Akira Ishii, Shinnosuke Okawa, Hokuto Uehara: Exceptional collections on $\Sigma_2$ does a very careful analysis of the exceptional sequences in the derived category of the second Hirzebruch surface. This is not a del Pezzo surface, only a weak del Pezzo surface, and an important question is to understand what goes through, and what fails, for the derived category of such surfaces. Theorem 1.1 summarises what is known for del Pezzo surfaces, Conjecture 1.3 which modifications one needs to do for weak del Pezzo surfaces. Aside from answering the conjecture for the second Hirzebruch surface, the preprint raises many interesting questions.
Nicolas Addington, Daniel Bragg: Hodge numbers are not derived invariants in positive characteristic proves what the title suggests it does (Nick is skilled with such titles). This is a super-cool result. With a positive characteristic counterexample to the derived invariance of Hodge numbers, what do you think about the derived invariance in characteristic zero?
Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süss, Nivedita Viswanathan: The Calabi problem for Fano threefolds is a large manuscript, going a large way in describing all K-polystable smooth Fano 3-folds. The proof is a tour de force (as many proofs involving the classification of Fano 3-folds are) with a mixture of generic methods and a case-by-case analysis.
I hope to at least add the data to Fanography soon, and maybe I'll write a bit more on this blog in order to more carefully understand the results.