# Fortnightly links (68)

First of all, my inside source told me that the ICRA award this year goes to Gustavo Jasso and Julian Külshammer (but the ICRA website doesn't mention this yet). Congratulations!

Giovanni Cerulli Irelli, Three lectures on quiver Grassmannians are the slides (and in the future also lecture notes) for a lecture series at this year's ICRA.

Dmitri Orlov, Derived noncommutative schemes, geometric realizations, and finite dimensional algebras is a mix of an overview article on geometricity for smooth and proper dg algebras, whilst also introducing a language to study morphisms between these objects. One could argue that a large portion of my research interests are inspired by questions related to and inspired by geometricity for such dg algebras. Though I'm still not a fan of his typographical conventions, using different calligraphic fonts to denote related categories.

After reading this preprint I also realised that I've discussed Krull–Schmidt partners before on this blog, see the two-part series on Bondal's example (part 1: exceptionality and part 2: non-extendability). Orlov has introduced this notion, to describe (possibly inequivalent, as Bondal's example shows) admissible subcategories which nevertheless have the same K-theory (and other additive invariants) as they are orthogonal to the same category inside a bigger category.

Elana Kalashnikov, Four-dimensional Fano quiver flag zero loci adds 139 previously unknown Fano 4-folds to the classification. She does this by giving new methods to construct the weak Landau–Ginzburg mirror (which is a Laurent polynomial) for zero loci in quiver flag varieties (previously this was only known for toric complete intersections and zero loci in usual flag varieties). This way at least 139 new families of Fano 4-folds are constructed.

One interesting fact is that up to dimension 3 every Fano variety is either a toric complete intersection, or a zero locus in a quiver flag variety. Could the same be true for Fano 4-folds? I'd love to see the classification of Fano 4-folds settled!

Is someone keeping track of the total number of distinct families constructed so far?