• Moritz Kerz, Florian Strunk, Georg Tamme: Algebraic K-theory and descent for blow-ups proves Weibel's conjecture on the vanishing of algebraic K-theory, which predicts that $\mathrm{K}_i(X)=0$ for $i\leq -d-1$ if $X$ is a $d$-dimensional noetherian scheme. The proof uses derived algebraic geometry in an essential way, but Weibel's conjecture significantly predates the advent of derived algebraic geometry, so it is a cool application of these abstract techniques!

  • Anton Fonarev, Alexander Kuznetsov: Derived categories of curves as components of Fano manifolds is a preprint I've been waiting for, except that I was rather expecting it from Narasimhan. But his paper has not appeared yet (it now has), and this preprint proves the result in a different (and from my point of view probably more interesting) way. Their result is that $\mathbf{D}^{\mathrm{b}}(C)$ (for a generic curve $C$) embeds fully faithfully in the derived category of its moduli space of vector bundles of rank two and odd degree. Awesome!

  • Asher Auel, Marcello Bernardara: Cycles, derived categories, and rationality is a really nice overview on derived categories and how semi-orthogonal decompositions are connected to rationality questions. I strongly suggest reading it if you wonder how these relate.

    If anything, you should check out table 1 on page 45, for an intimidating overview of rationality questions for 3- and 4-folds.