• Aleksandr Novikov: Lefschetz exceptional collections on isotropic Grassmannians is a master thesis, constructing new full (and even Lefschetz) exceptional collections for isotropic Grassmannians. The case $\mathrm{IGr}(3,10)$ (so the parabolic associated to the middle vertex in type $\mathrm{C}_5$) in particular, although one might hope that the methods generalise to $\mathrm{IGr}(3,2n)$.

This also means that another box on grassmannian.info now has a shading, when you toggle the exceptional collections view. Maybe soon (for some version of this word) all $G/P$ are known to have a full exceptional collection!

• Fabian Reede: The Fourier-Mukai transform of a universal family of stable vector bundles answers a (not too precise) question I (and presumably some others) have had for a while in the negative. If $X$ is a variety, and $Y$ is a nice moduli space of sheaves on $X$, such that $\mathbf{D}^{\mathrm{b}}(Y)$ looks like it might contain a copy of $\mathbf{D}^{\mathrm{b}}(X)$ embedded via Fourier–Mukai functor given by the universal sheaf, does it actually do?

Examples of this behavior are moduli of vector bundles of curves and Hilbert schemes of points for surfaces (and higher-dimensional varieties if $n=2$) if the structure sheaf if exceptional. But Fabian shows that this (admittedly naive) hope was too optimistic, by showing that $\Phi_{\mathcal{E}}\colon\mathbf{D}^{\mathrm{b}}(\mathbb{P}^2)\to\mathbf{D}^{\mathrm{b}}(\mathrm{M}_{\mathbb{P}^2}(4,1,3))$ is not fully faithful. This is very interesting!