Volumes of cones

I was talking to a friend today who does some tutoring in secondary school maths in the UK (roughly the same as the beginning of high school in the US) who pointed out that the volume of a 2- or 3-dimensional cone of height 1 over a line or polygon is respectively 1/2 or 1/3 times the volume of the base. He wondered if the same result held in higher dimensions with the dimension of the cone (equivalently, one more than the dimension of the base) as denominator. In other words, if $\mathcal{C}P$ denotes a cone of height 1 over a polytope $P$ of dimension $d$, then $Vol(\mathcal{C}P)=\frac{1}{d+1}Vol(P)$. This is indeed true, as may be known to some of you. As a Saturday afternoon diversion, I wrote two proofs of this equality: one direct proof after reducing to the case of simplices, and one using Ehrhart theory. The latter crops up regularly at the combinatorial end of algebraic geometry – naturally in toric geometry – where certain generating functions for lattice point counts are Hilbert series for toric varieties. You could try to recast the equality above in these terms if you were so inclined or desirous of diversion…

The del Pezzo surface of degree 5

The 10 del Pezzo surfaces are precisely the smooth Fano surfaces, and their geometry is classically elegant (involving root systems, iconic graphs, and line counting). For instance, the smooth cubic surface and its 27 lines occur as the degree 3 del Pezzo surface. There is some motivation in finding ‘nice’ effective representatives of the anticanonical divisor class on such surfaces since it allows one to express these surfaces in the framework of log Calabi-Yau pairs, which are central to the GHK approach to mirror symmetry via cluster varieties. Thus motivated (and after being prompted by a good friend) I decided to work out one of these examples in detail: the del Pezzo surface of degree 5. It seems like there a few appendices that could be added to this story – there is a tropical angle to it, and a good reason for the appearance of the Petersen graph is not apparent to me – but the intention is worked through!

Reconciling mutations

There are two main types of mutation that I know: cluster (or quiver) mutations, and mutations of polytopes (roughly equivalent to mutations of Laurent polynomials). If you haven’t had the joy of meeting these yet, see here and here for introductions (only one of which was written by me). A natural aim is to reconcile these two related notions. To me at least, a reasonable approach is to try to find a cluster algebra whose seeds biject with polytopes in a given mutation class, perhaps by associating some integer invariants to them which can then appear in a quiver. You may recall from a few posts ago that this was essentially the strategy for communicating the Markov spectrum into the cluster vocabulary. Indeed, the Markov cluster algebra is going to be a constant companion here as we encounter it in a more geometrical guise: the Markov triples parameterise the del Pezzo surfaces that degenerate to $\mathbb{P}^2$ via sending a triple $(a,b,c)$ to the weighted projective plane $\mathbb{P}(a^2,b^2,c^2)$. One of the hopes of mirror symmetry is to find a cluster algebra (or cluster variety) that somehow gathers together all the toric degenerations of a given Fano variety, which is partly accomplished for $\mathbb{P}^2$ by the Markov cluster algebra. As a result of this classification for $\mathbb{P}^2$ – due to Hacking-Prokhorov – that was extended to describe mutations between arbitrary weighted projective planes by Akhtar-Kasprzyk, some deformation invariants naturally appear. The blurb is that to each mutation class of weighted projective planes, one can associate a Diophantine equation (generalising the Markov equation) whose integer solutions are the weights for weighted projective planes in that class. The coefficients of this equation are expressible in terms of the weight data or, more concretely, in combinatorial terms. It is also interesting to see how other related quantities change (or don’t change) as one traverses the mutation class. Throughout, I am using the principle that $\mathbb{Q}$-Gorenstein deformations of toric Fanos correspond to mutations of the associated Fano polytopes. Anyway, my survey article can be found here and includes some comments about these deformation invariants that I don’t think you can find elsewhere.

As a last remark, let me describe how the smoothable and nonsmoothable cases differ with regard to their behaviour under mutations. One can mutate away $T$-singularities (i.e. the smoothable ones) and a toric Fano is smoothable iff all of its singularities are $T$s. In the case of weighted projective planes, which correspond to Fano triangles as discussed last time, if the wpp is smoothable then there are three possible mutations: one in each face. From Hacking-Prokhorov’s classification, it turns out that each of these $T$-singularities is elementary; here, it means that the $T$-singularity is as thin as possible and so only admits one mutation. The same is true of the resulting wpp after mutation. Hence, the graph of mutations is a 3-regular tree. In particular, all of the mutations are involutions. Suppose that a wpp has a nonelementary $T$-singularity (which means as a result that it will be nonsmoothable). This means that multiple mutations in the same direction are possible, which is very dissimilar to cluster mutations. And, furthermore, the result of mutation might not even be a simplex anymore! These difficulties – and others like them, for instance when the singularities are residual so have no smoothable part – render the translation between the geometric and cluster worlds very nontrivial, making the special cases like $\mathbb{P}^2$ where many things work… special.

A remark of Zagier

This is a short advertisement for cluster algebras via a remark of Don Zagier that was quoted on a Stack.Exchange post I came across this evening. Zagier invites one to consider a sequence with the first two values known and then with the $i$th term defined by requiring that the $i$th term multiplied by the $i-2$th term is $1$ plus the $i$th term. That is, $x_{i+1}=(1+x_i)/x_{i-1}$ or $x_{i-1}x_{i+1}=1+x_i$.

Let’s start with the values $x_1=3$ and $x_2=4$ as Zagier does. The resulting sequence is $3,4,5/3,2/3,1,3,4,5/3,2/3,1,...$. Indeed, it is not hard to prove that, for any pair of initial values, the sequence is 5-periodic: $x_{i+5}=x_i$.

The recurrence defining the sequence is a trinomial relation that one might hope to capture in mutations of a quiver; in other words, in a cluster algebra. This is indeed possible as I discuss. The associated quiver is an orientation of the Dynkin diagram $A_2$, whose cluster algebra has $5$ seeds as part of the classification for finite-type cluster algebras. This is another way of seeing the periodicity or finiteness of the situation as stemming from a quiver with strong finiteness properties. Advertisement over.

Fake weighted projective spaces

Partly to support my qual preparation, partly to support my DRP student this semester, and partly for the sake of exposition, I am shortly going to begin writing a series of posts on toric geometry; with especial focus on toric Fano varieties and the classification problems thereof. As a sort of prelude, I offer a note I wrote about fake weighted projective spaces. Traditional weighted projective spaces have simplices as their toric polytopes, but it is not the case that all simplices give rise to weighted projective spaces. The additional toric varieties needed are fake weighted projective spaces: finite quotients of weighted projective spaces by a natural group action. This is a pleasing story in its own right, so I thought that I would tell it.

Roots of Hilbert series

I recently came across a tidy little result I proved when at Imperial College a while ago and that, as it was peripheral to what I was primarily doing, has not seen the light of day since. Now that I have this new medium to output periphery into, I thought it might be a good idea to air it. It also provided a good opportunity to review complete intersections, which are a class of geometrically motivated rings with many appealing computational properties. I couldn’t find much content at the time (nor now) regarding the roots of Hilbert series, but there are some blatant connections between them and the structure of the rings from which they arise. Here is my note with some exposition and then said tidy little result.

Pythagorean triples and cluster algebras

This post is coincidentally a week after my previous one [I originally wrote the new year post on 01/04 and this a week later before being persuaded to convert to WordPress] however I doubt that such regularity will last once the semester starts. I’ve been having some interesting conversations with my good friend Alex – who has the same new year’s resolution as me: his blog is coming soon! – over the Christmas break about a most classical of subjects: Pythagorean triples. It was shown in the 1930s that primitive Pythagorean triples fall into the structure of a ternary tree linked by certain linear transformations that are guaranteed to produce new Pythagorean triples from old. This immediately reminded me of the Markov equation. See my recent talk notes for some details of how this seemingly innocuous equation often crops up in geometry. The Markov equation has a cluster algebra underpinning it in the sense that the transformations linking its solutions can be realised as quiver mutations, and that the seeds of the corresponding cluster algebra biject with solutions. For some winter entertainment, I decided to try to tell a similar story with Pythagoras’ equation $a^2+b^2=c^2$ as protagonist. There turned out to be some positivity issues that obstructed such a concise and complete tale in this case, however I made some decent progress! You can read my account of it here.

New year

Happy new year! In my case, this is the first that I’ve spent outside the UK. I have been at 10700 feet on a mountain in Colorado for the last two weeks and have some poetry to prove it. The four poems Halves, Vast, Commonality, and Cycle form my Colorado period’. I’ve also charted out the Dynkin correspondences that I am aware of for my own convenience. For those who haven’t suffered a monologue from me about them, Dynkin diagrams are a class of finite graphs/quivers that solve a remarkable number of classification problems. This common parameterisation allows one to link the various objects parameterised in an abstract’ way, however there is often something geometrical underlying these coincidences. The most recent Dynkin classification I know about is that of finite-type cluster algebras by Fomin and Zelevinsky. A natural thing to attempt, then, is to incorporate this cluster classification into the web of Dynkin correspondences. This is a note outlining the existing correspondences that I know of. You will observe that it is very incomplete. I am currently thinking about how to convincingly (= geometrically) extend the fingers of cluster algebras into many of the other pies admitting Dynkin classifications.