Today’s update to the type A catalogue brings 98 new geometric descriptions of Fano fourfolds! We have also made a few improvements to the website since the initial beta release.

#98 new descriptions

The highlight of this release is the new batch of geometric descriptions. Alongside the numerical invariants in the catalogue, these notes explain the geometry of the constructions and their relationships with other models. They include proofs and, where available, discussions of semiorthogonal decompositions.

For a taste, take family 2-144-36-c: a conic bundle over $\mathbb P^3$ with a quintic discriminant surface having exactly 20 ordinary double points. Its other map to $\mathbb P^3$ has fibres isomorphic to $\mathbb P^1$ away from exactly 15 points, where the fibres become $\mathbb P^2$. This second description also shows that the fourfold is rational.

You can explore these descriptions from the relevant family pages in the catalogue. We will keep adding to these notes as we work through the catalogue and understand more of the geometry behind its entries.

#A few website improvements

There are also some new ways to explore and compare the data:

  • Grouped quantum periods. In the quantum period table, you can group entries whose coefficients agree through degree 100 and sort by group size. This makes it easier to spot possible identifications worth investigating; agreement of periods alone does not prove that two constructions belong to the same family.
  • More period comparisons. The comparisons with published periods now include the Coates–Kasprzyk database, with matching record identifiers shown alongside the other literature references.
  • Search refinements. We have made numerical inputs easier to use and brought the results table into line with the main catalogue table. Result previews also show more of the available data.

#Still in beta, with more to come

Everything is still in beta. For the time being, data, descriptions, and website features may change, and things may occasionally break. If you spot an error or something is not working, please get in touch.

We will release more zero loci soon, and further releases will go beyond zero loci to include classical and orbital degeneracy loci. There is plenty more on the way!

The dataset will be made public soon, so that you can download it and use it independently of the website. If you would like access in the meantime, please contact us at info@fanobase.net.