Back in 2015, Johan Commelin and I made le superficie algebriche, an interactive picture of the geography of minimal complex algebraic surfaces, plotted by their Chern numbers $\mathrm{c}_1^2$ and $\mathrm{c}_2$ (see the original post and a 2019 update). It has just had its biggest update since.

A table. Next to the plane of Chern numbers there is now a table: every class of surface in the database with its Kodaira dimension and numerical invariants, sortable by any column, searchable by name, and with descriptions you can unfold in place.

More surfaces, with descriptions and references. A literature sweep added a long list of named classes, from fake quadrics and the ball quotients on the Bogomolov–Miyaoka–Yau line (Ishida, Cartwright–Steger, Yeung’s surface of maximal canonical degree) to product-quotient and Beauville-type surfaces. Each now comes with a short description, what it is and why it is interesting, and full citations resolved to MathSciNet, arXiv or DOI, using the same bibliography machinery I built for mgnbar.

The map, filled in. The whole interior between the Noether line and the signature-zero line is now populated (following Persson), so the picture genuinely shows which Chern numbers occur and, just as clearly, the thin strip below the Bogomolov–Miyaoka–Yau line that is still an open problem.

Smaller things: the holomorphic Euler characteristic of the tangent bundle $\chi(\mathrm{T}_S)$ is shown too, and every surface now has its own URL, so you can link straight to one, like superficie.info/2/4-8/fake-quadrics. As with my other websites it is now a static site built with Hugo.

Corrections and, especially, more surfaces are very welcome, on GitHub or by email. If you know a surface that belongs on the map, it is easy to add.

LLMs were used to build this, which made the whole process much faster.