Pieter Belmans

Rationality of hypersurfaces: a new website

Fri, 11 Sep 2026 00:00:00 +0000

After seeing John Christian Ottem, Stable irrationality of quartic sixfolds on the arXiv yesterday, I was reminded of my plans a little while ago to make a website like mgnbar.info surveying the state of the art on a precise question. Inspiration for this I found in

Without further ado, see rationality.fanography.info. It is just a single page, with features very much like mgnbar.info. I’m not sure I like the methods overview, maybe I will get rid of it at some point.

A noncommutatively rigid Fano fivefold

Wed, 09 Sep 2026 00:00:00 +0000

Here's a question I have had for a while, ever since reading Noncommutative rigidity of the moduli stack of stable pointed curves by Shinnosuke Okawa and Taro Sano.

They prove the cool result that $\mathrm{HH}^2(\overline{\mathcal{M}}_{g,n})=0$ (except when $(g,n)=(0,5)$, and possibly when $(g,n)=(4, 0), (3, 1), (3, 0), (2, 2), (2, 1), (2, 0), (1, 3), (1, 2)$). Here $\mathrm{HH}^2$ is the second Hochschild cohomology, and it measures the deformations of the category of coherent sheaves. This is the noncommutative rigidity in the title of their paper. Note that they are considering a smooth projective Deligne–Mumford stack whenever $g\geq 1$.

Finding noncommutatively rigid varieties

Because I like Fano varieties so much, I was wondering whether we can find a Fano variety $X$ for which the second Hochschild cohomology vanishes, i.e., does there exist a noncommutatively rigid Fano variety?

Note that it is in fact easy to find a non-Fano example of a noncommutatively rigid smooth projective surface: a fake projective plane is of this shape. But I wanted a Fano variety!

Finding noncommutatively rigid Fano varieties

In dimension 2, we have that $\overline{\mathcal{M}}_{0,5}$ is the del Pezzo surface of degree 5, and whilst it is rigid as a variety, it has a 6-dimensional space of Poisson structures, and thus lots of interesting noncommutative deformations. The same holds for the other rigid del Pezzo surfaces: lots of interesting noncommutative deformations, which are truly at the heart of noncommutative algebraic geometry.

In dimension 3, we can use Fanography, which contains the data from the paper Polyvector fields for Fano 3-folds that I wrote a few years ago with Enrico Fatighenti and Fabio Tanturri, to conclude that there are also no noncommutatively rigid Fano 3-folds.

In an ongoing project with many people on classifying Fano 4-folds we are also computing polyvector cohomology, but none of the cases studied so far seem to satisfy the required vanishings.

The example

Now, to dimension 5! PartialFlagVarieties.jl is a useful first selection mechanism, as it makes it possible to consider many zero loci in partial flag varieties at once when looking for candidates. This is how I found a complete intersection in $\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^3$, of degrees $(1,1,0,0)$, $(0,1,0,1)$, $(0,0,1,1)$, $(0,0,1,1)$, which had the potential of being an example. With a little help from an LLM, I could translate the remainder of the problem into a Macaulay2 computation, which confirmed that it is indeed an example.

I was originally writing this up as a blogpost, but it got longer and longer, so I felt like a pdf was more appropriate: A noncommutatively rigid Fano fivefold. It contains all the Julia and Macaulay2 code needed to verify the claims, so that you can check them yourself, and the code is also available in the repository noncommutatively-rigid-fano-fivefold. For now I have no plans to submit it for publication anywhere, but feel free to convince me otherwise.

Brauer groups of Fano 3-folds

Wed, 02 Sep 2026 00:00:00 +0000

After returning from holidays earlier this summer, I organized all the stuff on my desk a bit, and I ran into various half-finished computations and notes. Whilst some might eventually become (short) papers, most of them will (at least initially) just be fun blog posts.

The first topic I want to tackle is the Brauer group of Fano 3-folds. In the wonderful paper Fano varieties with torsion in the third cohomology group by John Christian Ottem and Jørgen Vold Rennemo, they construct even-dimensional Fano varieties (from dimension 4 onwards) for which $\mathrm{H}^3(X,\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}$. For a Fano variety, the torsion subgroup of $\mathrm{H}^3(X,\mathbb{Z})$ is its cohomological Brauer group.

For a del Pezzo surface there is nothing to be said about (torsion in) $\mathrm{H}^3(X,\mathbb{Z})$: we have that $\mathrm{H}^3(X,\mathbb{Z})=0$. But a priori there could be something interesting happening in dimension 3, as $\mathrm{H}^3(X,\mathbb{Z})$ can certainly be non-zero. The introduction of the Ottem–Rennemo paper states the following:

In dimension 3, there are 105 deformation classes of Fano varieties, and direct inspection shows that, in each class, the group $\mathrm{H}^3(X,\mathbb{Z})$ is torsion-free.

When asked about the case of Fano 3-folds during a talk, one of the authors suggested that they had not written down the details of the direct inspection, and certainly the paper does not contain these details. So let's just do it here! After all, to paraphrase Sir Mix-a-Lot: I like Fano 3-folds and I cannot lie.

A first reduction

We will do a case-by-case analysis to show that $\mathrm{H}^3(X,\mathbb{Z})$ is torsion-free. Kodaira vanishing and the exponential sequence give \[ \operatorname{Br}(X) \cong \mathrm{H}^3(X,\mathbb{Z})_{\mathrm{tors}}. \] On the other hand, the universal coefficient theorem gives \[ \mathrm{H}^3(X,\mathbb{Z})_{\mathrm{tors}} \cong \operatorname{Hom}\left( \mathrm{H}_2(X,\mathbb{Z})_{\mathrm{tors}}, \mathbb{Q}/\mathbb{Z} \right). \] In particular, $\operatorname{Br}(X)=0$ if and only if $\mathrm{H}_2(X,\mathbb{Z})$ is torsion-free. I guess it is a matter of taste and preference which of these groups is the most familiar to you. We will freely use these different perspectives.

We will also use that this torsion is constant in a smooth proper family. Indeed, by Ehresmann's theorem such a family is locally trivial in the differentiable category, so its integral cohomology groups, including their torsion subgroups, form a local system. It therefore suffices to check one convenient representative of each deformation family.

Our first goal is to significantly reduce the number of families to consider, using the following facts:

the Brauer group of $\mathbb{P}^3$ is trivial;
the Brauer group is a stable birational invariant of smooth projective varieties.

The birational links in the Iskovskikh–Mori–Mukai classification first reduce the problem from 105 deformation families to the 30 primitive ones. Of these primitive families, 17 are rational. Using the descriptions and rationality results collected in Fanography, we are thus left with the following 13 families:

ID	description
1–1	a double cover of $\mathbb{P}^3$ branched along a smooth sextic surface
1–2	a quartic hypersurface $X_4\subset\mathbb{P}^4$
1–3	a complete intersection $X_{2,3}\subset\mathbb{P}^5$ of a quadric and a cubic
1–4	a complete intersection $X_{2,2,2}\subset\mathbb{P}^6$ of three quadrics
1–5	a Gushel–Mukai 3-fold: a section of $\operatorname{Gr}(2,5)$ by a codimension-2 linear subspace and a quadric
1–7	a codimension-5 linear section of $\operatorname{Gr}(2,6)$ in its Plücker embedding
1–11	the double Veronese cone $V_1=X_6\subset\mathbb{P}(1,1,1,2,3)$
1–12	the quartic double solid: a double cover of $\mathbb{P}^3$ branched along a smooth quartic surface
1–13	a cubic hypersurface $V_3\subset\mathbb{P}^4$
2–2	a double cover of $\mathbb{P}^1\times\mathbb{P}^2$ branched along a smooth divisor of bidegree $(2,4)$
2–6	a Verra 3-fold: a double cover of the $(1,1)$-divisor $W_6\subset\mathbb{P}^2\times\mathbb{P}^2$, branched along a smooth anticanonical divisor
2–8	a double cover of $V_7=\operatorname{Bl}_p\mathbb{P}^3$, branched along a smooth anticanonical divisor meeting the exceptional divisor smoothly
3–1	a double cover of $(\mathbb{P}^1)^3$ branched along a smooth divisor of tridegree $(2,2,2)$

Awesome! Only 13 cases to consider.

Tools for the remaining cases

Let's tackle those remaining cases. Our first tool is the following.

Lefschetz in degree 2 for complete intersections. Let $M$ be a smooth projective variety, and let $X\subset M$ be obtained by successively taking smooth ample divisors, with $\dim X\geq 3$. Then $\mathrm{H}_2(X,\mathbb{Z})\cong\mathrm{H}_2(M,\mathbb{Z})$.

This is the integral Lefschetz hyperplane theorem applied successively; see, for instance, Theorem 3.1.17 in Positivity in Algebraic Geometry I.

Looking at our table, we see that quite a few of the interesting cases admit such a description, where $M$ is a partial flag variety. Such varieties have a Schubert cell decomposition and therefore torsion-free integral homology in every degree. In particular, this applies to 1–2, 1–3, 1–4, 1–5, 1–7, and 1–13.

There are now 7 families of Fano 3-folds we still have to deal with. A significant portion of them can be dealt with using the following variation of Lefschetz.

Lefschetz in degree 2 for cyclic covers. Let $X\to Y$ be a cyclic cover of smooth projective varieties of dimension at least 3, branched along a smooth ample divisor. Then $\mathrm{H}_2(X,\mathbb{Z})\cong\mathrm{H}_2(Y,\mathbb{Z})$.

This variation of the Lefschetz theorem for complete intersections is Proposition 1.11 in Topological properties of cyclic coverings branched along an ample divisor.

Again looking at our table, we see that quite a few of the remaining cases admit such a description. Namely, this tool applies to 1–1, 1–12, 2–2, 2–6, 2–8, and 3–1. We are left only with 1–11.

To deal with this remaining case, we want to have a version of Lefschetz that works for weighted complete intersections. Luckily, there is one!

Lefschetz in degree 2 for weighted complete intersections. Let $X\subset\mathbb{P}(a_0,\ldots,a_N)$ be a strongly smooth weighted complete intersection of dimension at least 3. Then \[ \mathrm{H}_2(X,\mathbb{Z})\cong\mathbb{Z}. \] Here strongly smooth means that $X$ is quasismooth and the greatest common divisor of the weights corresponding to the nonzero coordinates of a point is independent of the point.

This variation of the Lefschetz theorem for complete intersections is Proposition 6(i)–(ii) in Monodromy and Betti numbers of weighted complete intersections.

This applies to 1–11: if one of the weight-one coordinates is non-zero, the relevant greatest common divisor is 1; if they all vanish, the weight-two and weight-three coordinates are both non-zero and their weights are coprime.

Conclusion

We have thus proved that the Brauer group of a Fano threefold vanishes! It was indeed a direct inspection, but one must know which tools to use for the inspection.

New paper: The hyperkähler period-index conjecture is false

Tue, 11 Aug 2026 00:00:00 +0000

This post is about The hyperkähler period-index conjecture is false, joint with James Hotchkiss.

If I were still posting fortnightly links, Alexander Perry: The period-index conjecture is false would have prominently featured, because it disproves (as the title suggests) the period-index conjecture, which is famously a theorem due to de Jong in dimension 2, with a beautiful proof. But Alex found counterexamples in every dimension at least 3, with some assistance from an LLM.

The period-index conjecture

To understand the statement of the period-index conjecture, let us consider a field $K$ of transcendence degree $d$ over an algebraically closed field $k$. We attach two integers to a Brauer class $\alpha\in\mathop{\rm Br}(K)$:

the index $\mathop{\rm ind}(\alpha)$ (which measures the size of the unique division algebra in the Brauer class)
the period $\mathop{\rm per}(\alpha)$ (which is the order of $\alpha$ in the torsion abelian group $\mathop{\rm Br}(K)$)

The period-index conjecture, attributed to Colliot-Thélène, then predicts that \[ \mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{d-1}. \]

Using a bit of Galois cohomology, one can prove that

$\mathop{\rm per}(\alpha)\mid\mathop{\rm ind}(\alpha)$
the period and the index have the same prime factors

so certainly for every individual $\alpha$ there is some exponent for which the index divides the corresponding power of the period. The conjecture claims there is a uniform bound, which is moreover given in terms of the transcendence degree.

As a special case of this we can consider a smooth projective variety $X$ together with its function field $k(X)$, for which $\mathop{\rm Br}(X)\hookrightarrow\mathop{\rm Br}(k(X))$. Johan de Jong proved that it suffices to prove the period-index conjecture for $k(X)$ for the classes in the image of this inclusion, the unramified classes.

Alex constructed unramified classes in every dimension at least 3 which violate the period-index conjecture.

The hyperkähler period-index conjecture

Hyperkähler varieties (see also hyperkaehler.info) are a special kind of variety. They have rich geometry, strong constraints, and many tools to study them. In particular, their geometry is controlled to a large extent by $\mathrm{H}^2(X,\mathbb{Z})$, together with its Beauville–Bogomolov–Fujiki form.

Because of this, and various pieces of evidence, Huybrechts conjectured the stronger hyperkähler period-index conjecture, predicting that \[ \mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{(\dim X)/2} \] for unramified Brauer classes on smooth projective hyperkähler varieties.

Given that Alex started from an incorrect LLM-assisted counterexample, James and I independently tried constructing an LLM-assisted counterexample for the hyperkähler period-index conjecture, ending up with examples in different deformation types of hyperkähler fourfolds. Given the similarity in tools used (namely James's Hodge-theoretic index), it felt natural to write a joint paper.

So what are the counterexamples? First, observe that for the fourfolds we consider (whose integral cohomology is torsion-free) it is possible to write every Brauer class in terms of $\mathrm{H}^2(X,\mathbb{Z})$, as $\mathop{\rm Br}(X)\cong(\mathrm{H}^2(X,\mathbb{Z})/\mathop{\rm NS}(X))\otimes\mathbb{Q}/\mathbb{Z}$.

Then it becomes a matter of writing down classes in the lattice, giving us the polarization, the Brauer class, and a way to check the conditions for our two obstruction lemmas. For more details one is referred to the paper.

The results are

a period-2 Brauer class on certain hyperkähler fourfolds of type $\mathrm{Kum}^2$, whose index is divisible by 8
a period-2 Brauer class on certain hyperkähler fourfolds of type $\mathrm{K3}^{[2]}$, whose index is divisible by 8
a period-5 Brauer class on certain hyperkähler fourfolds of type $\mathrm{K3}^{[2]}$, whose index is divisible by 125

In each case the index is one factor too big: the conjecture predicts $\mathop{\rm ind}(\alpha)\mid 4$ resp. $\mathop{\rm ind}(\alpha)\mid 25$. The classical period-index conjecture only predicts $\mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^3$, so these are not, as far as we know, counterexamples to it.

This construction lives just outside the known cases of the hyperkähler period-index conjecture:

Huybrechts proved it for $X$ admitting a Lagrangian fibration, for classes whose period is coprime to an integer depending on $X$.
Huybrechts proved it for $X$ the Hilbert scheme of points on a K3 surface, for all classes, with a variant for generalised Kummer varieties.
Hotchkiss–Maulik–Shen–Yin–Zhang proved it for $X$ of $\mathrm{K3}^{[n]}$-type of Picard rank at least 2, for non-special coprime classes.
Bottini–Huybrechts removed the condition on the Picard rank, and also cover the special coprime classes whose period is squarefree.

Hyperelliptic varieties: quotients of complex tori by finite groups

Tue, 21 Jul 2026 00:00:00 +0000

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 hyperelliptic curves. The remarkable thing is that all the numerical invariants (the Hodge diamond, the order of the canonical bundle, the number of moduli, the irregularity, the polyvector fields, the twisted Hodge numbers, the Hochschild cohomology) depend only on the tangent representation $\rho\colon G \to \mathrm{GL}(V)$, and can be read off from its character theory. The website computes them all with OSCAR, for every group in the classifications of Uchida–Yoshihara, Lange and Catanese–Demleitner (dimension 3) and of Demleitner (the 79 groups in dimension 4).

This all started from a collaboration with Andreas Demleitner and Pedro Núñez on the Albanese morphism for these varieties. Working on that paper I learned a great many things about hyperelliptic varieties from Andreas, and this website is in a sense a place to keep all of it: the invariants we computed by hand, and many more, now generated automatically and cross-checked against the literature.

As with my other websites it is a static site built (using LLMs) with Hugo. Feature requests, corrections and contributions are very welcome, on GitHub or by email.

New paper: Fano 4-fold quiver moduli from subspace quivers

Wed, 15 Jul 2026 00:00:00 +0000

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.

Origin

It all started from Laurent Manivel's A four-dimensional cousin of the Segre cubic. In this paper, he studies a Fano fourfold, defined as the zero locus of a vector bundle on the product of two Grassmannians. Looking at the properties he finds for this Fano fourfold, my mind was applying the duck test in the following form:

If it looks like a quiver moduli space, swims like a quiver moduli space, and quacks like a quiver moduli space, then it probably is a quiver moduli space.

Quiver moduli are after all very special varieties: rational, Hodge–Tate, rigid. Laurent's Segre cubic cousin had all these properties. Through its relation to the Segre cubic, I also quickly guessed the quiver and dimension vector that should do the trick: the 6-subspace quiver, and $\mathbf{d}=(1,1,1,1,1,2;3)$. However, I couldn't pin down an isomorphism.

Then in February Markus came to Utrecht as the Springer visiting chair. Markus realized there is actually a fun classification problem hiding here: which other subspace quivers give rise to Fano 4-folds? The reason to be interested in subspace quivers is that, besides their moduli spaces being rigid, their moduli spaces also have no infinitesimal automorphisms, making them extra special.

It turns out that up to natural identifications, there are precisely 4 such Fano 4-fold subspace quiver moduli: one being the (expected) Segre cousin, another being the Fano model of $\mathop{\rm Bl}_6\mathbb{P}^4$, which is also $(\mathbb{P}^1)^7//\mathrm{PGL}_2$, the moduli space of 7 points on $\mathbb{P}^1$ (let's not spell out stability).

The other two are also very interesting: one admits a natural map to $\mathbb{P}^2$, exhibiting it as something called an involution surface bundle (the first time I encountered one in the wild), and another which looks a lot like the Segre cubic cousin.

And in the end, we also managed to prove that Manivel's Segre cubic cousin was indeed the subspace quiver moduli space we expected it to be all along. All's well that ends well.

Working with quiver moduli

What I like about this paper is that it is so very explicit with quiver moduli, and applies so many of the tools that have been developed for them. I consider it a bit of an advertisement, and I hope that it gets picked up by people who, a priori, would not work with quiver moduli, but now realize how useful this perspective can be!

Related to this is also the use of QuiverTools, which makes it possible to compute many invariants of quiver moduli. For the 4 cases in this paper, the code is available as FanoFourfoldSubspaceQuiverModuli.jl. And the tools we have used in this paper were also the inspiration for some of the new features for QuiverTools, more about those later.

Note that the duck test for quiver moduli doesn't always work so well! We are also working on a paper that explains how well the duck test works for Fano 3-folds. Stay tuned for that.

Also note that there is another work-in-progress (don't expect either of them before the start of my summer break, though!) where we take the involution surface bundle Fano 4-fold subspace quiver moduli, and take its connection to the extended Dynkin quiver $\widetilde{\mathrm{D}}_4$, to find 5 infinite series of even-dimensional Fano varieties attached to extended Dynkin quivers, with lots of interesting properties. Stay tuned for that too!

Cubic 4-folds: special cubic fourfolds and their K3 surfaces

Mon, 13 Jul 2026 00:00:00 +0000

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 special if it contains a surface not homologous to a complete intersection; these form the Hassett divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds.

The website is a table with one row for every discriminant $d$, collecting some things that are known about $\mathcal{C}_d$: the various notions of an associated (twisted) K3 surface, rationality, the Kodaira dimension, Fourier–Mukai partners, and which surfaces the generic member contains, with clickable references. It goes back to a chart of Nicolas Addington, whose comments shaped the website.

As with my other websites it is a static site built with Hugo. Feature requests, corrections and contributions are very welcome, on GitHub or by email.

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

le superficie algebriche: a table and more surfaces

Sun, 12 Jul 2026 00:00:00 +0000

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.

DOIs for my websites

Fri, 10 Jul 2026 00:00:00 +0000

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.

Mgnbar.info: two new layers

Thu, 09 Jul 2026 00:00:00 +0000

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 $\overline{\mathrm{M}}_{g,n}$, and of the open $\mathrm{M}_{g,n}$, is generated by the tautological classes, the natural $\psi$- and $\kappa$-classes and boundary strata, or whether there are genuinely more mysterious cycles.

Point counts. Whether the number of points of $\overline{\mathrm{M}}_{g,n}$ and $\mathrm{M}_{g,n}$ over a finite field $\mathbb{F}_q$ is a polynomial in $q$. The first place where this fails is $\overline{\mathrm{M}}_{1,11}$, where the culprit is the Ramanujan $\tau$-function, the coefficients of the weight-12 cusp form.

Both layers are transcribed from Hannah Larson’s recent survey arXiv:2606.29656, which collects the state of the art on precisely these questions. It is a wonderful read, and any errors in the transcription are of course mine.

More on the Kodaira dimension. Kodaira dimension $-\infty$ is only the coarsest sign that a moduli space is not of general type. The finer notions rational $\Rightarrow$ unirational $\Rightarrow$ rationally connected $\Rightarrow$ uniruled all imply it, and are now shown, as shaded blues, on the cells where the Kodaira dimension is $-\infty$. The baseline is Benzo’s 2014 survey table, extended with the more recent results of Agostini–Barros, Keneshlou–Tanturri, and (for genus 11-15) Verra and Bruno–Verra.

Along the way each layer got its own URL, and clicking a cell now updates the address bar too, so you can link straight to a single entry like mgnbar.info/tautological/#9,4. There is a small about page explaining how to cite the website, and, as with my other websites, Mgnbar.info is now a static site built with Hugo. I’m a big fan of Hugo nowadays, can you tell?

Suggestions and corrections are very welcome, on GitHub or by email.

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