This fortnight it's just 3 interesting preprints on the arXiv. But boy are they fun reads!
Geometry and categorification by Ben Webster. In case you've been wondering (just like I was) what people in representation theory are exactly doing with flag varieties, or just want to get a general overview of categorification.
Morphisms to Brauer-Severi varieties, with applications to del Pezzo surfaces by Christian Liedtke. It is based on the funny observation that one could try to replace projective spaces in the yoga of ample line bundle by their twisted forms (the Brauer-Severi varieties), and try to gain arithmetic data from this. Fun!
Moduli spaces of sheaves on K3 surfaces by Justin Sawon. Before this article I couldn't say I understood much about stability conditions in the context of moduli spaces of sheaves. After reading it I'm not saying that I do, but I am at least getting used to it as von Neumann would say.