This fortnightly links has a lot of catching up to do.

  • Max Lieblich, Martin Olsson: A reconstruction theorem for varieties is a very remarkable preprint. It explains that schemes are almost completely characterised by (what I up to now would call) a very coarse invariant, namely their underlying topological space. Of course, for curves this is very much false (compact Riemann surfaces with the same genus are homeomorphic) but in dimension 2 and above (and over an infinite field) all you need to remember is the rational equivalence relation on effective divisors.

  • Arend Bayer, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, Paolo Stellari: Stability conditions in families is a long-awaited paper, with a long list of authors, and it is long. It settles many important questions in the study of derived categories of cubic fourfolds, I think my favorite one at the moment is that having a Hodge-theoretic K3 is equivalent to having an actual K3 surface as the Kuznetsov component (i.e. Hassett = Kuznetsov, which was only known generically up to now). Alternative to the introduction, you can also read a related Oberwolfach report for a good view on what is happening here.

  • Izuru Mori, Shinnosuke Okawa, Kazushi Ueda: Moduli of noncommutative Hirzebruch surfaces does what the title suggests. But for me the truly interesting aspect of this paper is their classification of locally free sheaf bimodules of rank 2, and more probably more appealing to a general public: a noncommutative version of the McKay correspondence for the cyclic group $1/d(1,1)$, relating cyclic quotient singularities to Hirzebruch surfaces in the commutative case.

Lots of papers didn't make the somewhat arbitrary cut here, but so it goes.