Pieter Belmans

There is a new website, hyperelliptic.ncag.info, on the classification of hyperelliptic (or generalized hyperelliptic) varieties in complex dimensions 2, 3 and 4. For now it lives as a subdomain of ncag.info. A hyperelliptic variety is a quotient $X = T/G$ of a complex torus $T$ by a finite group $G$ acting freely and without translations. In dimension 1 these are the elliptic curves, in dimension 2 the seven bielliptic surfaces of Bagnera and De Franchis; the word has nothing to do with…

Whilst it might look like I am hyperproductive these few weeks, with a new blogpost coming up almost every day about something I have done, I am mostly just putting things online that were finished before, in preparation of the summer holidays. This post is about Fano 4-fold quiver moduli from subspace quivers, a new paper joint with Markus Reineke. Let's discuss the origin story of this paper in a way that is not appropriate for the paper's introduction, but that is perfect for a blog post.…

Since early 2023 there has been a website at cubics.fanography.info on the relationship between cubic fourfolds and K3 surfaces, but I never announced it I think: a first draft version was written in just 2 days and I didn’t know what to do next. I’m not sure it is in any state or form “finished”, but given that I’m on a website spree lately, I might as well get this one out too. So this is that announcement, three years late. Hassett divisors A smooth cubic fourfold $X\subseteq\mathbb{P}^5$ is…

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,…

Some quick housekeeping: several of my websites now have a Zenodo DOI, so they are easier to cite. This currently covers fanography.info, grassmannian.info, hyperkaehler.info and mgnbar.info, with more to follow. Each site's "how to cite" section now lists the DOI, and I use the version-independent one, so it always resolves to the latest release. If you use any of these in your work, please do cite them: it helps me explain to the people who fund my work that building these things is useful.

Back in 2023, together with Ignacio Barros, I made Mgnbar.info, a website about the geometry of $\overline{\mathrm{M}}_{g,n}$, the moduli space of stable $n$-pointed genus $g$ curves. For a long time it showed a single invariant, the Kodaira dimension. It now has two more layers, which you can pick from the selector above the table, and the Kodaira dimension layer itself has gained some extra information. The tautological ring. For each $(g,n)$ the table records whether the Chow ring of…

I'm happy to announce a new paper, Brauer groups of resolved quiver moduli via gerbes, joint with Gianni Petrella and Sebastian Torres. In this paper we give a new and very different proof of an old result due to Le Bruyn and Schofield (two mathematicians I admire greatly!), by translating a proof (due to Biswas–Hogadi–Holla) in the context of moduli of vector bundles on curves to that of quiver moduli. It is no secret that the quiver-curve dictionary is a favourite topic of mine, and this…

I want to congratulate Gianni Petrella, for his PhD defense today at the University of Luxembourg, and the wonderful thesis Windows, walls and invariants of quiver moduli that he has written. The thesis will be available online at some point, but you can also check out the papers it builds on: Rigidity and Schofield's partial tilting conjecture for quiver moduli, a joint paper with Ana-Maria Brecan, Hans Franzen, Markus Reineke, and myself, published in the Journal de l'École Polytechnique…

Almost eight years ago I launched Fanography. It has just received its biggest technical overhaul since then: fanography.info is now a static website, generated with Hugo and served from GitHub Pages. From Flask to Hugo Before, Fanography was a Flask application: a little Python server that assembled each page on the fly from the underlying classification data. That worked well, but it meant there was always a server that needed to keep running, be kept up to date, and be paid for in one way or…

I like Fano 3-folds. I like them so much that I created Fanography! One thing which I found intriguing in the story one aspect of the details of the Mori–Mukai classification that is not often mentioned. Namely, at the very end of their Classification of Fano threefolds with $\mathrm{b}_2\geq 2$, I, in paragraph (7.33), they introduce an invariant that they need to distinguish between various Fano 3-folds with otherwise the same numerical invariants. For instance, the families 2.22 and 2.24…

This post concerns Semisimple.jl, a Julia package for computations with semisimple Lie algebras. There is documentation. Over on the HomogeneousTools blog I wrote a post titled Verifying Kostant’s conjecture using Semisimple.jl, which describes a fun way to use computer algebra to experiment with conjectures.

This is a cross-post from the HomogeneousTools website, where package-specific updates about Lie.jl will also appear. I will occasionally post about Lie.jl here, but if you really want to keep up with development, you should head over to homogeneous.tools. This post concerns Lie.jl, a Julia package for computations with semisimple Lie algebras. There is documentation. This first stable release focuses on efficient computations with semisimple Lie algebras, of the type that are most useful to an…

Together with Javier Fernández Píriz, a PhD student at the University of Luxembourg, we are working on HomogeneousTools, a collection of packages meant to efficiently deal with homogeneous varieties and bundles on them. For now, our main joint project isn’t live yet (more on that in a few weeks), but the project’s website is now live two preliminary packages I created for our main work are also live More on those already live packages soon!

Almost ten years ago, I moved from WordPress to Jekyll. It felt like time for another change, so I have now moved the site to Hugo. There was no particular need for it, build times weren't too bad, but it is nice to see how much faster Hugo is in building my website. The migration is complete, and I have tried to make sure everything still works as intended. Let me know if something did break!

5 years ago, Jieao Song wrote a really nice intersection theory package in Julia, my new favourite programming language. It is similar to the famous Schubert2 package in Macaulay2, and other, lesser-known packages, like Chow for SageMath. It is extremely efficient (being significantly faster than Schubert2). One plan that had been around for almost 5 years was to integrate it into OSCAR, the Julia analogue of what SageMath is in Python, so that it becomes part of a stable ecosystem. I'm happy...