## Hilbert schemes and tableaux

I’m flying to Japan next week to spend the summer there and so this will likely be my final post on American soil for a while. My friend Jeremy mentioned to me a while ago that the Bridgeland-King-Reid (BKR) theorem – which is the derived version of the McKay correspondence – is used in Haiman’s proof of the Macdonald positivity conjecture. While I found this intriguing, most of the literature regarding the proof is rather lengthy and so I didn’t initially follow up on his comment. There were celebrations for the graduating PhD students in the department last week that involved short talks describing the theses of the graduates, one of which expounded on Macdonald polynomials and the geometry of Hilbert schemes, which are currently inseparable in terms of the proof methods in the field. During the talk, I suddenly realised how one can associate a tableau to the toric zero-strata (or, alternatively, the affine pieces) of $\text{Hilb}^n\mathbb{A}^2$, and hence found a much needed access-point into the literature.

Anecdotes aside, Haiman’s proof of the Macdonald positivity conjecture is long and complicated, but seems to have a nice narrative structure that his survey articles emphasise. I was reminded of comment of Ian Stewart that I read last week about the seven (or so) storylines that populate all of Hellenistic drama, and how this resembles the relatively few plots of math proofs, however long they are…

My notes above try to sketch the route the proof follows dwelling on the details that I found most interesting and that were reasonably quick to digest. The result of BKR – which deals with equivariant Hilbert schemes – is applicable because of the fact that the standard Hilbert scheme $\text{Hilb}^n\mathbb{A}^2$ is isomorphic to the equivariant Hilbert scheme $S_n\text{-Hilb}\,\mathbb{A}^{2n}$ where $S_n$ acts by permuting pairs of coordinates; in this sense, really $\mathbb{A}^{2n}=(\mathbb{A}^2)^n$. It’s not so hard to see this isomorphism: tuples in $\mathbb{A}^{2n}$ considered up to the action of $S_n$ are just sets of $n$ points in $\mathbb{A}^2$ with some care taken at tuples with multiple copies of the same point.

The Macdonald positivity conjecture states that the coefficients (which are only a priori rational functions with rational coefficients) in expressing certain combinatorially/algebraically interesting polynomials indexed by partitions as a linear combination of the Schur basis for the algebra of symmetric functions are actually polynomials with positive integral coefficients. A standard means of accomplishing such a feat is to realise the original polynomial as some sort of dimension count; in this case, as the Frobenius series of a bigraded $S_n$-module. This module turns out to be the fibre of the tautological bundle over $S_n\text{-Hilb}\,\mathbb{A}^{2n}$ at the zero-stratum corresponding to the relevant partition.

## Toric divisors in depth

In probably the final instalment of me filling in my understanding of toric varieties for a while, I cover some of the approaches to studying divisors on toric varieties. A key approximation result is the ability to find a torus-invariant divisor inside any rational equivalence class. This immediately transports essentially any interesting question about divisors to the arena of combinatorics.

This approach is effective for testing whether or not various adjectives can be attached to a given divisor. To a torus-invariant divisor $D$ one can associate a piecewise-linear function $\varphi_D$ that contains enough information to recover the divisor, but is often more amenable to study. The main entries in the dictionary between divisors and piecewise-linear functions are…

$\text{basepoint-free}\longleftrightarrow\text{convex}$

$\text{ample}\longleftrightarrow\text{strictly convex}$

It actually follows from the first correspondence that basepoint-free divisors are exactly the nef divisors on a toric orbifold, by observing that they are also the divisors whose associated piecewise-linear functions are convex.

Returning to Mori theory gives, in my opinion, the best way to view to these correspondences. The functions $\varphi_D$ can be used to calculate intersection pairings between torus-invariant divisors and curves and demanding convexity picks out the nef cone, while strict convexity restricts to the interior of the nef cone, which consists of the ample divisors.

## Endowing numbers with meaning

It seems to be a quirk of the company I find myself in that my friends like setting me linear algebra teasers. My friend Dan introduced me to (sub)stochastic matrices this week, which are characterised by having nonnegative entries and row sums all (at most) 1. The incentive for this definition from probability is quite clear; one thinks of the entries as being the transition probabilities for some Markov process. He mentioned the following fact:

If $A$ is a substochastic matrix, then $0\leq\text{det}(\text{Id}-A)\leq 1$.

It’s quite a fun linear algebra exercise to prove this! I wrote a proof here alongside some elaboration on the rest of this post, but I suggest you try it yourself first. Pitman – who you’ve probably heard of from his probability book – preaches the mantra that any ‘interesting’ number between 0 and 1 should be regarded as a probability. The open problem associated to the above fact is…

What probability does $\text{det}(\text{Id}-A)$ describe?

It’s quite a probabilistic setup, so it’s not so outrageous to hope that some cute interpretation might exist.

If you’ve ever talked to me or skimmed my past posts, you will realise how distant this sort of question is from my usual research environment. However, I was quite intrigued because of the similarity of Pitman’s philosophy with an agenda of Kontsevich and Zagier pertaining to ‘periods’. A period is the integral of a rational polynomial over a region in $\mathbb{R}^n$ described by rational polynomial inequalities. Rational here means with coefficients in $\mathbb{Q}$… It’s quite apparent from the setup that periods are widespread in geometry; the name really comes from period integrals in Hodge theory. I give some examples in the note that’s linked above. The suggestion of Kontsevich and Zagier is that any ‘interesting’ transcendental number should be a period, and then that presenting such a number as a period (which of course has many different expressions as an algebraic integral) might explain apparently surprising equalities that crop up in geometry or number theory.

## Intersection theory on toric varieties

Things are getting serious: I’m scheduling my qual for September, and so there is an extra incentive to dutifully fill in some of the details of the computations in toric intersection theory from my recent post on Hirzebruch surfaces.

I’ve often found that toric geometry is capable of introducing concepts from general algebraic geometry with the additional grounding of its proximity to combinatorics a valuable aid to visualisation that can help shed intuition on the original notion. I’m undecided as to whether or not intersection theory is such a topic. To me at least it is already quite tangible, and the extensive formulae appearing in the toric setting only really obscure what is happening. The existence of such formulae, on the other hand, is extremely powerful, especially since they depend exclusively on readily available combinatorial data.

There a few general principles that these arguments seem to follow. Of primary use is the fact that affine toric varieties have no nontrivial Cartier divisors, and so the open affine cover coming from the maximal cones of a fan provides a trivialisation for any Cartier divisor.

Another slightly stealthier theme is that there’s very little difference between the orbifold (or simplicial) and the smooth cases. Much like computations in weighted projective space where some weight adjustment is the only change needed to extend calculations done in usual projective space, some weight or ‘multiplicity’ adjustment is all that’s needed.

## The nicest spheres

I was showing off my blog to a nonmathematician recently and realised that I haven’t posted anything designed to be accessible to that audience for a while. Around the same time, an undergraduate asked me to read a class paper they wrote on the Hopf fibration, which made me decide to write a post on spheres.

The classification of spheres that are also Lie groups (or Lie loops, if you relax the need for associativity) is one of my favourite stories. I love it when questions like this with no expectation of finitude in the statement turn out to have very finite answers. I am personally of the view that such finitude is typically for a good reason that is worth exploring. In the question at hand, the only spheres which are H-spaces (more general than Lie loops and groups) are…

$S^0,S^1,S^2,\text{and }S^7$

which correspond to the only real division algebras

$\mathbb{R},\mathbb{C},\mathbb{H},\text{and }\mathbb{O}$

In each case, the sphere is the set of unit length elements in the corresponding division algebra under the usual norm. Since the octonions are not associative, one must allow loops to include $S^7$. This result is closely related to the ‘Hopf invariant one problem’ that was believed for many years until Adams finally completed the intricate details of the proof inside $K$-theory. This completely classifies the spheres that are $H$-spaces, remarkably giving the same list as for spheres that are Lie loops or parallelisable.

For a recent exhibition of finitude, Papanikolopoulos and Siksek have proved that every cubic hypersurface of dimension at least $48$ is ‘fake-cyclic’. There is a similar secant construction to that for elliptic curves allows one to produce new rational points from a pair of old ones, and fake-cyclic means that it suffices to start with a single point and repeatedly apply the construction to it to obtain all rational points on the cubic hypersurface. It’s fake because the construction doesn’t give a group structure. Cubic is needed to get a unique point out by Bezout. Fake-cyclicity of course fails radically for elliptic curves, so there must be a nontrivial threshold after which all cubic hypersurfaces are fake-cyclic. I haven’t read the paper yet, so maybe this is established there or perhaps their proof just doesn’t work in dimensions lower than $48$.

## Ruled surfaces II: toric aspects

The Hirzebruch surfaces – geometrically ruled surfaces over $\mathbb{P}^1$ – are very classical and well-studied objects that in addition turn out to be toric. I suppose that an intuitive reason for why this is the case is that there is at least a 2-torus action on such a surface: $\text{PGL}_2(\mathbb{C})$ acts on both the base and on each fibre and contains a 1-torus as the classes of diagonal matrices.

Apparently they were first studied as embedded surfaces known as ‘scrolls’ depending on two parameters, which can be symplectified into a moment polytope and then dualised to give a toric fan, causing one to realise that only one parameter was needed after all to determine the surface as an abstract variety.

The elementary transformations that move one between the different Hirzebruch surfaces have a toric incarnation too. They also exhibit the main combinatorial features of intersection theory for toric varieties, by which one can distinguish the different Hirzebruch surfaces. They also carry the combinatorics associated to toric projective bundles; both of these are topics I plan to discuss in more detail soon. In the meantime, here are my notes.

## Toric models of LCYs

As stated last time, one of the motivations for learning the very classical story of elementary transformations was to better appreciate the role they play in GHK mirror symmetry. In turn, apart it from being a radical development in cluster theory and mirror symmetry itself that I want learn more about, I’d very much like to understand how the other versions of mirror symmetry that I’ve encountered (and some that I haven’t) can be reconciled with the GHK mentality.

It seems that much of the philosophy for mirror symmetry for Fano varieties or, roughly more generally, log Calabi-Yau (LCY) pairs is based on their proximity to toric varieties. In the Fano case as studied by Coates-Corti-Galkin-Golyshev-Kasprzyk-… the mirror is built from the toric degenerations of the Fano you started with. I’ve posted before about the conjecture that ‘smooth implies TG’. In the LCY case studied by GHK(K), it is toric models that play the part of mirror builders.

LCY is the log version of CY: these are pairs $(Y,D)$ consisting of a smooth projective variety and a normal crossings divisor $D\subset Y$ such that $D\in|{-K_Y}|$. Normal crossings is important for certain calculations to depend only on the interior $U=Y\setminus D$, which is also often referred to as an [open] LCY. There’s a good source of these: if $Y$ is a toric variety and $D$ is its toric boundary – that is, $D=Y\setminus T$ removing the big torus, or it’s the divisor from the rays of the polytope defining $Y$ – then $(Y,D)$ is a LCY. A toric LCY is a ‘toric model’ of a LCY is it can be obtained by a series of suitable blowdowns.

The point is that for each toric model $(\overline{Y},\overline{D})$ of $(Y,D)$ one gets an open torus inside $U=Y\setminus D$. Gathering all the tori from all the toric models of $(Y,D)$ covers $U$ up to codimension 2, which is good enough to study regular functions by Hartogs’ lemma (which is what one ends up doing). Of course, like collections of toric degenerations, the collection of toric models comes with much more structure than I’ve stated so far. One can ‘mutate’ between toric models via elementary transformations.

Here lies the problem with naively comparing GHK to CCGGK via comparing toric models and toric degenerations. I discussed the classical example last time of the elementary transformation from $\mathbb{F}_0=\mathbb{P}^1\times\mathbb{P}^1$ to $\mathbb{F}_1=\text{dP}_8$. These are then both toric models for each other. However, they are not deformation-equivalent! Hence, they get conflated in GHK and distinguished in CCGGK.

## Ruled surfaces I

The title is slightly misleading: I’ve actually been learning about ‘geometrically ruled surfaces’ this week mainly from some lecture notes written by Vakil. The motivation partly came from Gross-Hacking-Keel, where certain ‘elementary transformations’ provide the appropriate notion of mutation in their presentation of cluster varieties. These were classically studied by Hirzebruch among others first in the situation of $\mathbb{P}^1$-bundles over curves.

A geometrically ruled surface is a surface $X\to C$ over a curve whose fibres are all rational curves (equivalently, the generic fibre is rational). A ready source of these is $\mathbb{P}^1$-bundles over curves, which turns out to be all of them (the Noether-Enriques theorem). Like projective bundles in general, one can perform very explicit and comprehensive calculations of their intersection theory. This then allows one to distinguish different geometrically ruled surfaces over the same base; for instance the Hirzebruch surfaces

$\mathbb{F}_n:=\mathbb{P}\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n)$

over $\mathbb{P}^1$. These are the only Fano geometrically ruled surfaces, and are also toric. They are also known as ‘scrolls’, which is maybe a more romantic name befitting something that feels so classical.

Classifying projective bundles is a rather more modern activity: these are locally trivial fibrations by projective spaces, or $\text{PGL}_n$-torsors, which then are classified by $H^1(\text{PGL}_n)$. In the case when the base is a curve, it’s immediate to extract from this the classification of projective bundles as projectivisations of vector bundles modulo the action of the Picard group by tensoring, about which much of the computation pivots.