# Fortnightly links (115)

Lorenzo De Biase, Enrico Fatighenti, Fabio Tanturri: Fano 3-folds from homogeneous vector bundles over Grassmannians gives alternative descriptions of the 105 families of Fano 3-folds, all in terms of zero loci of homogeneous vector bundles on products of Grassmannians. The nice thing about this approach is that it allows for extensions to the case of Fano 4-folds, which the authors are working on. They told me they are finding hundreds of unique combinations of invariants, which should then be cross-checked with the existing constructions, and hopefully yield many new Fano 4-folds!

This is the third way of describing Fano 3-folds, after the birational approach due to Mori–Mukai and the approach used by Coates–Corti–Galkin–Kasprzyk (using a combination of toric geometry and vector bundles, to which known quantum cohomological methods could be applied).

Only the Mori–Mukai approach is featured on Fanography, meaning that I should add these two alternative descriptions soon!

Will Donovan: Stringy Kähler moduli for the Pfaffian-Grassmannian correspondence explains the suggestion from physics that the monodromy of the Kähler moduli space gives autoequivalences of Calabi-Yau varieties. For the case of the Pfaffian-Grassmannian equivalence (a derived equivalence between non-birational Calabi-Yau threefolds, meaning that they are "double mirrors") this leads to beautiful visualisations of the moduli space and the autoequivalences it induces. Cool stuff!

Building the Mathematical Library of the Future is a Quanta article about how Lean is being used to set up a library of results and proofs, on which more complicated proofs can then be built. There exists an overview of what mathlib knows. Especially the

`algebraic_geometry`section is interesting, and we hope that at some point there will be some interaction between mathlib and the Stacks project.

Maybe I should start mentioning in fortnightly links when Kerodon was updated. Today, a section on $(\infty,2)$-categories was added.