This installment of fortnightly links is the second part of a long overdue batch of preprints and links I found interesting.
Eleonore Faber, Colin Ingalls, Shinnosuke Okawa, Matthew Satriano: On stacky surfaces and noncommutative surfaces shows how a nice enough sheaf of noncommutative algebras on a surface (also known as an order) is "the same" as an Azumaya algebra on a nice Deligne–Mumford stack. So when doing noncommutative algebra, you are secretly doing algebraic stacks, or vice versa. Cool!
Jonas Bayer, Christoph Benzmüller, Kevin Buzzard, Marco David, Leslie Lamport, Yuri Matiyasevich, Lawrence Paulson, Dierk Schleicher, Benedikt Stock, Efim Zelmanov: Mathematical Proof Between Generations is a collection of essays, on the nature of a proof in mathematics. With all the recent progress in proof verification, I often wonder what this means for "geometric" proofs in algebraic geometry, and how the in my eyes inevitable shift to include more and more formal proof verification into our research will play out. I have no answers to this.
Completion of the Liquid Tensor Experiment announces that Scholze's Liquid Tensor Experiment is now solved, i.e. a full formalised proof of one of the foundational results in condensed mathematics exists. This is a formalisation of a complicated statement about complicated mathematical objects, and is truly a landmark in mathematical formalisation.
There is also an interesting visualisation on the structure of the proof, somewhat similar to what the Stacks project used to have (and I should reimplement this at some point...)
Alexander Kuznetsov, Evgeny Shinder: Categorical absorptions of singularities and degenerations is another banger of a preprint, introducing something "opposite" of a categorical resolution of singularities, which exists in certain natural settings. The introduction gives a great outline of what is to come in later work too.
Samuel Stark: Deformations of the Fano scheme of a cubic hypersurface shows that the deformation theory of a cubic hypersurface of dimension at least 5 is equivalent to that of its Fano variety of lines, a natural moduli space associated to it. It's no secret probably I'm a sucker for this type of results, and I wonder whether a proof using a Fourier–Mukai functor from the derived category of the hypersurface towards the derived category of the Fano variety works, just like one can do for Hilbert schemes of points in joint work with Lie Fu and Theo Raedschelders. In fact, I think derived categories of these Fano varieties of lines should be pretty interesting to study, as they are Fano varieties (in the usual sense) themselves.