introduction to derived categories of coherent sheaves is a nice introduction to one of my favourite objects in mathematics (i.e. $\mathbf{D}^{\mathrm{b}}(\mathop{\rm coh} X)$). It seems to be an excellent starting point for a student to get into this topic, complementing other introductions nicely.