• Lena Ji, Fumiaki Suzuki: Arithmetic and birational properties of linear spaces on intersections of two quadrics is a really cool preprint, which studies $\mathrm{F}_r(Q_1\cap Q_2)$ for $Q_1\cap Q_2\subset\mathbb{P}^{2g+1}$ and $\mathbb{P}^{2g}$. This is a very well-behaved moduli space attached to the intersection of 2 quadrics, whose geometry is a source of great fun, as it is closely related to the geometry of a hyperelliptic (resp. stacky) curve. Ji and Suzuki obtain some strong results on the rationality of these moduli spaces, putting some of the earlier results into a single pretty and coherent picture.

    It does not appear in their introduction, but I in particular like Theorem 4.13, which relates the existence of a maximal linear subspace on $Q_1\cap Q_2\subset\mathbb{P}^{2g+1}$ to the vanishing of a natural Brauer class on this associated hyperelliptic curve.

  • Reflections on moduli space is a collection of comments, musings, answers to the question "Why are you interested in moduli spaces" asked by Rahul Pandharipande to several of his colleagues. There are some really interesting comments in there! One which I particularly like as it directly applies to me, but I hadn't realized it before, is

    When I learned about moduli spaces, it was at a time when I generally liked all sorts of stuff (literature/cinema) that was self-referential, so a statement like the set of all geometric objects of a certain type is itself a geometric object was really music to my ears.

  • Yujiro Kawamata: On formal non-commutative deformations of smooth varieties is a really cool preprint which explains how to do deformations of $\operatorname{coh}X$ without having to deal with those pesky gerby deformations. In other works, one is only concerned with the piece $\mathrm{H}^0(X,\bigwedge^2\mathrm{T}_X)\oplus\mathrm{H}^1(X,\mathrm{T}_X)\subset\mathrm{HH}^2(X)$. The corresponding deformation-obstruction calculus is worked out in a way that is immediately applicable by an algebraic geometer.