## Root systems and del Pezzo surfaces revisited

After talking with my good friend Tom and reading the paper by Iqbal, Neitzke, and Vafa, I believe that I have access to two interpretations of the appearance of root systems inside the Picard lattice of del Pezzo surfaces. The first is pleasingly simple and touches on my recent exploration of the MMP. For del Pezzo surfaces $X=dP_d$ of degree $d$ less than 8, the (-1)-curves generate the Picard lattice of $X$ and in particular are the extremal rays generating the Mori cone. Since $X$ is Fano, $-K_X$ is ample and so $K_X$ pairs negatively with every curve on $X$ by Kleiman’s criterion for ampleness. This argument shows that, instead of being only partly locally polyhedral, the Mori cone of a Fano variety is globally polyhedral and contained within the halfspace on which $K_X$ is negative. In particular, it only meets the orthogonal hyperplane to $K_X$ at the origin. The root systems constructed last time are exactly the shortest integral vectors in this hyperplane, and their Weyl reflections can be verified to preserve the Mori cone and hence permute the (-1)-curves while preserving intersections: that is, induce automorphisms of their incidence graph.

Having only gingerly dabbled in string theory before, I took the physics interpretation largely on faith with reassurance from the computational similarities that arise. However, it is interesting to see how caring about algebraic automorphisms of del Pezzo surfaces naturally arises in what is really only a smooth context from wanting a fixed Planck length, and how an explicit identification between the Picard lattice of $X$ and the coweight lattice of the corresponding root system emerges from regarding the relevant moduli problem in string theory.

## Root systems and del Pezzo surfaces

I got a copy of Manin’s classic treatise ‘Cubic forms’ out the library today in order to run through the details of a pleasing episode in the study of del Pezzo surfaces. The 27 lines on a cubic surface (aka $dP_3$, the blowup of $\mathbb{P}^2$ in 6 general points) are intimately connected with the root system $E_6$ and accompanying Lie-theoretic paraphernalia. Something similar is true for the other del Pezzo surfaces, at least when 2 or more points are blown up to ensure that the Picard lattice is big enough for something interesting to happen. More precisely, one can find a [usually irreducible] root system inside the Picard lattice of a del Pezzo surface whose Weyl group is isomorphic to the automorphism group of the incidence graph of the lines (or (-1)-curves) on the del Pezzo.

I worked out an example and stated the main result from Manin’s book with some discussion. I came across a paper by Iqbal, Neitzke, and Vafa this evening that seems to claim to interpret and enrich Manin’s result from a string theoretic point of view, which I plan to spend some time reading. I’ll keep you posted.

## Another reason why homotopy groups are infuriating

I’ve dabbled in higher homotopy groups before mostly to define things like higher $K$-theory but have never spent much time getting to know them. I was intrigued when I heard that $\pi_n(X)$ is naturally an integral representation of $\pi_1(X)$ via the standard monodromy action. This means that, if you tensor with some field of characteristic zero, you end up with a ‘proper’ representation of the fundamental group, or, equivalently, a local system on $X$. These objects arose quite naturally, so maybe they’re interesting…

I know of no interesting examples. First, higher homotopy groups are often torsion so the tensored representation is frequently going to be zero. Second, in many cases where there are nice descriptions of things like $\pi_n(X)\otimes\mathbb{C}$ the monodromy action is trivial. This is the case for Lie groups, generalising the fact that $\pi_1(G)$ is abelian and so the monodromy $=$ conjugation action is trivial.

Serre proved in the 1950s that the action of $\pi_1(X)$ is trivial when $X$ is an $H$-space: a topological unital magma (yep, I just wanted to use the word ‘magma’). Here’s my slightly sorrowful note proving Serre’s result (from which one can deduce that $\pi_1(X)$ is abelian for $X$ an $H$-space and so, in particular, the case for Lie groups).

## Singularities of the MMP

To get in the mood for my MMP talk on Monday, I decided to take a more detailed look at the singularities that naturally occur as a result of the basic operations of the MMP: canonical and terminal singularities, and their log versions. The definitions are in terms of discrepancy, which I’ve conveniently just spent three posts describing in the case of cyclic quotient singularities.

In low dimensions…

• the surface canonical singularities are exactly the Du Val singularities, which have a Dynkin classification via the McKay correspondence,
• there are no terminal surface singularities (i.e. terminal implies smooth for surfaces)
• the 3-fold canonical and terminal singularities are classified, with some surprises. For instance, the only cyclic quotient singularities that are terminal are those of the form $\frac{1}{r}(1,a-a)$, the proof of which depends on some cryptic applications of $L$-series.

## Resolving quotient singularities III

To conclude this miniseries on resolutions of quotient singularities, I present the full toric picture of what is happening, including gory computations of the self-intersections of exceptional curves in terms of Jung-Hirzebruch continued fractions. The intricacy of the computations maybe conceals the conciseness of the toric lexicon: these three pages of computations are best summarised by the following picture for the quotient singularity $\frac{1}{7}(1,5)$

From it one can immediately read that its [minimal] resolution has three exceptional curves with discrepancies -1/7, -2/7, and -3/7. From the continued fraction expansion of $\frac{7}{5}=[2,2,3]$ one gains the equivalent information that there are three exceptional curves with self-intersection -2,-2, and -3.

## Resolving quotient singularities II

One of the first tales of algebraic geometry that I was told was the story of Jung-Hirzebruch continued fractions and their role in resolving cyclic quotient singularities. The Jung-Hirzebruch continued fraction expansion of a rational number $q$ is the expression

$q=b_0-\frac{1}{b_1-\frac{1}{b_2-\frac{1}{\ddots}}}$

which one often condenses to $[b_0,b_1,\dots]$. These are always finite expansions, at least for rational numbers as contemplated here. As promised, for the second instalment of this miniseries on quotient singularities, I will explain the application of these expansions to resolving quotient singularities of the form $\frac{1}{r}(1,a)$ as well as why one is justified in just considering these singularities to represent all of those arising from quotients of smooth varieties by finite group actions.

Here are my notes, which are largely adapted from Kollar’s book on resolutions. Like many things, this can all be conveniently viewed via toric geometry where the continued fractions also make a ready appearance. It is also easier there to sight the final few unjustified claims in my notes, such as the fact that the continued fraction expansion knows the intersection pairing between exceptional curves.

## Symplectic embeddings I

I’ve been participating in the Berkeley-Tokyo winter school on geometry, representation theory, and topology for the last week – you can find the notes from my talk last Thursday on my webpage – and have found it unexpectedly interesting being around many people interested in 4-dimensional topology.

A highlight was a killer talk by Michael Hutchings (whose blog I recently discovered thanks to Alex) on symplectic embeddings. I tend not to take notes at such events, but jotted down what I could remember and scoured the immediate references to start to fill in some of the gaps. The problem is essentially as follows: given two symplectic manifolds of the same dimension, can you tell when there is a symplectic embedding between the two? As usual, ‘symplectic’ means that it respects the symplectic forms on each space. Hutchings’ talk addressed a result that follows from an aggregation of his work and that of Dusa McDuff describing when two 4-dimensional ellipsoids can be embedded into one another. Such an ellipsoid is, in complex coordinates, described by two positive real numbers $a,b$ giving the axial radii. Denote the corresponding ellipsoid by $E(a,b)$. The result is a beautiful fusion of symplectic geometry and the geometry of numbers based on the sequence of numbers of the form $ma+nb$ for $m,n\in\mathbb{N}$ listed in ascending order and including repeats. These turn out to be precisely a stream of invariants from contact homology called ECH capacities defined by Hutchings.

As someone who works in mirror symmetry, I’ve often felt guilty about my lack of exposure to ‘authentic’ symplectic geometry. I intend to use the next few blog posts – intermingled with some talk prep for upcoming talks on intersection theory and ribbon graphs (separately…) – to begin to amend that. This compelling story also has an algebraic and a combinatorial twist: these invariants very clearly have something to do with lattice point counts and hence Ehrhart theory (which was made rigorous in some cases by some students at Berkeley a few years ago), and, less obviously, continued fractions and homology of blowups of $\mathbb{P}^2$ past the Fano-threshold. As my posts to date (and general interests) suggest, these latter two topics are very close to my heart and so might be good anchors to use when approaching the more unfamiliar symplectic side of the story. I hope your interest is as piqued as mine is.

## Incarnations of the MMP

As was alluded to in my last post, my MMP talk is nearly upon me (if President’s day doesn’t intervene, that is). I think that I will make a habit of posting my talk notes here around the time I am giving each talk so, as a first instalment of this practice, here is the extended edition of my notes on the MMP.

I’m intrigued by this idea generally ascribed to Mori and Reid to allow a ‘Goldilocks’ class of singularities – wild enough to encompass what contractions in high dimensions could do, mild enough to be manageable by the machinery of the MMP – into the picture in order to develop a reasonable existence theory for minimal models. I hope to write something soon about the singularities appearing in the MMP – terminal singularities, canonical singularities, log versions of each of these,… – soon.

There are also a few words at the end of my notes on the ‘homological MMP’ developed by Michael Wemyss among others. I haven’t had much time to explore the topic deeply, but I find the ties between quiver GIT and the MMP enticing; especially after hearing in several talks in the UK how the former can be used to attack and interpret generalisations of the McKay correspondence.

## Some comments on K3 surfaces

I’m preparing a talk on the minimal model program with some emphasis on flops: certain birational maps that carry one between minimal models. One of the first to be constructed was the Atiyah flop, which has a pleasing toric interpretation and carries interesting consequences for moduli of K3 surfaces. K3 surfaces are the surface version of elliptic curves: smooth projective surfaces with trivial canonical bundle (and the technical constraint that the first Betti number vanishes). I have some overflow notes from those that I’m currently refining down for my talk and so I thought that I would post them here to ensure that they do not go untold!

## Resolving quotient singularities I

There is one particular omitted detail in my previous post that I would like to address now. I gave a very classically-styled construction adapted from Corti and Heuberger’s paper of an orbifold del Pezzo surface with four $\frac{1}{3}(1,1)$ singularities and degree 1/3, which turned out to be what was needed to move outside class TG. I stated that contracting the four (-3)-curves on the blown up surface produced four $\frac{1}{3}(1,1)$ singularities without proof; I supply one here in slightly greater generality: that contracting a (-r)-curve produces a $\frac{1}{r}(1,1)$ singularity.

This initial foray sets things up nicely to treat the sumptuous connection between Jung-Hirzebruch continued fractions and resolutions of surface quotient singularities. I’ve used this many times in my work to date but have never gone through the derivation in detail. As the watchmaker once said, it’s about time.