Pieter Belmans

Intersection theory in OSCAR

Tue, 17 Mar 2026 00:00:00 +0000

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 to say that this integration is now done! This was a joint effort of Wolfram Decker and myself. I want to thank Wolfram for the great help (and patience). Aside from simply integrating the infrastructure, we have added lots of documentation, examples, tests, and some minor features. That said, it cannot be overstated how great the foundation laid by Jieao was. I feel like standing on the shoulder of a giant here, and I hope that this will be a great tool for many people in the future.

It is not yet in the latest version (which is 1.7.0 at the time of writing) of OSCAR, but you can use the development version per these instructions.

If you want to see what it is capable of, see the documentation (which for now still refers to the development branch; in the latest branch it is still the older version), e.g., check out

it can work with.

It is still in the experimental phase, and we are eager to hear your feedback and comments!

Hyperkaehler.info: an update

Mon, 09 Mar 2026 00:00:00 +0000

Last week I updated Hyperkaehler.info a bit, and now it also contains information on:

Especially the latter is really cool, it is a refinement of the Hodge decomposition using the representation theory of a finite-dimensional Lie algebra attached to a hyperkähler variety. Jieao Song created an interactive visualization of this decomposition, and this is now integrated into the website. You can explore it for type K3^[4]. Notice how each type now also has its own canonical URL!

Full disclosure: I have used LLMs (more precisely, Claude Opus and Sonnet, via GitHub Copilot) to implement these additions. What would probably have taken me more than a day of work (including various unrelated fixes) ended up taking only one or two hours, split between some implementation work of my own and a larger amount of reviewing LLM-generated code.

It seems that LLMs have reached sufficient maturity to be useful for these things (but one has to carefully review things! they are basically very good autocomplete machines behaving like slot machines, or vice versa) and I'll be describing some more successful use cases later.

Welcome to the new members of the research group

Fri, 06 Feb 2026 00:00:00 +0000

It is all good to post job announcements, but I've come to realize that it is also good to actually publicly welcome the people who have joined the research group!

I am really glad to say that I'm being joined by

dr. Yuki Mizuno, who is joining me as a postdoc for 3 years, working on noncommutative algebraic geometry
Matthijs Holstege, who is joining me as a PhD student, working on quiver moduli and noncommutative algebraic geometry
Javier Fernández Piriz, who is joining Karin Melnick and me as a PhD student (so joint with the University of Luxembourg), working on partial flag varieties

I'm looking forward to all the interesting mathematics that we will do together!

Postdoc position in algebraic geometry

Thu, 05 Feb 2026 00:00:00 +0000

I am excited to announce a postdoc position in algebraic geometry .

The summary:

starting date is October 1 2026 (or as soon as possible thereafter)
it is for 3 years (with a 1-year probation)
the focus is (derived categories of) Hilbert schemes of points on surfaces, so some experience with these or topics closely related to it is required
it is part of a joint grant between Andreas Krug and myself, and regular visits to Hannover are anticipated
we will start reviewing and interviewing candidates after the application deadline of February 22

There is a really active research group on algebraic geometry in Utrecht, and the Netherlands is a great place in general for algebraic geometry, with many interesting activities.

The position is funded by the NWO, through the NWO+DFG Weave project HilbertMoVieS.

All information on how to apply can be found on the Utrecht University job portal. If you have any questions, please get in touch!

An assistant professorship at Utrecht University

Tue, 03 Feb 2026 00:00:00 +0000

The previous such post is only a few months old, but here is another position!

Utrecht University is hiring an assistant professor in pure mathematics, not tied to specific area. I can attest it is a great place to work, so if you fit the (broad) profile, and would like to work at Utrecht University, please consider applying!

If you have questions, you can ask me, or in case of more official questions, the head of the committee, which is Gijs Heuts.

Events related to Markus Reineke as Springer Chair

Fri, 16 Jan 2026 00:00:00 +0000

This academic year, the Springer Chair (a visiting professorship at Utrecht University) has been given to Markus Reineke. He will be visiting us in February, and in this context we are organizing several events:

a colloquium talk on Tuesday, February 3, from 16 to 17, in Minaert 201
a lecture series on February 4, 9, and 10 (see the webpage for more details)
a problem session on Wednesday, February 11, from 13h15 to 15h00, in HFG 611
a 2-day workshop on February 23 and 24 (see the webpage for more details, note that capacity is very limited)

Everyone is cordially invited to the colloquium, lecture series, and problem session!

Whilst the workshop isn't invitation-only, we could only get a small room, so if you plan to attend it, please get in touch with me at p.belmans@uu.nl.

Change of host for Hyperkaehler.info

Sun, 28 Dec 2025 00:00:00 +0000

I've been hosting all interactive websites on PythonAnywhere, but recently I felt the urge to try and host them on my Synology DS920+. Four years ago, I bought it with the possibility in mind of starting to self-host various things aside from using it as a NAS. After a failed attempt a few weeks ago (despite port forwarding being set up in a seemingly correct fashion, my ISP was blocking communication with the Let's Encrypt servers) I figured out that I could use CloudFlare Tunnels instead.

I have now done this for Hyperkaehler.info, as a little experiment. You shouldn't notice anything different, except that response times might even be a little quicker, and the SSL certificate comes from somewhere else. But in case it does go down for a prolonged period, now you know why.

If this approach works well for the next few months, I'll probably move my other interactive websites over as well (the static ones can simply continue running via GitHub Pages for the time being)

New paper: Generic non-semisimplicity of small quantum cohomology of Kronecker moduli

Wed, 26 Nov 2025 00:00:00 +0000

It's not quite a new paper, rather it is a new appendix (pdf of a standalone version), to be included in A-D-E diagrams, Hodge–Tate hyperplane sections and semisimple quantum cohomology by Sergey Galkin, Naichung Conan Leung, Changzheng Li, and Rui Xiong.

It is a short appendix, in which we observe that one of the main results in the Galkin–Leung–Li–Xiong paper is also sufficiently strong to deduce that the small quantum cohomology of certain Kronecker moduli is not semisimple. Here, Kronecker moduli is shorthand for moduli spaces of stable quiver representations of the Kronecker quiver; this class of smooth projective varieties has many extremely nice properties, and many strong tools to study them.

I refer to the introduction of the main paper for additional context motivating the study of quantum cohomology. Let me just point out that

combining the Schofield and Dubrovin conjectures predicts that the big quantum cohomology should be generically semisimple;
special cases of Kronecker moduli are Grassmannians, for which the small quantum cohomology is generically semisimple;
Meng has shown that the small quantum cohomology for the Kronecker moduli space for the 3-Kronecker quiver and dimension vector $(2,3)$ (the smallest case which is not a Grassmannian) has generically simple quantum cohomology.

One can experimentally (using QuiverTools) check that the obstruction from Galkin–Leung–Li–Xiong's Theorem 1.2 vanishes for many examples of Kronecker moduli. But Markus and I noticed that it does not vanish for the following infinite family of Kronecker moduli:

Theorem Let $\mathrm{M}_{(2,m)}^m$ be the Kronecker moduli space for the $m$-Kronecker quiver and dimension vector $(2,m)$, and $m\geq 5$ odd. Then the small quantum cohomology of $\mathrm{M}_{(2,m)}^m$ is not generically semisimple.

It's a fun, combinatorial proof, that fits in 1.5 page, so I won't comment further on it.

Updated paper: Moduli spaces of semiorthogonal decompositions in families

Mon, 17 Nov 2025 00:00:00 +0000

This update is concerned with Moduli spaces of semiorthogonal decompositions in families, written together with Shinnosuke Okawa, and Andrea Ricolfi. I already alluded to it in the previous post: this paper has seen a significant update.

What's new?

Following a very useful referee report, we have:

streamlined the entire paper;
referred to the literature for étale descent of semiorthogonal decompositions, thereby getting rid of a characteristic-zero assumption;
included a second proof of the functor being limit-preserving, following a suggestion by the referee;
included the referee's proof for uniqueness of deformations (with the original proof moved to a separated paper);
everything now works in a general smooth and proper setting, so the standing assumptions are always the same;
we have expanded the discussion on our conjecture regarding the subspace of non-trivial semiorthogonal decompositions.

I believe the paper is much improved now, so if you didn't like the previous version, you can give this one a shot!

New paper: Deformation theory for a morphism in the derived category with fixed lift of the codomain

Fri, 14 Nov 2025 00:00:00 +0000

This blogpost is concerned with Deformation theory for a morphism in the derived category with fixed lift of the codomain, written together with Wendy Lowen, Shinnosuke Okawa, and Andrea Ricolfi. If you want to go straight to the details, you can directly start reading the paper, this blogpost is just to give a bit of background and history.

Origin story

Following a referee report and subsequent revision for Moduli spaces of semiorthogonal decompositions in families, we have performed an appendectomy on what was formerly Appendix A in that paper (whose v3 will be online on Monday, I'll describe those significant changes in a separate blogpost), turning it into a standalone paper. After all, the main result proved in this former appendix is of independent interest. However, it being buried in an appendix was not an ideal situation.

So now it is a standalone paper which also includes our original proof that semiorthogonal decompositions deform uniquely in smooth and proper families. The referee has suggested a more self-contained proof for this uniqueness, which will be featured in v3 of the original paper, but more on that in a later blogpost.

What's in the paper?

The paper establishes a deformation theory of morphisms, explaining when they lift to derived categories of deformations of abelian categories. The case of objects was worked out by Lowen in 2005. One important conclusion from the results for objects is a precise theorem that explains the heuristic that exceptional objects should lift uniquely along deformations.

To do the same lifting for morphisms, you have to say first what you precisely want to do with the objects that are the domain and codomain of the morphism. As the title suggests, we want to fix a lift of the codomain, and then we try to understand how the domain and morphism itself can be lifted.

The reason for this choice of setup is that it is exactly what is needed for our application, namely to understand the deformation theory of semiorthogonal decompositions. After all, a semiorthogonal decomposition can be described using a decomposition triangle of projection functors, with the middle term being the identity functor, encoded using the structure sheaf of the diagonal. We therefore know which lift of the codomain we want to take: the structure sheaf of the diagonal.

And thus, that is what we have done:

Theorem A gives the technical result for lifting morphisms, identifying the obstruction class and torsor structure for the set of lifts;
Corollary B gives the algebro-geometric version, which requires a restriction argument, because in algebraic geometry one uses tensor products for pullbacks, but those are a priori not compatible with using Grothendieck categories and injective objects;
Theorem C shows that semiorthogonal decompositions deform uniquely in smooth and proper families.

I hope that this abstract and general machinery can be useful for you too!